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j) \ \hbox{and} \vspace*{4 pt} \\ \hspace*{100 pt}
d (j, \eta_{t-} (x + 1 )) = s_+ \ \hbox{and} \ x - 1 \to_t x) \vspace*{4 pt} \\ \hspace*{25 pt} = \
\card \{i_+ : d (i_-, i_+) = s \ \hbox{and} \ d (i_+, j) = s_+ \} / \card \{i_+ : d (i_+, j) = s_+ \} \end{array}
\end{equation}
where the last equality follows from~\eqref{eq:uniform} and~\eqref{eq:collision-2 } which, together, imply that the opinion at~$x + 1 $ just before
the jump is independent of the other opinions on its left and chosen uniformly at random from the set of opinions at distance~$s_+$ of opinion~$j$. Assuming in addition that the underlying opinion graph is distance-regular~\eqref{eq:dist-reg-1 }, we also have
\begin{equation}
\label{eq:collision-4 }
\begin{array}{l}
\card \{i_+ : d (i_-, i_+) = s \ \hbox{and} \ d (i_+, j) = s_+ \} \vspace*{4 pt} \\ \hspace*{50 pt} = \
N (\Gamma, (i_-, s), (j, s_+)) \ = \ f (s_-, s_+, s) \vspace*{8 pt} \\
\card \{i_+ : d (i_+, j) = s_+ \} \ = \ N (\Gamma, (j, s_+)) \ = \ h (s_+). \end{array}
\end{equation}
In particular, the conditional probability in~\eqref{eq:collision-3 } does not depend on the particular choice of the pair of opinions~$i_-$ and~$j$ from
which it follows that
\begin{equation}
\label{eq:collision-5 }
\begin{array}{l}
P \, (\xi_t (e) = s \, | \, \xi_{t-} (e - 1 ) = s_- \ \hbox{and} \ \xi_{t-} (e) = s_+ \ \hbox{and} \ x - 1 \to_t x) \vspace*{4 pt} \\ \hspace*{40 pt} = \
P \, (\xi_t (e) = s \, | \, B_{t-} (i_-, j) \ \hbox{and} \ \xi_{t-} (e) = s_+ \ \hbox{and} \ x - 1 \to_t x) \end{array}
\end{equation}
The lemma is then a direct consequence of~\eqref{eq:collision-3 }--\eqref{eq:collision-5 }. \end{proof} \\ \\
As previously mentioned, it follows from Lemma~\ref{lem:collision} that, provided the opinion model starts from a product measure in which
the density of each opinion is constant across space and the opinion graph is distance-regular, the system of piles itself is a Markov process. Another important consequence is the following lemma, which gives bounds for the probabilities that the jump of an active pile onto a frozen
pile results in a reduction or an increase of its order. \begin{lemma} --
\label{lem:jump}
Let~$x = e - 1 /2 $. Assume~\eqref{eq:uniform} and~\eqref{eq:dist-reg-1 }. Then,
$$ \begin{array}{l}
P \, (\ceil{\xi_t (e) / \tau} < \ceil{\xi_{t-} (e) / \tau} \, | \, (\xi_{t-} (e - 1 ), \xi_{t-} (e)) = (s_-, s_+) \ \hbox{and} \ x - 1 \to_t x) \ \leq \ p_n \vspace*{4 pt} \\
P \, (\ceil{\xi_t (e) / \tau} > \ceil{\xi_{t-} (e) / \tau} \, | \, (\xi_{t-} (e - 1 ), \xi_{t-} (e)) = (s_-, s_+) \ \hbox{and} \ x - 1 \to_t x) \ \geq \ q_n \end{array} $$
whenever~$0 < \ceil{s_- / \tau} = 1 $ and~$\ceil{s_+ / \tau} = n > 1 $. \end{lemma}
\begin{proof}
Let~$p (s_-, s_+, s)$ be the conditional probability
$$ P \, (\xi_t (e) = s \, | \, (\xi_{t-} (e - 1 ), \xi_{t-} (e)) = (s_-, s_+) \ \hbox{and} \ x - 1 \to_t x) $$
in the statement of Lemma~\ref{lem:collision}. Then, the probability that the jump of an active pile onto the pile of order~$n$ at edge~$e$ results in a reduction of its order is smaller than
\begin{equation}
\label{eq:jump-1 }
\begin{array}{l} \max \, \{\sum_{s : \ceil{s / \tau} = n - 1 } \, p (s_-, s_+, s) : \ceil{s_- / \tau} = 1 \ \hbox{and} \ \ceil{s_+ / \tau} = n \} \end{array}
\end{equation}
while the probability that the jump of an active pile onto the pile of order~$n$ at edge~$e$ results in an increase of its order is larger than
\begin{equation}
\label{eq:jump-2 }
\begin{array}{l} \min \, \{\sum_{s : \ceil{s / \tau} = n + 1 } \, p (s_-, s_+, s) : \ceil{s_- / \tau} = 1 \ \hbox{and} \ \ceil{s_+ / \tau} = n \}. \end{array}
\end{equation}
But according to Lemma~\ref{lem:collision}, we have
$$ p (s_-, s_+, s) \ = \ f (s_-, s_+, s) / h (s_+) $$
therefore~\eqref{eq:jump-1 }--\eqref{eq:jump-2 } are equal to~$p_n$ and~$q_n$, respectively. \end{proof} \\ \\
We refer to Figure~\ref{fig:coupling} for a schematic illustration of the previous lemma. In order to prove the theorem, we now use Lemmas~\ref{lem:collision}--\ref{lem:jump} to find a stochastic lower bound for the contribution of each edge. To express this lower bound, we let~$X_t$ be the discrete-time birth and death Markov chain with transition probabilities
$$ p (n, n - 1 ) \ = \ p_n \qquad p (n, n) \ = \ 1 - p_n - q_n \qquad p (n, n + 1 ) \ = \ q_n $$
for all~$1 < n < M := \ceil{\mathbf{d} / \tau}$ and boundary conditions
$$ p (1, 1 ) \ = \ 1 \quad \hbox{and} \quad p (M, M - 1 ) \ = \ 1 - p (M, M) \ = \ p_M. $$
This process will allow us to retrace the history of a frozen pile until time~$T_e$ when it becomes an active pile. To begin with, we use a first-step analysis to compute explicitly the expected value of the first hitting time to state~1 of the birth and death process. \begin{lemma} --
\label{lem:hitting}
Let~$T_n := \inf \, \{t : X_t = n \}$. Then,
$$ E \, (T_1 \, | \, X_0 = k) \ = \ 1 + \mathbf W (k) \quad \hbox{for all} \quad 0 < k \leq M = \ceil{\mathbf{d} / \tau}. $$
\end{lemma}
\begin{proof}
Let~$\sigma_n := E \, (T_{n - 1 } \, | \, X_0 = n)$. Then, for all~$1 < n < M$,
$$ \begin{array}{rcl}
\sigma_n & = & p (n, n - 1 ) + (1 + \sigma_n) \, p (n, n) + (1 + \sigma_n + \sigma_{n + 1 }) \, p (n, n + 1 ) \vspace*{3 pt} \\
& = & p_n + (1 + \sigma_n)(1 - p_n - q_n) + (1 + \sigma_n + \sigma_{n + 1 }) \, q_n \vspace*{3 pt} \\
& = & p_n + (1 + \sigma_n)(1 - p_n) + q_n \, \sigma_{n + 1 } \vspace*{3 pt} \\
& = & 1 + (1 - p_n) \, \sigma_n + q_n \, \sigma_{n + 1 } \end{array} $$
from which it follows, using a simple induction, that
\begin{equation}
\label{eq:hitting-1 }
\begin{array}{rcl}
\sigma_n & = & 1 / p_n + \sigma_{n + 1 } \, q_n / p_n \vspace*{4 pt} \\
& = & 1 / p_n + q_n / (p_n \, p_{n + 1 }) + \sigma_{n + 2 } \, (q_n \, q_{n + 1 }) / (p_n \, p_{n + 1 }) \vspace*{4 pt} \\
& = & \sum_{n \leq m < M} \, (q_n \cdots q_{m - 1 }) / (p_n \cdots p_m) + \sigma_M \, (q_n \cdots q_{M - 1 }) / (p_n \cdots p_{M - 1 }). \end{array}
\end{equation}
Since~$p (M, M - 1 ) = 1 - p (M, M) = p_M$, we also have
\begin{equation}
\label{eq:hitting-2 }
\sigma_M \ = \ E \, (T_{M - 1 } \, | \, X_0 = M) \ = \ E \, (\geometric (p_M)) \ = \ 1 / p_M. \end{equation}
Combining~\eqref{eq:hitting-1 }--\eqref{eq:hitting-2 }, we deduce that
$$ \begin{array}{l} \sigma_n \ = \ \sum_{n \leq m \leq M} \, (q_n \, q_{n + 1 } \cdots q_{m - 1 }) / (p_n \, p_{n + 1 } \cdots p_m), \end{array} $$
which finally gives
$$ \begin{array}{rcl}
E \, (T_1 \, | \, X_0 = k) & = & \sum_{1 < n \leq k} \, E \, (T_{n - 1 } \, | \, X_0 = n) \ = \ \sum_{1 < n \leq k} \, \sigma_n \vspace*{4 pt} \\
& = & \sum_{1 < n \leq k} \, \sum_{n \leq m \leq M} \, (q_n \cdots q_{m - 1 }) / (p_n \cdots p_m) \ = \ 1 + \mathbf W (k). \end{array} $$
This completes the proof. \end{proof} \\ \\
The next lemma gives a lower bound for the contribution~\eqref{eq:contribution-frozen} of an edge~$e$ that keeps track of the number
of active piles that jump onto~$e$ before the pile at~$e$ becomes active. The key is to show how the number of jumps relates to the birth and death process. Before stating our next result, we recall that~$T_e$ is the first time the pile of particles at edge~$e$ becomes active. \begin{figure}[t]
\centering
\scalebox{0.45 }{\input{coupling. pstex_t}}
\caption{\upshape{Schematic illustration of the coupling between the opinion model and the system of piles along with their evolution rules. In our example, the threshold~$\tau = 2 $, which makes piles with three or more particles frozen piles and piles with one or two particles active piles. }}
\label{fig:coupling}
\end{figure}
\begin{lemma} --
\label{lem:coupling}
Assume~\eqref{eq:uniform} and~\eqref{eq:dist-reg-1 }. Then, for~$1 < k \leq \ceil{\mathbf{d} / \tau}$,
$$ \begin{array}{l} E \, (\cont (e \, | \, T_e < \infty)) \ \geq \ \mathbf W (k) \quad \hbox{when} \quad \ceil{\xi_0 (e) / \tau} = k. \end{array} $$
\end{lemma}
\begin{proof}
Since active piles have at most~$\tau$ particles, the triangle inequality~\eqref{eq:triangle} implies that the jump of an active pile onto
a frozen pile can only increase or decrease its size by at most~$\tau$ particles, and therefore can only increase or decrease its order by at most one. In particular,
$$ \begin{array}{l}
P \, (|\ceil{\xi_t (e) / \tau} - \ceil{\xi_{t-} (e) / \tau}| > 2 \, | \, x - 1 \to_t x) \ = \ 0. \end{array} $$
This, together with the bounds in Lemma~\ref{lem:jump} and the fact that the outcomes of consecutive jumps of active piles onto a frozen pile are independent
as explained in the proof of Lemma~\ref{lem:collision}, implies that the order of a frozen pile before it becomes active dominates stochastically the
state of the birth and death process~$X_t$ before it reaches state~1. In particular,
$$ \begin{array}{l} E \, (\cont (e \, | \, T_e < \infty)) \ \geq \ - 1 + E \, (T_1 \, | \, X_0 = k) \quad \hbox{when} \quad \ceil{\xi_0 (e) / \tau} = k. \end{array} $$
Using Lemma~\ref{lem:hitting}, we conclude that
$$ \begin{array}{l} E \, (\cont (e \, | \, T_e < \infty)) \ \geq \ - 1 + (1 + \mathbf W (k)) \ = \ \mathbf W (k) \end{array} $$
whenever~$\ceil{\xi_0 (e) / \tau} = k$. \end{proof} \\ \\
We now have all the necessary tools to prove the theorem. The key idea is the same as in the proof of Lemma~\ref{lem:expected-weight} but relies on the previous lemma in place of Lemma~\ref{lem:deterministic}. \\ \\
\begin{demo}{Theorem~\ref{th:dist-reg}} --
Assume~\eqref{eq:uniform} and~\eqref{eq:dist-reg-1 } and
$$ \begin{array}{l} S_{\reg} (\Gamma, \tau) \ = \ \sum_{k > 0 } \, (\mathbf W (k) \, \sum_{s : \ceil{s / \tau} = k} \, h (s)) \ > \ 0.
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_0 (e - 1 /2 ) = i) \vspace*{4 pt} \\
& = & \sum_{i \in V} \, F^{-1 } \, h (s) \, P \, (\eta_0 (e - 1 /2 ) = i) \ = \ F^{-1 } \, h (s). \end{array} $$
Using also Lemma~\ref{lem:coupling}, we get
$$ \begin{array}{rcl}
E \, (\cont (e \, | \, T_e < \infty)) & \geq & \sum_{k > 0 } \, \mathbf W (k) \, P \, (\ceil{\xi_0 (e) / \tau} = k) \vspace*{4 pt} \\
& = & \sum_{k > 0 } \, \mathbf W (k) \, P \, ((k - 1 ) \tau < \xi_0 (e) \leq k \tau) \vspace*{4 pt} \\
& = & \sum_{k > 0 } \, \mathbf W (k) \, \sum_{s : \ceil{s / \tau} = k} \, F^{-1 } \, h (s) \vspace*{4 pt} \\
& = & F^{-1 } \, S_{\reg} (\Gamma, \tau) \ > \ 0. \end{array} $$
Now, let~$\mathbf W_e$ be the collection of random variables
$$ \begin{array}{l} \mathbf W_e \ := \ \sum_{k > 0 } \, \mathbf W (k) \, \mathbf{1 } \{\xi_0 (e) = k \} \quad \hbox{for all} \quad e \in \mathbb{Z} + 1 /2. \end{array} $$
Using Lemma~\ref{lem:weight} and the fact the number of collisions to turn a frozen pile into an active pile is independent for different
frozen piles, we deduce that there exists~$c_{11 } > 0 $ such that
$$ \begin{array}{l}
P \, (\sum_{e \in (0, N)} \cont (e \, | \, T_e < \infty) \leq 0 ) \ \leq \
P \, (\sum_{e \in (0, N)} \mathbf W_e \leq 0 ) \vspace*{4 pt} \\ \hspace*{40 pt} = \
P \, (\sum_{e \in (0, N)} \, (\mathbf W_e - E \mathbf W_e) \notin (- \epsilon N, \epsilon N)) \ \leq \ \exp (- c_{11 } N) \end{array} $$
for all~$N$ large. This, together with~\eqref{eq:inclusion-1 }, implies that
$$ \begin{array}{rcl}
P \, (H_N) & \leq &
P \, (\sum_{e \in (l, r)} \cont (e \, | \, T_e < \infty) \leq 0 \ \hbox{for some~$l < - N$ and~$r \geq 0 $}) \vspace*{4 pt} \\ & \leq &
\sum_{l < - N} \, \sum_{r \geq 0 } \, \exp (- c_{11 } \, (r - l)) \ \to \ 0 \end{array} $$
as~$N \to \infty$. In particular, it follows from Lemma~\ref{lem:fixation-condition} that the process fixates. \end{demo}
\section{Proof of Corollaries~\ref{cor:path}--\ref{cor:hypercube}}
\label{sec:graphs}
\indent This section is devoted to the proof of Corollaries~\ref{cor:path}--\ref{cor:hypercube} that give sufficient conditions
for fluctuation and fixation of the infinite system for the opinion graphs shown in Figure~\ref{fig:graphs}. To begin with, we prove the fluctuation part of all the corollaries at once. \\ \\
\begin{demo}{Corollaries~\ref{cor:path}--\ref{cor:hypercube} (fluctuation)} --
We start with the tetrahedron. In this case, the diameter equals one therefore, whenever the threshold is positive, the system reduces to a four-opinion voter model,
which is known to fluctuate according to~\cite{arratia_1983 }. To deal with paths and stars, we recall that combining Theorem~\ref{th:fluctuation}a and Lemma~\ref{lem:partition} gives fluctuation
when~$\mathbf{r} \leq \tau$. Recalling also the expression of the radius from Table~\ref{tab:summary} implies fluctuation when
$$ \begin{array}{rl}
F \leq 2 \tau + 1 & \hbox{for the path with~$F$ vertices} \vspace*{3 pt} \\
r \leq \tau & \hbox{for the star with~$b$ branches of length~$r$}. \end{array} $$
For the other graphs, it suffices to find a partition that satisfies~\eqref{eq:fluctuation}. For the remaining four regular polyhedra and the hypercubes, we observe that there is a unique vertex at distance~$\mathbf{d}$ of any
given vertex. In particular, fixing an arbitrary vertex~$i_-$ and setting
$$ V_1 \ := \ \{i_-, i_+ \} \quad \hbox{and} \quad V_2 \ := \ V \setminus V_1 \quad \hbox{where} \quad d (i_-, i_+) = \mathbf{d} $$
defines a partition of the set of opinions such that
$$ d (i, j) \ \leq \ \mathbf{d} - 1 \quad \hbox{for all} \quad (i, j) \in V_1 \times V_2. $$
Recalling the expression of the diameter from Table~\ref{tab:summary} and using~Theorem~\ref{th:fluctuation}a give the fluctuation
parts of Corollaries~\ref{cor:polyhedron} and~\ref{cor:hypercube}. Using the exact same approach implies fluctuation when the opinion graph is a cycle with an even number of vertices and~$F \leq 2 \tau + 2 $. For cycles with an odd number of vertices, we again use Lemma~\ref{lem:partition} to deduce fluctuation if
$$ \integer{F / 2 } = \mathbf{r} \leq \tau \quad \hbox{if and only if} \quad F \leq 2 \tau + 1 \quad \hbox{if and only if} \quad F \leq 2 \tau + 2, $$
where the last equivalence is true because~$F$ is odd. \end{demo} \\ \\
We now prove the fixation part of the corollaries using Theorems~\ref{th:fixation} and~\ref{th:dist-reg}. The first two classes of graphs, paths and stars, are not distance-regular therefore, to study the behavior of the
model for these opinion graphs, we rely on the first part of Theorem~\ref{th:fixation}. \\ \\
\begin{demo}{Corollary~\ref{cor:path} (path)} --
Assume that~$4 \tau < \mathbf{d} = F - 1 \leq 5 \tau$. Then,
$$ \begin{array}{rcl}
S (\Gamma, \tau) & = & \sum_{k > 0 } \, ((k - 2 ) \, \sum_{s : \ceil{s / \tau} = k} \, N (\Gamma, s)) \vspace*{4 pt} \\
& = & \sum_{0 < k \leq 4 } \, ((k - 2 ) \, \sum_{s : \ceil{s / \tau} = k} \,2 \, (F - s)) + 3 \, \sum_{4 \tau < s \leq d} \,2 \, (F - s) \vspace*{4 pt} \\
& = & \sum_{0 < k \leq 4 } \, ((k - 2 )(2 F \tau - (k \tau)(k \tau + 1 ) + ((k - 1 ) \, \tau)((k - 1 ) \, \tau + 1 )) \vspace*{4 pt} \\ && \hspace*{50 pt} + \
3 \, (2 F \, (F - 4 \tau - 1 ) - F \, (F - 1 ) + 4 \tau \, (4 \tau + 1 )) \vspace*{4 pt} \\
& = & 4 F \tau + \tau \, (\tau + 1 ) + 2 \tau \, (2 \tau + 1 ) + 3 \tau \, (3 \tau + 1 ) \vspace*{4 pt} \\ && \hspace*{50 pt} + \
4 \tau \, (4 \tau + 1 ) + 6 F \, (F - 4 \tau - 1 ) - 3 F \, (F - 1 ) \vspace*{4 pt} \\
& = & 3 F^2 - (20 \tau + 3 ) \, F + 10 \, (3 \tau + 1 ) \, \tau. \end{array} $$
Since the largest root~$F_+ (\tau)$ of this polynomial satisfies
$$ 4 \tau \leq F_+ (\tau) - 1 = (1 /6 )(20 \, \tau + 3 + \sqrt{40 \, \tau^2 + 9 }) - 1 \leq 5 \tau \quad \hbox{for all} \quad \tau \geq 1 $$
and since for any fixed~$\tau$ the function~$F \mapsto S (\Gamma, \tau)$ is nondecreasing, we deduce that fixation occurs under the assumptions of the lemma
according to Theorem~\ref{th:fixation}. \end{demo} \\ \\
The case of the star with~$b$ branches of equal length~$r$ is more difficult mainly because there are two different expressions for the number of pairs of vertices at a
given distance of each other depending on whether the distance is smaller or larger than the branches' length. In the next lemma, we compute the number of pairs of vertices at a given distance of each other, which we then use to find a condition for fixation when
the opinion graph is a star. \begin{lemma} --
\label{lem:star}
For the star with~$b$ branches of length~$r$,
$$ \begin{array}{rclcl}
N (\Gamma, s) & = & b \, (2 r + (b - 3 )(s - 1 )) & \hbox{for all} & s \in (0, r] \vspace*{3 pt} \\
& = & b \, (b - 1 )(2 r - s + 1 ) & \hbox{for all} & s \in (r, 2 r]. \end{array} $$
\end{lemma}
\begin{proof}
Let~$n_1 (s)$ and~$n_2 (s)$ be respectively the number of directed paths of length~$s$ embedded in a given branch of the star and the total
number of directed paths of length~$s$ embedded in a given pair of branches of the star. Then, as in the proof of the corollary for paths,
$$ n_1 (s) = 2 \, (r + 1 - s) \quad \hbox{and} \quad n_2 (s) = 2 \, (2 r + 1 - s) \quad \hbox{for all} \quad s \leq r. $$
Since there are~$b$ branches and $(1 /2 )(b - 1 ) \, b$ pairs of branches, and since self-avoiding paths embedded in the star cannot
intersect more than two branches, we deduce that
$$ \begin{array}{rcl}
N (\Gamma, s) & = & b \, n_1 (s) + ((1 /2 )(b - 1 ) \, b)(n_2 (s) - 2 n_1 (s)) \vspace*{4 pt} \\
& = & 2 b \, (r + 1 - s) + b \, (b - 1 )(s - 1 ) \vspace*{4 pt} \\
& = & b \, (2 r + 2 \, (1 - s) + (b - 1 )(s - 1 )) \ = \ b \, (2 r + (b - 3 )(s - 1 )) \end{array} $$
for all~$s \leq r$. To deal with~$s > r$, we let~$o$ be the center of the star and observe that there is no vertex at distance~$s$ of vertices which are
close to the center whereas there are~$b - 1 $ vertices at distance~$s$ from vertices which are far from the center. More precisely,
$$ \begin{array}{rclcl}
\card \{j \in V : d (i, j) = s \} & = & 0 & \quad \hbox{when} & d (i, o) < s - r \vspace*{3 pt} \\
\card \{j \in V : d (i, j) = s \} & = & b - 1 & \quad \hbox{when} & d (i, o) \geq s - r. \end{array} $$
The number of directed paths of length~$s$ is then given by
$$ \begin{array}{rcl}
N (\Gamma, s) & = & (b - 1 ) \, \card \{i \in V : d (i, o) \geq s - r \} \vspace*{4 pt} \\
& = & b \, (b - 1 )(r - (s - r - 1 )) \ = \ b \, (b - 1 )(2 r - s + 1 ) \end{array} $$
for all~$s > r$. This completes the proof of the lemma. \end{proof} \\ \\
\begin{demo}{Corollary~\ref{cor:star} (star)} --
Assume that~$3 \tau < \mathbf{d} = 2 r \leq 4 \tau$. Then,
$$ \begin{array}{rcl}
S (\Gamma, \tau) & = & \sum_{k > 0 } \, ((k - 2 ) \, \sum_{s : \ceil{s / \tau} = k} \, N (\Gamma, s)) \vspace*{4 pt} \\
& = & - \ \sum_{0 < s \leq \tau} \, N (\Gamma, s) + \sum_{2 \tau < s \leq 3 \tau} \, N (\Gamma, s) + 2 \, \sum_{3 \tau < s \leq 2 r} \, N (\Gamma, s).
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2 r} \, b \, (b - 1 )(2 r - s + 1 ) \vspace*{4 pt} \\
& = & - \ b \, (2 r - b + 3 ) \, \tau - (b/2 )(b - 3 ) \, \tau \, (\tau + 1 ) \vspace*{4 pt} \\ &&
+ \ b \, (b - 1 )(2 r + 1 ) \, \tau + (b/2 )(b - 1 )(2 \tau \, (2 \tau + 1 ) - 3 \tau \, (3 \tau + 1 )) \vspace*{4 pt} \\ &&
+ \ 2 b \, (b - 1 )(2 r + 1 )(2 r - 3 \tau) + b \, (b - 1 )(3 \tau \, (3 \tau + 1 ) - 2 r \, (2 r + 1 )). \end{array} $$
Expanding and simplifying, we get
$$ (1 /b) \, S (\Gamma, \tau) \ = \ 4 \, (b - 1 ) \, r^2 + 2 \, ((4 - 5 b) \, \tau + b - 1 ) \, r + (6 b - 5 ) \, \tau^2 + (1 - 2 b) \, \tau. $$
As for paths, the result is a direct consequence of Theorem~\ref{th:fixation}. \end{demo} \\ \\
The remaining graphs in Figure~\ref{fig:graphs} are distance-regular, which makes Theorem~\ref{th:dist-reg} applicable. Note that the conditions for fixation in the last three corollaries give minimal values for the confidence threshold that lie between one third and
one half of the diameter. In particular, we apply the theorem in the special case when~$\ceil{\mathbf{d} / \tau} = 3 $. In this case, we have
$$ \mathbf W (1 ) \ = \ - 1 \qquad \mathbf W (2 ) \ = \ \mathbf W (1 ) + (1 / p_2 )(1 + q_2 / p_3 ) \qquad \mathbf W (3 ) \ = \ \mathbf W + 1 / p_3 $$
so the left-hand side of~\eqref{eq:th-dist-reg} becomes
\begin{equation}
\label{eq:common}
\begin{array}{rcl}
S_{\reg} (\Gamma, \tau) & = & \sum_{0 < k \leq 3 } \, (\mathbf W (k) \, \sum_{s : \ceil{s / \tau} = k} \, h (s)) \vspace*{4 pt} \\ & = &
- \ (h (1 ) + h (2 ) + \cdots + h (\mathbf{d})) \vspace*{4 pt} \\ &&
+ \ (1 /p_2 )(1 + q_2 / p_3 )(h (\tau + 1 ) + h (\tau + 2 ) + \cdots + h (\mathbf{d})) \vspace*{4 pt} \\ &&
+ \ (1 /p_3 )(h (2 \tau + 1 ) + h (2 \tau + 2 ) + \cdots + h (\mathbf{d})). \end{array}
\end{equation}
This expression is used repeatedly to prove the remaining corollaries. \\ \\
\begin{demo}{Corollary~\ref{cor:polyhedron} (cube)} --
When~$\Gamma$ is the cube and~$\tau = 1 $, we have
$$ p_2 \ = \ f (1, 2, 1 ) / h (2 ) \ = \ 2 /3 \quad \hbox{and} \quad q_2 \ = \ f (1, 2, 3 ) / h (2 ) \ = \ 1 /3 $$
which, together with~\eqref{eq:common} and the fact that~$p_3 \leq 1 $, implies that
$$ \begin{array}{rcl}
S_{\reg} (\Gamma, 1 ) & \geq & - \ (h (1 ) + h (2 ) + h (3 )) + (1 /p_2 )(1 + q_2 )(h (2 ) + h (3 )) + h (3 ) \vspace*{4 pt} \\
& = & - \ (3 + 3 + 1 ) + (3 /2 )(1 + 1 /3 )(3 + 1 ) + 1 \ = \ 2 \ > \ 0. \end{array} $$
This proves fixation according to Theorem~\ref{th:dist-reg}. \end{demo} \\ \\
\begin{demo}{Corollary~\ref{cor:polyhedron} (icosahedron)} --
When~$\Gamma$ is the icosahedron and~$\tau = 1 $,
$$ p_2 \ = \ f (1, 2, 1 ) / h (2 ) \ = \ 2 /5 \qquad \hbox{and} \qquad q_2 \ = \ f (1, 2, 3 ) / h (2 ) \ = \ 1 /5. $$
Using in addition~\eqref{eq:common} and the fact that~$p_3 \leq 1 $, we obtain
$$ \begin{array}{rcl}
S_{\reg} (\Gamma, 1 ) & \geq & - \ (h (1 ) + h (2 ) + h (3 )) + (1 /p_2 )(1 + q_2 )(h (2 ) + h (3 )) + h (3 ) \vspace*{4 pt} \\
& = & - \ (5 + 5 + 1 ) + (5 /2 )(1 + 1 /5 )(5 + 1 ) + 1 \ = \ 8 \ > \ 0 \end{array} $$
which, according to Theorem~\ref{th:dist-reg}, implies fixation. \end{demo} \\ \\
\begin{demo}{Corollary~\ref{cor:polyhedron} (dodecahedron)} --
Fixation of the opinion model when the threshold equals one directly follows from Theorem~\ref{th:fixation} since in this case
$$ \begin{array}{rcl}
F^{-1 } \, S (\Gamma, 1 ) & = & (1 /20 )(- h (1 ) + h (3 ) + 2 \, h (4 ) + 3 \, h (5 )) \vspace*{3 pt} \\
& = & (1 /20 )(- 3 + 6 + 2 \times 3 + 3 \times 1 ) \ = \ 3 /5 \ > \ 0. \end{array} $$
However, when the threshold~$\tau = 2 $,
$$ \begin{array}{rcl}
F^{-1 } \, S (\Gamma, 2 ) & = & (1 /20 )(- h (1 ) - h (2 ) + h (5 )) \vspace*{3 pt} \\
& = & (1 /20 )(- 3 - 6 + 1 ) \ = \ - 2 /5 \ < \ 0 \end{array} $$
so we use Theorem~\ref{th:dist-reg} instead: when~$\tau = 2 $, we have
$$ \begin{array}{rcl}
p_2 & = & \max \, \{\sum_{s = 1, 2 } f (s_-, s_+, s) / h (s_+) : s_- = 1, 2 \ \hbox{and} \ s_+ = 3, 4 \} \vspace*{4 pt} \\
& = & \max \, \{f (1, 3, 2 ) / h (3 ), (f (2, 3, 2 ) + f (2, 3, 1 )) / h (3 ), f (2, 4, 2 ) / h (4 ) \} \vspace*{4 pt} \\
& = & \max \, \{2 /6, (2 + 1 ) / 6, 1 /3 \} \ = \ 1 /2. \end{array} $$
In particular, using~\eqref{eq:common} and the fact that~$p_3 \leq 1 $ and~$q_2 \geq 0 $, we get
$$ \begin{array}{rcl}
S_{\reg} (\Gamma, 2 ) & \geq & - \ (h (1 ) + h (2 ) + h (3 ) + h (4 ) + h (5 )) \vspace*{4 pt} \\ && \hspace*{25 pt} + \
(1 /p_2 )(h (3 ) + h (4 ) + h (5 )) + h (5 ) \vspace*{4 pt} \\
& = & - \ (3 + 6 + 6 + 3 + 1 ) + 2 \times (6 + 3 + 1 ) + 1 \ = \ 2 \ > \ 0, \end{array} $$
which again gives fixation. \end{demo} \\ \\
\begin{demo}{Corollary~\ref{cor:cycle} (cycle)} --
Regardless of the parity of~$F$,
\begin{equation}
\label{eq:cycle-1 }
\begin{array}{rclclcl}
f (s_-, s_+, s) & = & 0 & \hbox{when} & s_- \leq s_+ \leq \mathbf{d} & \hbox{and} & s > s_+ - s_- \vspace*{2 pt} \\
f (s_-, s_+, s) & = & 1 & \hbox{when} & s_- \leq s_+ \leq \mathbf{d} & \hbox{and} & s = s_+ - s_- \end{array}
\end{equation}
while the number of vertices at distance~$s_+$ of a given vertex is
\begin{equation}
\label{eq:cycle-2 }
h (s_+) = 2 \ \ \hbox{for all} \ \ s_+ < F/2 \quad \hbox{and} \quad h (s_+) = 1 \ \ \hbox{when} \ \ s_+ = F/2 \in \mathbb{N}. \end{equation}
Assume that~$F = 4 \tau + 2 $. Then, $\mathbf{d} = 2 \tau + 1 $ so it follows from~\eqref{eq:cycle-1 }--\eqref{eq:cycle-2 } that
$$ \begin{array}{rcl}
p_2 & = & \max \, \{\sum_{s : \ceil{s / \tau} = 1 } f (s_-, s_+, s) / h (s_+) : \ceil{s_- / \tau} = 1 \ \hbox{and} \ \ceil{s_+ / \tau} = 2 \} \vspace*{3 pt} \\
& = & \max \, \{f (s_-, s_+, s_+ - s_-) / h (s_+) : \ceil{s_- / \tau} = 1 \ \hbox{and} \ \ceil{s_+ / \tau} = 2 \} \vspace*{3 pt} \\
& = & \max \, \{f (s_-, s_+, s_+ - s_-) / h (s_+) : \ceil{s_+ / \tau} = 2 \} \ = \ 1 /2. \end{array} $$
Using in addition that~$p_3 \leq 1 $ and~$q_2 \geq 0 $ together with~\eqref{eq:common}, we get
$$ \begin{array}{rcl}
S_{\reg} (\Gamma, \tau) & \geq & - \ (h (1 ) + h (2 ) + \cdots + h (2 \tau + 1 )) \vspace*{4 pt} \\ &&
+ \ (1 /p_2 )(h (\tau + 1 ) + h (\tau + 2 ) + \cdots + h (2 \tau + 1 )) + h (2 \tau + 1 ) \vspace*{4 pt} \\
& = & - \ (4 \tau + 1 ) + 2 \times (2 \tau + 1 ) + 1 \ = \ 2 \ > \ 0. \end{array} $$
In particular, the corollary follows from Theorem~\ref{th:dist-reg}. \end{demo} \\ \\
\begin{demo}{corollary~\ref{cor:hypercube} (hypercube)} --
The first part of the corollary has been explained heuristically in~\cite{adamopoulos_scarlatos_2012 }. To turn it into a proof, we first observe that opinions on the hypercube can be represented by vectors with coordinates equal to zero or one while the distance
between two opinions is the number of coordinates the two corresponding vectors disagree on. In particular, the number of opinions at distance~$s$ of a given opinion, namely~$h (s)$, is equal to the number of subsets of size~$s$ of a set of size~$d$. Therefore, we have the symmetry property
\begin{equation}
\label{eq:hypercube-1 }
h (s) \ = \ {d \choose s} \ = \ {d \choose d - s} \ = \ h (d - s) \quad \hbox{for} \quad s = 0, 1, \ldots, d,
\end{equation}
from which it follows that, for~$d = 3 \tau + 1 $,
$$ \begin{array}{rcl}
2 ^{-d} \, S (\Gamma, \tau) & = & - \ h (1 ) - \cdots - h (\tau) + h (2 \tau + 1 ) + \cdots + h (d - 1 ) + 2 \, h (d) \vspace*{3 pt} \\
& = & h (d - 1 ) - h (1 ) + h (d - 2 ) - h (2 ) + \cdots + h (d - \tau) - h (\tau) + 2 \, h (d) \vspace*{3 pt} \\
& = & 2 \, h (d) \ = \ 2 \ > \ 0. \end{array} $$
Since in addition the function~$d \mapsto S (\Gamma, \tau)$ is nondecreasing, a direct application of Theorem~\ref{th:fixation} gives the first part of the corollary. The second part is more difficult. Note that, to prove this part, it suffices to show that, for any fixed~$\sigma > 0 $, fixation occurs when
\begin{equation}
\label{eq:hypercube-2 }
d \ = \ (2 + 3 \sigma) \, \tau \quad \hbox{and} \quad \tau \ \ \hbox{is large}. \end{equation}
The main difficulty is to find a good upper bound for~$p_2 $ which relies on properties of the hypergeometric random variable. Let~$u$ and~$v$ be two opinions at distance~$s_-$ of each other. By symmetry, we may assume without loss of generality that both vectors disagree on their first~$s_-$ coordinates. Then, changing each of the first~$s_-$ coordinates in either one vector or the other vector and changing each of the remaining coordinates in either both vectors
simultaneously or none of the vectors result in the same vector. In particular, choosing a vector~$w$ such that
$$ d (u, w) \ = \ s_+ \quad \hbox{and} \quad d (v, w) \ = \ s $$
is equivalent to choosing~$a$ of the first~$s_-$ coordinates and then choosing~$b$ of the remaining~$d - s_-$ coordinates with the following constraint:
$$ a + b \ = \ s_+ \quad \hbox{and} \quad (s_- - a) + b \ = \ s.
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2 $. Then,
$$ P \, (Z \geq K) \ = \ \sum_{a = K}^{s_-} {s_- \choose a}{d - s_- \choose s_+ - a}{d \choose s_+}^{-1 } \leq \ 1 /2. $$
\end{lemma}
\begin{proof}
The proof is made challenging by the fact that there is no explicit expression for the cumulative distribution function of the hypergeometric random variable
and the idea is to use a combination of symmetry arguments and large deviation estimates.
Symmetry is used to prove the result when~$s_-$ is small while large deviation estimates are used for larger values.
Note that the result is trivial when~$s_+ > s_- + \tau$ since in this case the sum in the statement of the lemma is empty so equal to zero.
To prove the result when the sum is nonempty, we distinguish two cases. \vspace*{5 pt} \\
{\bf Small active piles} -- Assume that~$s_- < \sigma \tau$. Then,
\begin{equation}
\label{eq:hypergeometric-1 }
\begin{array}{rcl}
s_+ & \leq & s_- + \tau < (1 + \sigma) \, \tau \ = \ (1 /2 )(d - \sigma \tau) \ < \ (1 /2 )(d - s_-) \vspace*{3 pt} \\
K & \geq & (1 /2 )(s_- + s_+ - \tau) \ > \ s_- / 2 \ > \ s_- - K \end{array}
\end{equation}
from which it follows that
\begin{equation}
\label{eq:hypergeometric-2 }
{s_- \choose a}{d - s_- \choose s_+ - a} \ \leq \ {s_- \choose a}{d - s_- \choose s_+ - s_- + a} \quad \hbox{for all} \quad K \leq a \leq s_-.
\end{equation}
Using~\eqref{eq:hypergeometric-2 } and again the second part of~\eqref{eq:hypergeometric-1 }, we deduce that
$$ \begin{array}{rcl}
h (s_+) \, P \, (Z \geq K) & = & \displaystyle \sum_{a = K}^{s_-} {s_- \choose a}{d - s_- \choose s_+ - a}
\ \leq \ \displaystyle \sum_{a = K}^{s_-} {s_- \choose a}{d - s_- \choose s_+ - s_- + a} \vspace*{4 pt} \\
& = & \displaystyle \sum_{a = 0 }^{s_- - K} {s_- \choose s_- - a}{d - s_- \choose s_+ - a}
\ \leq \ \displaystyle \sum_{a = 0 }^{K - 1 } {s_- \choose a}{d - s_- \choose s_+ - a}. \end{array} $$
In particular, we have~$P \, (Z \geq K) \leq P \, (Z < K)$, which gives the result. \vspace*{5 pt} \\
{\bf Larger active piles} -- Assume that~$\sigma \tau \leq s_- \leq \tau$.
In this case, the result is a consequence of the following large deviation estimates for the hypergeometric random variable:
\begin{equation}
\label{eq:hypergeometric-3 }
P \, \bigg(Z \geq \bigg(\frac{s_-}{d} + \epsilon \bigg) \, s_+ \bigg) \ \leq \ \bigg(\bigg(\frac{s_-}{s_- + \epsilon d} \bigg)^{s_- / d + \epsilon} \bigg(\frac{d - s_-}{d - s_- - \epsilon d} \bigg)^{1 - s_- / d - \epsilon} \bigg)^{s_+}
\end{equation}
for all~$0 < \epsilon < 1 - s_- / d$, that can be found in~\cite{hoeffding_1963 }.
Note that
$$ \begin{array}{rcl}
d \, (s_+ + s_- - \tau) - 2 s_+ \, s_- & = & (d - 2 s_-) \, s_+ + d \, (s_- - \tau) \vspace*{3 pt} \\
& \geq & (d - 2 s_-)(\tau + 1 ) + d \, (s_- - \tau) \ \geq \ (d - 2 \tau) \, s_- \vspace*{3 pt} \\
& = & 3 \sigma \tau s_- \ = \ (3 \sigma \tau / 2 s_+)(2 s_+ \, s_-) \ \geq \ (3 \sigma / 4 )(2 s_+ \, s_-) \end{array} $$
for all~$\tau < s_+ \leq 2 \tau$.
It follows that
$$ K \ \geq \ \frac{s_+ + s_- - \tau}{2 } \ \geq \ \bigg(1 + \frac{3 \sigma}{4 } \bigg) \, \frac{s_+ \, s_-}{d} \ = \ \bigg(\frac{s_-}{d} + \frac{3 \sigma s_-}{4 d} \bigg) \, s_+ \ \geq \ \bigg(\frac{s_-}{d} + \frac{\sigma^2 }{3 } \bigg) \, s_+ $$
which, together with~\eqref{eq:hypergeometric-3 } for~$\epsilon = \sigma^2 / 3 $, gives
$$ \begin{array}{rcl}
P \, (Z \geq K) & \leq &
\displaystyle P \, \bigg(Z \geq \bigg(\frac{s_-}{d} + \epsilon \bigg) \, s_+ \bigg) \ \leq \ \bigg(\frac{s_-}{s_- + \epsilon d} \bigg)^{s_+ s_- / d} \vspace*{8 pt} \\ & \leq &
\displaystyle \bigg(\frac{3 s_-}{3 s_- + \sigma^2 d} \bigg)^{s_+ s_- / d} \leq \ \bigg(\frac{3 }{3 + 2 \sigma^2 } \bigg)^{(\sigma / 3 ) \, s_+} \leq \ \bigg(\frac{3 }{3 + 2 \sigma^2 } \bigg)^{(\sigma / 3 ) \, \tau}. \end{array} $$
Since this tends to zero as~$\tau \to \infty$, the proof is complete.
\end{proof} \\ \\
It directly follows from the lemma that
$$ \begin{array}{l}
p_2 \ = \ \max \, \{\sum_{s : \ceil{s / \tau} = 1 } f (s_-, s_+, s) / h (s_+) : \ceil{s_- / \tau} = 1 \ \hbox{and} \ \ceil{s_+ / \tau} = 2 \} \ \leq \ 1 /2. \end{array} $$
This, together with~\eqref{eq:common} and~$p_3 \leq 1 $ and~$q_2 \geq 0 $, implies that
$$ \begin{array}{rcl}
S_{\reg} (\Gamma, \tau) & \geq & - \ h (1 ) - \cdots - h (d) + (1 /p_2 ) \, h (\tau + 1 ) + \cdots + (1 /p_2 ) \, h (d) \vspace*{3 pt} \\
& \geq & - \ h (1 ) - \cdots - h (d) + 2 \, h (\tau + 1 ) + \cdots + 2 \, h (d) \vspace*{3 pt} \\
& = & - \ h (1 ) - \cdots - h (\tau) + h (\tau + 1 ) + \cdots + h (d). \end{array} $$
Finally, using again~\eqref{eq:hypercube-1 } and the fact that~$d > 2 \tau$, we deduce that
$$ \begin{array}{rcl}
S_{\reg} (\Gamma, \tau) & \geq & - \ h (1 ) - \cdots - h (\tau) + h (\tau + 1 ) + \cdots + h (d) \vspace*{3 pt} \\
& \geq & h (d - 1 ) - h (1 ) + h (d - 2 ) - h (2 ) + \cdots + h (d - \tau) - h (\tau) + h (d) \vspace*{3 pt} \\
& = & h (d) \ = \ 1 \ > \ 0. \end{array} $$
The corollary follows once more from Theorem~\ref{th:dist-reg}.
\end{demo}
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\section{Proof of the First Zonklar Equation}
\ifCLASSOPTIONcaptionsoff
\newpage
\fi
\input{. /section/6 _Bibliography}
\begin{IEEEbiography}[{\includegraphics[width=1 in, height=1.25 in, clip, keepaspectratio]{. /fig/6 _Bibliography/alessandro_betti. jpg}}]{Alessandro Betti}
Dr. Alessandro Betti received the M. S. degree in Physics and the Ph. D. degree in Electronics Engineering from the University of Pisa, Italy, in 2007 and 2011, respectively. His main field of research was modeling of noise and transport in quasi-one dimensional devices. His work has been published in 10 papers in peer-reviewed journals in the field of solid state electronics and condensed matter physics and in 16 conference papers, presenting for 3 straight years his research at the top International Conference IEEE in electron devices, the International Electron Device Meeting in USA. In September 2015 he joined the company i-EM in Livorno, where he currently works as a Senior Data Scientist developing power generation forecasting, predictive maintenance and Deep Learning models, as well as solutions in the electrical mobility fields and managing a Data Science Team. \end{IEEEbiography}
\begin{IEEEbiography}[{\includegraphics[width=1 in, height=1.25 in, clip, keepaspectratio]{. /fig/6 _Bibliography/Emanuele_Crisostomi}}]{Emanuele Crisostomi}(M'12 -SM'16 ) received the B. Sc. degree in computer science engineering, the M. Sc. degree in automatic control, and the Ph. D. degree in automatics, robotics, and bioengineering from the University of Pisa, Pisa, Italy, in 2002, 2005, and 2009, respectively. He is currently an Associate Professor of electrical engineering in the Department of Energy, Systems, Territory and Constructions Engineering, University of Pisa. He has authored tens of publications in top refereed journals and he is a co-author of the recently published book on ``Electric and Plug-in Vehicle Networks: Optimization and Control'' (CRC Press, Taylor and Francis Group, 2017 ). His research interests include control and optimization of large scale systems, with applications to smart grids and green mobility networks. \end{IEEEbiography}
\begin{IEEEbiography}[{\includegraphics[width=1 in, height=1.25 in, clip, keepaspectratio]{. /fig/6 _Bibliography/gianluca_paolinelli. jpg}}]{Gianluca Paolinelli}
Gianluca Paolinelli received the bachelor and master degree in electrical engineering from the University of Pisa, Pisa, Italy, in 2014 and 2018 respectively. His research interests included big data analysis and computational intelligence applied in on-line monitoring and diagnostics. Currently, he is an electrical software engineer focused in the development of electric and hybrid power-train controls for Pure Power Control S. r. l. \end{IEEEbiography}
\begin{IEEEbiography}[{\includegraphics[width=1 in, height=1.25 in, clip, keepaspectratio]{. /fig/6 _Bibliography/antonio_piazzi. jpg}}]{Antonio Piazzi}
Antonio Piazzi received the M. Sc. degree in electrical engineering from the University of Pisa, Pisa, Italy, 2013. Electrical engineer at i-EM since 2014, he gained his professional experience in the field of renewable energies. His research interests include machine learning and statistical data analysis, with main applications on modeling and monitoring the behaviour of renewable power plants. Currently, he is working on big data analysis applied on hydro power plants signals. \end{IEEEbiography}
\begin{IEEEbiography}[{\includegraphics[width=1 in, height=1.25 in, clip, keepaspectratio]{. /fig/6 _Bibliography/FabrizioRuffini_Linkedin}}]{Fabrizio Ruffini}
Fabrizio Ruffini received the Ph. D. degree in Experimental Physics from University of Siena, Siena, Italy, in 2013. His research-activity is centered on data analysis, with particular interest in multidimensional statistical analysis. During his research activities, he was at the Fermi National Accelerator Laboratory (Fermilab), Chicago, USA, and at the European Organization for Nuclear Research (CERN), Geneva, Switzerland. Since 2013, he has been working at i-EM as data scientist focusing on applications in the renewable energy sector, atmospheric physics, and smart grids. Currently, he is senior data scientist with focus on international funding opportunities and dissemination activities. \end{IEEEbiography}
\begin{IEEEbiography}[{\includegraphics[width=1 in, height=1.25 in, clip, keepaspectratio]{. /fig/6 _Bibliography/Mauro_Tucci}}]{Mauro Tucci} received the Ph. D. degree in applied electromagnetism from the University of Pisa, Pisa, Italy, in 2008. Currently, he is an Associate Professor with the Department of Energy, Systems, Territory and Constructions Engineering, University of Pisa. His research interests include computational intelligence
and big data analysis, with applications in electromagnetism, non destructive testing, powerline communications, and smart grids. \end{IEEEbiography}
\end{document}
\section{Introduction}
\subsection{Motivation}
\IEEEPARstart{A}{s} power generation from renewable sources is increasingly seen as a fundamental component in a joint effort to support decarbonization strategies, hydroelectric power generation is experiencing a new golden age. In fact, hydropower has a number of advantages compared to other types of power generation from renewable sources. Most notably,
hydropower generation can be ramped up and down, which provides a valuable source of flexibility for the power grid, for instance, to support the integration of power generation from other renewable energy sources, like wind and solar. In addition, water in hydropower plants' large reservoirs may be seen as an energy storage resource in low-demand periods and transformed into electricity when needed \cite{Helseth2016, Hjelmeland2019 }. Finally, for large turbine-generator units, the mechanical-to-electrical energy conversion process can have a combined efficiency of over 90 \% \cite{Bolduc2014 }. Accordingly, in 2016, around 13 \% of the world's consumed electricity was generated from hydropower\footnote{\url{https://www. iea. org/statistics/balances/}}. In addition, hydropower plants have provided for more than 95 \% of energy storage for all active tracked storage installations worldwide\footnote{\url{http://www. energystorageexchange. org/projects/data_visualization}}. In addition to the aforementioned advantages, hydroelectric power plants are also typically characterized by a long lifespan and relatively low operation and maintenance costs, usually around 2.5 \% of the overall cost of the plant. However, according to \cite{IHA_db}, by 2030 over half of the world's hydropower plants will be due for upgrade and modernization, or will have already been renovated. Still according to \cite{IHA_db}, the main reason why major works seem around the corner is that most industries in this field wish to adopt best practices in operations and asset management plans, or in other words, share a desire for optimized performance and increased efficiency. In combination with the quick pace of technological innovation in hydropower operations and maintenance, together with the increased ability to handle and manage big amounts of data, a technological revolution of most hydropower plants is expected to take place soon. \subsection{State of the art}
In hydropower plants, planned periodic maintenance has been for a long time the main, if not the only one, adopted maintenance method. Condition monitoring procedures have been often reserved for protection systems, leading to shutting down the plants when single monitored signals exceeded pre-defined thresholds (e. g., bearings with temperature and vibration protection). In this context, one of the earliest works towards predictive maintenance has been \cite{Jiang2008 }, where Artificial Neural Networks (ANNs) were used to monitor, identify and diagnose the dynamic performance of a prototype of system. Predictive maintenance methods obviously require the measurement and storage of all the relevant data regarding the power plant. An example of early digitalization is provided in \cite{Li2009 } where a Wide Area Network for condition monitoring and diagnosis of hydro power stations and substations of the Gezhouba Hydro Power Plant (in China) was established. Thanks to measured data, available in real-time, more advanced methods that combine past history and domain knowledge can provide more efficient monitoring services, advanced fault prognosis, short- and long-term prediction of incipient faults, prescriptive maintenance tools, and residual lifetime and future risk assessment. Benefits of this include, among other things, preventing (possibly severe) faults from occurring, avoiding unnecessary replacements of components, more efficient criteria for scheduled maintenance. The equipment required for predictive maintenance in hydro generators is also described in \cite{Ribeiro2014 }, where the focus was to gain the ability to detect and classify faults regarding the electrical and the mechanical defects of the generator-turbine set, through a frequency spectrum analysis. More recent works (e. g., \cite{Selak2014 }) describe condition monitoring and fault-diagnostics (CMFD) software systems that use recent Artificial Intelligence (AI) techniques (in this case a Support Vector Machine (SVM) classifier) for fault diagnostics. In \cite{Selak2014 } a CMFD has been implemented on a hydropower plant with three Kaplan units. Another expert system has been developed for an 8 -MW bulb turbine downstream irrigation hydropower plant in India, as described in \cite{Buaphan2017 }. An online temperature monitoring system was developed in \cite{Milic2013 }, and an artificial neural network based predictive maintenance system was proposed in \cite{Chuang2004 }. The accuracy of early fault detection system is an important feature for accurate reliability modeling \cite{Khalilzadeh2014 }. \subsection{Paper contribution}
In this paper we propose a novel Key Performance Indicator (KPI) based on an appropriately trained Self-Organizing Map (SOM) for condition monitoring in a hydropower plant. In addition to detecting faulty operating conditions, the proposed indicator also identifies the component of the plant that most likely gave rise to the faulty behaviour. Very few works, as from the previous section, address the same problem, although there is a general consensus that this could soon become a very active area of research \cite{IHA_db}. In this paper we show that the proposed KPI performs better than a standard multivariate process control tool like the Hotelling control chart, over a test period of more than one year (from April 2018 to July 2019 ). This paper is organized as follows: Section \ref{case_study} describes more in detail the case study of interest, and the data used to tune the proposed indicator. Section \ref{Methodologies} illustrates the proposed indicator. Also, the basic theory of the Hotelling multivariate control chart is recalled, as it is used for comparison purposes. The obtained results are provided and discussed in Section \ref{Results}. Finally, in Section \ref{Conclusion} we conclude our paper and outline our current lines of research in this topic. \section{Case study}
\label{case_study}
Throughout the paper, we will refer to two hydroelectric power plants, called plant A and plant B, which have an installed power of 215 MW and 1000 MW respectively. The plants are located in Italy as shown in Figure \ref{fig:HPP_Plant_Soverzene_Presenzano_location}, and both use Francis turbines. Plant A is of type reservoir, while plant B is of type pumped-storage. More details are provided in the following subsections. \begin{figure}[ht]
\centering
\includegraphics[width=0.3 \textwidth]{. /fig/2 _Case_study/italy. png}
\caption{Location of the two considered hydropower plants in Italy. }
\label{fig:HPP_Plant_Soverzene_Presenzano_location}
\end{figure}
\subsection{Hydropower plants details}
Plant A is located in Northern Italy and consists of four generation units moved by vertical axes Francis turbines, with a power of 60 MVA for each unit. The machinery room is located 500 meters inside the mountain. The plant is powered by two connected basins: the main basin, with a daily regulation purpose with a capacity of 5.9 $\times 10 ^6 $ m$^3 $, and a second basin, limited by a dam, with a seasonal regulation capability. The plant is part of a large hydraulic system and it has been operative since 1951. At full power, the plant employs a flow of 88 m$^3 /$s, with a net head of 284 m; in nominal conditions (1 unit working 24 /7, 3 units working 12 /7 ) the main basin can be emptied in about 24 hours. The 2015 net production was of 594 GWh, serving both the energy and the service market thanks to the storage capability. Plant B is representative of pumped-storage power plants; power is generated by releasing water from the upper reservoir down to the power plant which contains four reversible 250 MW Francis pump-turbine-generators. After power production, the water is sent to the lower reservoir. The upper reservoir, formed by an embankment dam, is located at an elevation of 643 meters in the Province of Isernia. Both the upper and lower reservoirs have an active storage capacity of 60 $\times 10 ^6 $ m$^3 $. The difference in elevation leads to a hydraulic head of 495 meters. The plant has been in operation since 1994, and its net production in 2015 was 60 GWh. This plant is strategical for its pumping storage capability and sells mainly to the services market. \subsection{Dataset}
The dataset of plant A consists of 630 analog signals with a sampling time of 1 minute. The dataset of plant B consists of 60 analog signals, since a smaller number of sensors is installed in this system. The signals are collected from several plant-components, for instance, the water intake, penstocks, turbines, generators, and HV transformers. The acquisition system has been in service since the 1 $^{st}$ of May 2017. In this work, we used data from the 1 $^{st}$ of May 2017 to the 31 $^{st}$ of March 2018, as the initial training set. Then, the model was tested online from the 1 $^{st}$ of April 2018 until the end of July 2019. During the online testing phase, we retrained the model every two months to include the most recent data. As usual in this kind of applications, before starting the training phase, an accurate pre-processing was required to improve the quality of measured data. In particular:
\begin{enumerate}
\item
several measured signals had a large number of not regular data, such as missing, or ``frozen'' samples (i. e., where the signal measured by the sensor does not change in time), values out of physical or operative limits, spikes, and statistical outliers in general;
\item
the training dataset did not contain information about historical anomalies occurred in the plants; similarly, Operation and Maintenance (O\&M) logs were not available. \end{enumerate}
Accordingly, we first implemented a classic procedure of data cleaning (see for instance \cite{Data_Cleaning_Book}). This procedure has two advantages itself: first of all, it allows any data-based condition monitoring methodology to be tuned upon nominal data corresponding to a correct functioning of the plant; in addition, the plants' operators were informed of those signals whose percentage of regular samples was below a given threshold, so that they could evaluate whether it could be possible to mitigate the noise with which data were recorded, or whether the sensor was actually broken.
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shall see in greater detail in the next section, the output of our proposed procedure is a newly proposed KPI, which monitors the functioning of the hydropower plant. In particular, a warning is triggered when the KPI drops below a threshold and an automatic notification is sent to the operator. The operator, guided by the provided warning, checks and possibly confirms the nature of the detected anomaly. Then, the sequence of data during the fault is eventually removed from the log, so that it will not be included in any historical dataset and it will not be used in the future retraining stages. \section{Methodologies}
\label{Methodologies}
The proposed approach consists in training a self-organizing map (SOM) neural network in order to build a model of the nominal behaviour of the system, using a historical dataset comprehending nominal state observations. The new state observations are then classified as ``in control'' or ``out-of-control'' after comparing their distortion measure to the average distortion measure of the nominal states used during the training, as it is now illustrated in more detail. \subsection{Self-Organizing Map neural network based Key Performance Indicator}
Self-organizing maps (SOMs) are popular artificial neural network algorithms, belonging to the unsupervised learning category, which have been frequently used in a wide range of applications \cite{Kohonen1990 }-\cite{Tuc_2010 }. Given a high-dimensional input dataset, the SOM algorithm produces a topographic mapping of the data into a lower-dimensional output set. The SOM output space consists of a fixed and ordered bi-dimensional grid of cells, identified by an index in the range $1, \dots, D$, where a distance metric $d(c, i)$ between any two cells of index $c$ and $i$ is defined \cite{Kohonen1990 }. Each cell of index $i$ is associated with a model vector $\mathbf{m}_i\in \mathbb{R}^{1 \times n}$ that lies in the same high-dimensional space of the input patterns $\mathbf{r}\in \Delta$, where the matrix $\Delta\in \mathbb{R}^{N \times n}$ represents the training dataset to be analyzed, containing $N$ observations of row vectors $\mathbf{r}\in \mathbb{R}^{1 \times n}$. After the training, the distribution of the model vectors resembles the distribution of the input data, with the additional feature of preserving the grid topology: model vectors that correspond to neighbouring cells shall be neighbours in the high-dimensional input space as well. When a new input sample $\mathbf{r}$ is presented to the network, the SOM finds the best matching unit (BMU) $c$, whose model vector $\mathbf{m}_c$ has the minimum Euclidean distance from $\mathbf{r}$:
\[c = \mathbf{arg min}_{i} \{\| \mathbf{r} - \mathbf{m}_i\|\}. \]
It is known that the goal of the SOM training algorithm is to minimize the following distortion measure:
\begin{equation}\label{DM_average}
DM_{\Delta}=\frac{1 }{N} \sum_{\mathbf{r} \in \Delta} \sum_{i=1 }^{D} w_{ci} \|\mathbf{r}-\mathbf{m}_i\|,
\end{equation}
where the function
\begin{equation}
w_{ci} = exp\left( \frac{-d(c, i)^2 }{2 \sigma^2 } \right),
\end{equation}
is the neighbourhood function, $c$ is the BMU corresponding to input sample $\mathbf{r}$, and $\sigma$ is the neighbourhood width. The distortion measure indicates the capacity of the trained SOM to fit the data maintaining the bi-dimensional topology of the output grid. The distortion measure relative to a single input pattern $\mathbf{r}$ is computed as:
\begin{equation}\label{DM_single}
DM(\mathbf{r})= \sum_{i=1 }^{D} w_{ci} \|\mathbf{r}-\mathbf{m}_i\|,
\end{equation}
from which it follows that $DM_{\Delta}$, as defined in (\ref{DM_average}), is the average of the distortion measures of all the patterns in the training data $\mathbf{r}\in \Delta$. In order to assess the condition of newly observed state patterns $\mathbf{r}$ to be monitored, we introduce the following KPI:
\begin{equation}
KPI(\mathbf{r})=\frac{1 }{1 +\|1 - \frac{DM(\mathbf{r} )}{DM_{\Delta}} \|}. \end{equation}
Roughly speaking, the rationale behind the previous KPI definition is as follows: if the acquired state $\mathbf{r}$ corresponds to a normal behaviour, its distortion measure $DM(\mathbf{r}) $ should be similar to the average distortion measure of the nominal training set $DM_{\Delta}$ (which consists of non-faulty states), and the ratio $ \frac{DM(\mathbf{r}) }{DM_{\Delta}} $ should be close to one, which in turn gives a $KPI( \mathbf{r})$ value to be close to one as well. On the other hand, if the acquired state $\mathbf{r}$ actually corresponds to an anomalous behaviour, $DM(\mathbf{r}) $ should substantially differ from $DM_{\Delta}$, leading to values of $KPI( \mathbf{r})$ considerably smaller than one. In this way, values of the KPI near to one indicate a nominal functioning, while smaller values indicate that the plant is going out of control. A critical aspect is the choice of the threshold to discriminate a correct and a faulty functioning: for this purpose, we compute the average value $\mu_{KPI}$ and the variance $\sigma^2 _{KPI}$ of the filtered KPI values of all the points in the training set $\Delta$, and we define the threshold as a lower control limit (LCL) as follows:
\begin{equation}
LCL_{kpi}= \mu_{kpi}-3 \sigma_{kpi}. \end{equation}
If the measured data are noisy, the proposed KPI may present a noisy nature as well. For this purpose, in our work we filtered the KPI using an exponentially weighted average filter over the last 12 hours of consecutive KPI values. \subsection{The contribution of individual variables to the SOM-based KPI}
When the SOM-based KPI deviates from its nominal pattern, it is desired to identify the individual variables that most contribute to the KPI variation. This allows the operator not only to identify a possible malfunctioning in the hydropower plant, but also the specific cause, or location, of such a malfunctioning. For this purpose, we first calculate an average contribution to DM of individual variables using the data in the nominal dataset $\Delta$. We then compare the contribution of individual variables of newly acquired patterns to the average contribution of nominal training patterns. For each pattern $\mathbf{r} \in \Delta$, we calculate the following average distance vector $\mathbf{d(r)}\in \mathbb{R}^{1 \times n}$ :
\begin{equation}
\mathbf{d(r)} =\frac{1 }{D} \sum_{i=1 }^{D} w_{ci} ( \mathbf{r}-\mathbf{m}_i ), \forall \mathbf{r} \in \Delta. \end{equation}
Then we calculate the vector of squared components of $\mathbf{d(r)}$ normalized to have norm 1, named $\mathbf{d_n(r)} \in \mathbb{R}^{1 \times n}$, as
\begin{equation}\label{vet_dist}
\mathbf{d_n(r)} = \frac{\mathbf{d(r)}\circ \mathbf{d(r)} }{\| \mathbf{d(r)}\|^2 }, \forall \mathbf{r} \in \Delta,
\end{equation}
where the symbol $\circ$ denotes the Hadamard (element-wise) product. Finally, we compute the average vector of the normalized squared distance components for all patterns in $\Delta$:
\begin{equation}
\mathbf{d_{n_{\Delta}}} =\frac{1 }{N} \sum_{r\in\Delta} \mathbf{d_n(r)} . \end{equation}
When a new pattern $\mathbf{r}$ is acquired during the monitoring phase, we calculate the following Hadamard ratio:
\begin{equation}
\mathbf{d_n(r)}\div \mathbf{d_{n_{\Delta}}} = [ cr_1 cr_2 \dots cr_n]
\end{equation}
where the contribution ratios $cr_i$ , $i=1 \dots n$ represent how individual variables of the new pattern influence the DM compared to their influence in non-faulty conditions. If the new pattern actually corresponds to a non-faulty condition, $cr_i$ takes values close to one. If the new pattern deviates from the nominal behaviour, some of the $cr_i$ exceed the unitary value. An empirical threshold 1.3 was selected, as a trade-off between false positives and true positives. \subsection{Hotelling multivariate control chart}
As a term of comparison for our SOM-based KPI indicator, we consider the Hotelling multivariate control chart \cite{Hotelling1947 }. While very few works may be found for our hydropower plant application of interest, Hotelling charts are quite popular for multivariable process control problems in general, and we take it as a benchmark procedure for comparison. The Hotelling control chart performs a projection of the multivariate data to a scalar parameter denoted as $t^2 $ statistics,
which is defined as the square of the Mahalanobis distance \cite{Maesschalck2000 } between the
observed pattern and the vector containing the mean values of
the variables in nominal conditions. The Hotelling $t^2 $ statistics is able to capture the changes in
multivariate data, revealing the deviations from the nominal
behaviour, and for these reasons the
Hotelling control chart is widely used for early detection of
incipient faults, see for instance\cite{Aparisi2009 }. The construction of the control chart includes two phases: in the
first phase, historical data are analyzed and the control limits
are computed; phase two corresponds to the monitoring of the newly acquired state patterns. \subsubsection{Phase one}
Let the nominal historic dataset be represented by the matrix ${{\Delta}} \in {\mathbb{R} ^{N \times n}}$, containing N observations of row vectors $ {{\bf{r}}}\in {\mathbb{R} ^{1 \times n}}$. The sample mean vector ${{\boldsymbol{\mu }}}\in {\mathbb{R} ^{1 \times n}}$ of the data is defined as:
\begin{equation}
\label{mean_vector_phase_one}
{{{\boldsymbol{\mu }}}} = \frac{1 }{N}\sum\limits_{\mathbf{r} \in \Delta} {{{\bf{r}}}} . \end{equation}
The covariance matrix is defined by means of the zero-mean data matrix ${{\Delta}_{0 }} \in {\mathbb{R} ^{N \times n}}$:
\begin{equation}
{{\Delta}_{0 }} = \Delta- \bf{1 } \cdot \boldsymbol{\mu},
\end{equation}
where ${\bf{1 }} \in {\mathbb{R} ^{N \times 1 }}$ represents a column vector with all entries equal to one. Then, the covariance matrix ${{\bf{C}}} \in {\mathbb{R} ^{n \times n}}$ of the data is defined as:
\begin{equation}
{{\bf{C}}} = \frac{1 }{{N - 1 }}{\Delta}_{0 }^T{{\Delta}_{0 }},
\end{equation}
where ${\left( {} \right)^T}$ denotes vector transpose operation. The multivariate statistics ${{\boldsymbol{\mu }}}$ and $\bf{C}$ represent the multivariate distribution of nominal observations, and we assume that $\bf{C}$ is full rank. The scalar ${t^2 }$ statistics is defined as a function of a single state pattern ${\bf{r}}\in \mathbb{R} ^{1 \times n}$:
\begin{equation}
\label{t_2 _def}
t^2 (\bf{r}) = \left( {{{\bf{r}}} - {{\boldsymbol{\mu }}}} \right){\bf{C}}^{ - 1 }{\left( {{{\bf{r}}} - {{\boldsymbol{\mu }}}} \right)^T}. \,
\end{equation}
The ${t^2 }$ statistics is small when pattern vector ${\bf{r}}$ represents nominal states, while it increases when the pattern vector ${\bf{r}}$ deviates from the nominal behaviour. In order to define the control limits $UCL$ and $LCL$ of the control chart, in this first phase we calculate the mean value $\mu_{t^2 }$ and standard deviation $\sigma_{t^2 }$ of the $t^2 $ values obtained with all the observations of the historical dataset ${\bf{r}}\in{\Delta}$. Then we define the safety thresholds as:
\begin{equation}
\label{Safety_Thresholds}
\left\{
\begin{array}{lll}
UCL_{t^2 } & = & \mu_{t^2 } + 3 \sigma_{t^2 }\\
& & \\
LCL_{t^2 } & = & max(\mu_{t^2 } - 3 \sigma_{t^2 }, 0 )\\
\end{array}\right.. \end{equation}
\subsubsection{Phase two}
In the second phase, new observation vectors are measured, and the corresponding $t^2 $ values are calculated as in Equation (\ref{t_2 _def}). The Hotelling control chart may be seen as a monitoring tool that plots the $t^2 $ values as consecutive points in time and compares them against the control limits. The process is considered to be ``out of control'', when the $t^2 $ values continuously exceed the control limits. \section{Results}
\label{Results}
After a prototyping phase, the condition monitoring system has been operating since April 2018 on several components of the plants described in Section \ref{case_study}, with a total of more than 600 input signals. As an example, some of the most critical components are shown in Table \ref{tab:20190208 _analyzed_components}. As can be seen from the last column of the table, there is usually a large redundancy of sensors measuring the same, or closely related, signals. The components listed in Table \ref{tab:20190208 _analyzed_components} are among the most relevant components for the plant operators, as it is known that their malfunctioning may, in some cases, lead to major failures or plant emergency stops. \begin{table}[ht. ]
\centering
\begin{tabular}{l|l|c}
\hline
\textbf{Component name} & \textbf{Measured Signals} & \textbf{Number of sensors} \\
\hline
Generation Units & Vibrations & 34 \\
\hline
HV Transformer & Temperatures & 27 \\
& Gasses levels & \\
\hline
Turbine & Pressures & 27 \\
& Flows & \\
& Temperatures & \\
\hline
Oleo-dynamic system & Pressures & 20 \\
& Temperatures & \\
\hline
Supports & Temperatures & 54 \\
\hline
Alternator & Temperatures & 43 \\
\hline
\end{tabular}
\vspace{0.05 cm}
\caption{List of main components analyzed for the hydro plants. }
\label{tab:20190208 _analyzed_components}
\end{table}
Since April 2018, our system detected more than $20 $ anomalous situations, the full list of whose occurrences is given in Table \ref{occurrences}, with different degrees of severity, defined with plant operators, ranging from ``no action needed'' to ``severe malfunction'', leading to plant emergency stop. It is worth noting that these events were not reported by the standard condition-monitoring systems operative in the plants, and in some cases would not have been well identified by the multivariate Hotelling control chart either, thus emphasizing the importance of the more sophisticated KPI introduced in this paper. In addition, we also see how the ability to identify non-nominal operations improves over time, as new information is acquired.
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7, .855 } A & 2 & 01 /25 /2018 & Efficiency Parameters & Low & Weather anomalies effects \\
\hline
\rowcolor[rgb]{ .886, .937, .855 } A & 1 & 07 /02 /2019 & Supports Temperature & Low & Under investigation \\
\hline
\rowcolor[rgb]{ .886, .937, .855 } A & 3 & 03 /03 /2019 & Francis Turbine & Low & Anomaly related to ongoing maintenance activities\\
\hline
\rowcolor[rgb]{ .988, .894, .839 } A & 1 & 06 /11 /2018 & Efficiency Parameters & Medium-Low & Not relevant anomaly \\
\hline
\rowcolor[rgb]{ .988, .894, .839 } A & 1 & 06 /29 /2018 & Francis Turbine & Medium-Low & Weather anomalies effects on turbine's components temperature \\
\hline
\rowcolor[rgb]{ .988, .894, .839 } A & 4 & 06 /30 /2018 & Efficiency Parameters & Medium-Low & Not relevant anomaly \\
\hline
\rowcolor[rgb]{ .988, .894, .839 } A & 2 & 08 /07 /2018 & Francis Turbine & Medium-Low & Weather anomalies effects on turbine's components temperature \\
\hline
\rowcolor[rgb]{ .988, .894, .839 } A & 2 & 09 /14 /2018 & Generator Vibrations & Medium-Low & Not relevant anomaly \\
\hline
\rowcolor[rgb]{ .988, .894, .839 } A & 3 & 01 /10 /2019 & Generator Temperature & Medium-Low & Under investigation \\
\hline
\rowcolor[rgb]{ .988, .894, .839 } A & 3 & 01 /16 /2019 & Transformer Temperature & Medium-Low & Weather anomalies effects on transformer temperature \\
\hline
\rowcolor[rgb]{ .988, .894, .839 } A & 2 & 03 /03 /2019 & Generator Temperature & Medium-Low & Anomaly related to ongoing maintenance activities\\
\hline
\rowcolor[rgb]{ .988, .894, .839 } A & 3 & 06 /17 /2019 & Generator Temperature & Medium-Low & Under investigation \\
\hline
\rowcolor[rgb]{ .988, .894, .839 } A & 4 & 06 /21 /2019 & Generator Temperature & Medium-Low & Weather anomalies effects on generator temperature \\
\hline
\rowcolor[rgb]{ .886, .937, .855 } A & 1 & 04 /22 /2018 & HV transformer gasses & Medium-High & Monitoring system anomaly \\
\hline
\rowcolor[rgb]{ .886, .937, .855 } B & 2 & 06 /27 /2018 & Generator Vibrations & Medium-High & Data Quality issue \\
\hline
\rowcolor[rgb]{ .886, .937, .855 } A & 1 & 10 /29 /2018 & Generator Vibrations & Medium-High & Data Quality issue \\
\hline
\rowcolor[rgb]{ .886, .937, .855 } A & 3 & 11 /20 /2018 & Generator Vibrations & Medium-High & Under investigation \\
\hline
\rowcolor[rgb]{ .886, .937, .855 } A & 1 & 03 /24 /2019 & Francis Turbine & Medium-High & Under investigation \\
\hline
\rowcolor[rgb]{ .886, .937, .855 } A & 2 & 04 /07 /2019 & Oleo-Dynamic System & Medium-High & Under investigation \\
\hline
\rowcolor[rgb]{ .988, .894, .839 } B & 2 & 10 /01 /2018 & Generator Temperature & High & Sensor anomaly on block channel signal\\
\hline
\rowcolor[rgb]{ .988, .894, .839 } B & 2 & 11 /01 /2018 & Generator Temperature & High & Sensor anomaly on block channel signal\\
\hline
\rowcolor[rgb]{ .988, .894, .839 } A & 2 & 03 /01 /2019 & HV transformer gasses & High & Under investigation \\
\hline
\end{tabular}%
\caption{List of anomalous behaviours that have been noticed during 16 months of test on the two hydropower plants. Lines 14 and 20 correspond to the two faults that have been illustrated in this paper. }
\label{occurrences}%
\end{table*}%
We now describe more in detail two different anomalies belonging to the two different plants, as summarized in Table \ref{tab:20190131 _SOM_results}. \begin{table}[ht. ]
\centering
\begin{tabular}{c|c|c|l}
\hline
\textbf{Case Study} &\textbf {Plant} & \textbf{Warning name} &\textbf{ Warning date} \\
\hline
1 & B & Generator Temperature & 10 /01 /2018 \\
\hline
2 & A & HV transformer gasses & 04 /22 /2018 \\
\hline
\end{tabular}%
\caption{Two failures reported by the predictive system and discussed in detail. }
\label{tab:20190131 _SOM_results}%
\end{table}%
\subsection{Case Study 1 : plant B - generator temperature signals}
In October 2018, an anomalous behavior was reported by our system, which indicated an anomaly regarding a sensor measuring the iron temperature of the alternator. It was then observed that the temperature values of such a sensor were higher than usual, as shown in Figure \ref{fig:Temperatura_1 _Ferro_Alternatore_2 _warning_id1 _zoom. }, and also higher than the measurements taken by other similar sensors. However the temperature values did not yet exceed the warning threshold of the condition-monitoring systems already operative in the plant. From an inspection of the sensor measurements it was possible to establish that the exact day when this anomaly started occurring, the proposed SOM based KPI sharply notified a warning, as shown in Figure \ref{fig:KPI_temp_IDRT_subplot_SOM_T2 }; for this case, the time-extension of the training dataset was nine months, from 1 January 2018 to 1 September 2018. After receiving the warning alert, the plant operators checked the sensor and confirmed the event as a relevant anomaly. In particular, they acknowledged that this was a serious problem, since a further degradation of the measurement could have eventually led to the stop of the generation unit. For this reason, timely actions were taken: operators restored the nominal and correct behavior of the sensor starting on the 12 $^{th}$ of October 2018. The saved costs related to the prevented stopping of the generation unit were estimated in the range between 25 k\euro\ and 100 k\euro. While also the Hotelling control chart noticed an anomalous behaviour of the sensor in the same time frame, still it would have given rise to several false positives in the past, as shown in Figure \ref{fig:KPI_temp_IDRT_subplot_SOM_T2 }. \begin{figure}[. ht]
\centering
\includegraphics[trim=3 cm 0 cm 2 cm 0 cm, width=1 \linewidth]
{. /fig/4 _Results/Presenzano_Generator_v2. eps}
\caption{Measurement of the temperature sensor in the alternator of plant B. Anomalous values were detected by our system (start warning), timely actions were taken and the correct functioning was restored (end warning) }
\label{fig:Temperatura_1 _Ferro_Alternatore_2 _warning_id1 _zoom. }
\end{figure}
\begin{figure*}[. ht]
\centering
\includegraphics[trim=3 cm 0 cm 2 cm 0 cm, width=\linewidth]{. /fig/4 _Results/Presenzano_Generator_con_T2 _Mauro. eps}
\caption{Plant A: KPI as a function of time. SOM-based results (top) are compared with $t^2 $-based results (bottom). }
\label{fig:KPI_temp_IDRT_subplot_SOM_T2 }
\end{figure*}
\subsection{Case study 2 : Plant A - HV Transformer anomaly}
The SOM-based KPI detected an anomaly on the HV transformer of one of the Generation Units at the beginning of June 2018 as shown in fig. \ref{fig:KPI_hydran_IDCA_subplot_SOM_T2 }. After inspection of the operators, they informed us that similar anomalous situations had occurred in the past as well, but had not been tagged as faulty behaviours. As soon as the time occurrences of the similar past anomalies had been notified by a plant operator, we proceeded to remove the corresponding signals from the training set. Then we recomputed our KPI based on the revised corrected historical dataset, and the KPI retrospectively found out that the ongoing faulty pattern had actually started one month earlier. This updated information was then validated by analyzing the output of an already installed gas monitoring system, that continuously monitors a composite value of various fault gases in ppM (Parts per Million) and tracks the oil-moisture. The gas monitoring system had been measuring increasing values, with respect to the historical ones, since the 22 $^{nd}$ of April 2018, but no warning had been generated by that system. In this case, this was not however a critical fault, as the level of gasses in the transformer oil was not exceeding the maximum feasible limit. However, the warning triggered by our system was used to schedule maintenance actions that restored the nominal operating conditions. In addition, the feedback from the plant operators was very useful in order to tune the SOM-based monitoring system, and to increase its early detection capabilities. In this case, the Hotelling control chart realized of the anomalous behaviour only 20 days after our SOM-based KPI. \begin{figure*}[. ht]
\centering
\includegraphics[trim=4 cm 0 cm 2 cm 0 cm, width=\linewidth]
{. /fig/4 _Results/Soverzene_Transormer_con_T2. eps}
\caption{Plant B: KPI as a function of time. SOM-based results (top) are compared with $t^2 $-based results (bottom). }
\label{fig:KPI_hydran_IDCA_subplot_SOM_T2 }
\end{figure*}
\section{Conclusion}
\label{Conclusion}
Driven by rapidly evolving enabling technologies, most notably Internet-of-Thing sensors and communication tools, together with more powerful artificial intelligent algorithms, condition monitoring, early diagnostics and predictive maintenance methodologies and tools are becoming some of the most interesting areas of research in the power community. While some preliminary examples can be found in many fields, solar and wind plants being one of them, fewer applications can be found in the field of hydropower plants. In this context, this paper is one of the first examples, at least up to the knowledge of the authors, that results of a newly proposed KPI are validated over a 1 -year running test field in two hydropower plants. \\
\newline
This paper provided very promising preliminary results, that encourage further research on this topic. In particular the proposed procedure can not be implemented in a fully unsupervised fashion yet, but some iterations with plant operators still take place when alarm signals are alerted. Also, the proposed condition monitoring strategy is a first step towards fully automatic predictive maintenance schemes, where faults are not just observed but are actually predicted ahead of time, possibly when they are only at an incipient stage. In the opinion of the authors this is a very promising area of research and there is a general interest towards the development of such predictive strategies. \section{Proof of the First Zonklar Equation}
\ifCLASSOPTIONcaptionsoff
\newpage
\fi
\input{. /section/6 _Bibliography}
\begin{IEEEbiography}[{\includegraphics[width=1 in, height=1.25 in, clip, keepaspectratio]{. /fig/6 _Bibliography/alessandro_betti. jpg}}]{Alessandro Betti}
Dr. Alessandro Betti received the M. S. degree in Physics and the Ph. D. degree in Electronics Engineering from the University of Pisa, Italy, in 2007 and 2011, respectively. His main field of research was modeling of noise and transport in quasi-one dimensional devices. His work has been published in 10 papers in peer-reviewed journals in the field of solid state electronics and condensed matter physics and in 16 conference papers, presenting for 3 straight years his research at the top International Conference IEEE in electron devices, the International Electron Device Meeting in USA. In September 2015 he joined the company i-EM in Livorno, where he currently works as a Senior Data Scientist developing power generation forecasting, predictive maintenance and Deep Learning models, as well as solutions in the electrical mobility fields and managing a Data Science Team. \end{IEEEbiography}
\begin{IEEEbiography}[{\includegraphics[width=1 in, height=1.25 in, clip, keepaspectratio]{. /fig/6 _Bibliography/Emanuele_Crisostomi}}]{Emanuele Crisostomi}(M'12 -SM'16 ) received the B. Sc. degree in computer science engineering, the M. Sc. degree in automatic control, and the Ph. D. degree in automatics, robotics, and bioengineering from the University of Pisa, Pisa, Italy, in 2002, 2005, and 2009, respectively. He is currently an Associate Professor of electrical engineering in the Department of Energy, Systems, Territory and Constructions Engineering, University of Pisa. He has authored tens of publications in top refereed journals and he is a co-author of the recently published book on ``Electric and Plug-in Vehicle Networks: Optimization and Control'' (CRC Press, Taylor and Francis Group, 2017 ). His research interests include control and optimization of large scale systems, with applications to smart grids and green mobility networks. \end{IEEEbiography}
\begin{IEEEbiography}[{\includegraphics[width=1 in, height=1.25 in, clip, keepaspectratio]{. /fig/6 _Bibliography/gianluca_paolinelli. jpg}}]{Gianluca Paolinelli}
Gianluca Paolinelli received the bachelor and master degree in electrical engineering from the University of Pisa, Pisa, Italy, in 2014 and 2018 respectively.
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and statistical data analysis, with main applications on modeling and monitoring the behaviour of renewable power plants. Currently, he is working on big data analysis applied on hydro power plants signals.
\end{IEEEbiography}
\begin{IEEEbiography}[{\includegraphics[width=1 in, height=1.25 in, clip, keepaspectratio]{. /fig/6 _Bibliography/FabrizioRuffini_Linkedin}}]{Fabrizio Ruffini}
Fabrizio Ruffini received the Ph. D. degree in Experimental Physics from University of Siena, Siena, Italy, in 2013. His research-activity is centered on data analysis, with particular interest in multidimensional statistical analysis. During his research activities, he was at the Fermi National Accelerator Laboratory (Fermilab), Chicago, USA, and at the European Organization for Nuclear Research (CERN), Geneva, Switzerland. Since 2013, he has been working at i-EM as data scientist focusing on applications in the renewable energy sector, atmospheric physics, and smart grids. Currently, he is senior data scientist with focus on international funding opportunities and dissemination activities.
\end{IEEEbiography}
\begin{IEEEbiography}[{\includegraphics[width=1 in, height=1.25 in, clip, keepaspectratio]{. /fig/6 _Bibliography/Mauro_Tucci}}]{Mauro Tucci} received the Ph. D. degree in applied electromagnetism from the University of Pisa, Pisa, Italy, in 2008. Currently, he is an Associate Professor with the Department of Energy, Systems, Territory and Constructions Engineering, University of Pisa. His research interests include computational intelligence
and big data analysis, with applications in electromagnetism, non destructive testing, powerline communications, and smart grids.
\end{IEEEbiography}
\end{document}
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\section{ Introduction}
Nuclear collision experiments, performed at ion accelerators, are a very
powerful tool to study nuclear properties at low and intermediate
energies. In order to interpret accumulated experimental data appropriate
theoretical methods are necessary enabling the simultaneous description of the
available elastic, rearrangement and breakup reactions. Regardless of its importance, the theoretical description of quantum-mechanical
collisions turns out to be one of the most complex and slowly advancing problems
in theoretical physics. If during the last decade accurate solutions for the
nuclear bound state problem became
available, full solution of the scattering problem (containing elastic,
rearrangement and breakup channels) remains limited to the three-body case. The main difficulty is related to the fact that, unlike the
bound state wave functions, scattering wave functions are not localized. In
configuration space one is obliged to solve multidimensional
differential equations with extremely complex boundary conditions; by
formulating the quantum-mechanical scattering problem in momentum space
one has to deal with non-trivial singularities in the
kernel of multivariable integral equations. A rigorous mathematical formulation of the quantum mechanical three-body
problem in the framework of non relativistic dynamics has been introduced
by Faddeev in the early sixties~\cite{Fad_60 },
in the context of the three-nucleon system with short range
interactions. In momentum space these equations might be slightly
modified by formulating
them in terms of three-particle transition operators that
are smoother functions compared to the system wave functions. Such a modification was proposed by Alt, Grassberger, and
Sandhas~\cite{alt:67 a} (AGS). Solutions of the AGS equations with short range interactions were readily obtained in the early seventies. As large computers became available progress followed leading, by the end eighties, to fully converged solutions of these equations for neutron-deuteron ($n$-$d$) elastic scattering and breakup using realistic short range nucleon-nucleon ($N$-$N$) interactions. Nevertheless the inclusion of the long range Coulomb force in momentum space calculations of proton-deuteron ($p$-$d$) elastic scattering and breakup with the same numerical reliability as calculations with short range interactions alone, only become possible in the last decade. Significant progress has been achieved~\cite{deltuva:05 a, deltuva:05 d}
by developing the screening and renormalization
procedure for the Coulomb interaction in momentum space using a smooth
but at the same time sufficiently rapid screening. This technique permitted
to extend the calculations to the systems of three-particles with arbitrary
masses above the breakup threshold~\cite{deltuva:06 b, deltuva:07 d}. However it has taken some time to formulate the
appropriate boundary conditions in configuration space for
the three-body problem~\cite{Merkuriev_71, Merkuriev_74, MGL_76 }
and even longer to reformulate the original Faddeev equations to allow the
incorporation of long-range Coulomb like interactions~\cite{Merkuriev_80, Merkuriev_81 }. Rigorous solution of the three-body problem with short range interactions has
been achieved just after these theoretical developments, both below and above
breakup threshold. On the other hand the numerical solution for the three-body
problem including charged particles above
the three-particle breakup threshold has been achieved only recently. First it has been done by using approximate Merkuriev
boundary conditions in configuration space~\cite{kievsky:97 }. Nevertheless this approach proved to be a rather complex task numerically,
remaining unexplored beyond the $p$-$d$ scattering case, but not yet for the $p$-$d$ breakup. Finally, very recently configuration space method based on complex scaling have been
developed and applied for $p$-$d$ scattering~\cite{lazauskas:11 a}. This method allows to treat
the scattering problem using very simple boundary conditions, equivalent
to the ones employed to solve the bound-state problem. \bigskip
The aim of this lecture is to present these two recently developed
techniques, namely the momentum-space method based on screening and renormalization
as well as the configuration-space complex scaling method. This lecture is structured as follows: the first part serves to
introduce theoretical formalisms for momentum space and configuration
space calculations;
in the second part we present some selected calculations with an
aim to test the performance and validity of the two presented methods. \section{Momentum-space description of three-particle scattering}
\label{sec:p}
We describe the scattering process in a system of three-particles
interacting via pairwise short-range potentials
$v_\alpha$, $\alpha=1,2,3 $; we use the odd-man-out notation,
that is, $v_1 $ is the potential between particles 2 and 3. In the framework of nonrelativistic quantum mechanics the center-of-mass
(c. m. ) and the internal motion can be separated by introducing
Jacobi momenta
\begin{eqnarray}\label{eq:Jacobi}
\vec{p}_\alpha & = &\frac{m_{\gamma} \vec{k}_\beta - m_{\beta} \vec{k}_\gamma }
{m_{\beta} + m_{\gamma} }, \\
\vec{q}_\alpha & = &
\frac{m_{\alpha} (\vec{k}_\beta + \vec{k}_\gamma) -
(m_{\beta} + m_{\gamma}) \vec{k}_\alpha }
{m_{\alpha} + m_{\beta} + m_{\gamma} },
\end{eqnarray}
with ($\alpha \beta \gamma $) being cyclic permutations of (123 );
$\vec{k}_\alpha$ and $m_{\alpha}$ are the individual particle
momenta and masses, respectively. The c. m. motion is free and in the
following we consider only the internal motion; the corresponding
kinetic energy operator is $H_0 $ while the full Hamiltonian is
\begin{equation} \label{eq:H}
H = H_0 + \sum_{\alpha=1 }^3 v_\alpha . \end{equation}
\subsection{Alt, Grassberger, and Sandhas equations}
We consider the particle $\alpha$ scattering from the pair $\alpha$
that is bound with energy $ \epsilon_\alpha$. The initial channel state $|b_{\alpha}\vec{q}_\alpha\rangle$
is the product of the bound state wave function $|b_\alpha \rangle$
for the pair $\alpha$ and a plane wave with the
relative particle-pair $\alpha$ momentum $\mathbf{q}_\alpha$;
the dependence on the discrete quantum numbers is suppressed
in our notation. $|b_{\alpha}\vec{q}_\alpha\rangle$
is the eigenstate of the corresponding channel Hamiltonian
$H_\alpha = H_0 + v_\alpha$
with the energy eigenvalue $E= \epsilon_\alpha + q^2 _\alpha/2 M_\alpha$
where
$M_\alpha$ is the particle-pair $\alpha$ reduced mass. The final channel state is the particle-pair state in the same or
different configuration $|b_{\beta}\vec{q}_\beta\rangle$
in the case of elastic and rearrangement scattering
or, in the case of breakup,
it is the state of three free particles
$|\vec{p}_{\gamma}\vec{q}_\gamma\rangle$ with the same
energy $E= p_\gamma^2 /2 \mu_\gamma + q_\gamma^2 /2 M_\gamma $ and pair $\gamma$
reduced mass $\mu_\gamma$;
any set of Jacobi momenta can be used equally well for the breakup state. The stationary scattering states~\cite{schmid:74 a, gloeckle:83 a}
corresponding to the above channel states are eigenstates of the full
Hamiltonian; they are obtained from the channel states using
the full resolvent $G = (E+i0 -H)^{-1 }$, i. e.,
\begin{eqnarray} \label{eq:psi_a}
|b_\alpha \vec{q}_\alpha \rangle^{(+)} & = &
i0 G |b_\alpha \vec{q}_\alpha \rangle, \\
\label{eq:psi_0 }
|\vec{p}_\alpha\vec{q}_\alpha \rangle^{(+)} & = &
i0 G |\vec{p}_\alpha\vec{q}_\alpha \rangle. \end{eqnarray}
The full resolvent $G$ may be decomposed into the channel resolvents
$G_\beta = (E+i0 -H_\beta)^{-1 }$ and/or free resolvent $G_0 = (E+i0 -H_0 )^{-1 }$
as
\begin{equation}
G = G_\beta + G_\beta \bar{v}_\beta G ,
\end{equation}
with $\beta=0,1,2,3 $ and
$ \bar{v}_\beta = \sum_{\gamma=1 }^3 \bar{\delta}_{\beta \gamma} v_\gamma$
where $\bar{\delta}_{\beta \gamma} = 1 -{\delta}_{\beta \gamma}$. Furthermore, the channel resolvents
\begin{equation}
G_\beta = G_0 + G_0 T_\beta G_0 ,
\end{equation}
can be related to the corresponding two-particle transition operators
\begin{equation}
T_\beta = v_\beta + v_\beta G_0 T_\beta ,
\end{equation}
embedded into three-particle Hilbert space. Using these definitions
Eqs. ~(\ref{eq:psi_a}) and (\ref{eq:psi_0 })
can be written as triads of Lippmann-Schwinger equations
\begin{eqnarray} \label{eq:psi_LS}
|b_\alpha \vec{q}_\alpha \rangle^{(+)} & = {} &
\delta_{\beta \alpha} |b_\alpha \vec{q}_\alpha \rangle
+ G_\beta \bar{v}_\beta |b_\alpha \vec{q}_\alpha \rangle^{(+)} , \\
|\vec{p}_\alpha\vec{q}_\alpha \rangle^{(+)} & = {} &
(1 + G_0 T_\beta ) |\vec{p}_\alpha\vec{q}_\alpha \rangle
+ G_\beta \bar{v}_\beta |\vec{p}_\alpha\vec{q}_\alpha \rangle^{(+)} ,
\end{eqnarray}
with $\alpha$ being fixed and $\beta =1,2,3 $; they are necessary and sufficient
to define the states $|b_\alpha \vec{q}_\alpha \rangle^{(+)}$ and
$|\vec{p}_\alpha\vec{q}_\alpha \rangle^{(+)}$ uniquely. However, in scattering problems it may be more convenient to work with
the multichannel transition operators $U_{\beta \alpha}$ defined such that
their on-shell elements yield scattering amplitudes, i. e.,
\begin{equation} \label{eq:U-V}
U_{\beta \alpha} |b_\alpha \vec{q}_\alpha \rangle =
\bar{v}_\beta |b_\alpha \vec{q}_\alpha \rangle^{(+)}. \end{equation}
Our calculations are based on the AGS version~\cite{alt:67 a} of
three-particle scattering theory. In accordance with Eq. ~(\ref{eq:U-V})
it defines the multichannel transition operators $U_{\beta \alpha}$
by the decomposition of the full resolvent $ G$ into channel
and/or free resolvents as
\begin{equation} \label{eq:G-U}
G = \delta_{\beta \alpha} G_\alpha + G_\beta U_{\beta \alpha} G_\alpha . \end{equation}
The multichannel transition operators $U_{\beta \alpha}$
with fixed $\alpha$ and $\beta = 1,2,3 $ are solutions
of three coupled integral equations
\begin{equation} \label{eq:AGSnsym_a}
U_{\beta \alpha} = \bar{\delta}_{\beta \alpha} G_0 ^{-1 } +
\sum_{\gamma=1 }^3 \bar{\delta}_{\beta \gamma} T_{\gamma} G_0 U_{\gamma \alpha}. \end{equation}
The transition matrix $U_{0 \alpha} $ to final states with three free
particles can be obtained from the solutions of Eq. ~(\ref{eq:AGSnsym_a})
by quadrature, i. e.,
\begin{equation} \label{eq:AGSnsym_b}
U_{0 \alpha} = G_0 ^{-1 } + \sum_{\gamma=1 }^3 T_{\gamma} G_0 U_{\gamma \alpha}. \end{equation}
The on-shell matrix elements
$\langle b_{\beta} \vec{q}'_\beta |U_{\beta \alpha} |b_\alpha \vec{q}_\alpha \rangle$
are amplitudes (up to a factor) for elastic
($\beta = \alpha$) and rearrangement ($\beta \neq \alpha$) scattering. For example, the differential cross section for the
$\alpha + (\beta\gamma) \to \beta + (\gamma\alpha)$ reaction in the c. m. system is given by
\begin{equation} \label{eq:dcsab}
\frac{d \sigma_{\alpha \to \beta}}{d \Omega_\beta} =
(2 \pi)^4 M_\alpha M_\beta \frac{q'_\beta}{q_\alpha}
| \langle b_{\beta} \vec{q}'_\beta |U_{\beta \alpha}
|b_\alpha \vec{q}_\alpha \rangle|^2. \end{equation}
The cross section for the breakup is determined by the on-shell matrix elements
$\langle \vec{p}'_{\gamma} \vec{q}'_\gamma |U_{0 \alpha}
|b_\alpha \vec{q}_\alpha \rangle$. Thus, in the AGS framework all elastic, rearrangement, and breakup reactions
are calculated on the same footing. Finally we note that the AGS equations can be extended to include
also the three-body forces as done in Ref. ~\cite{deltuva:09 e}. \subsection{Inclusion of the Coulomb interaction}
The Coulomb potential $w_C$, due to its long range, does not satisfy the
mathematical properties required for the formulation of standard scattering theory as given in the previous subsection
for short-range interactions $v_\alpha$. However,
in nature the Coulomb potential is always screened at large distances. The comparison of the data from typical nuclear physics experiments
and theoretical predictions with full Coulomb is meaningful
only if the full and screened Coulomb become physically indistinguishable. This was proved in Refs. ~\cite{taylor:74 a, semon:75 a}
where the screening and renormalization method for the scattering of two
charged particles was proposed. We base our treatment of the Coulomb interaction on that idea. Although we use momentum-space framework, we first choose the
screened Coulomb potential in configuration-space representation as
\begin{equation} \label{eq:wr}
w_R(r) = w_C(r)\; e^{-(r/R)^n} ,
\end{equation}
and then transform it to momentum-space. Here $R$ is the screening radius
and $n$ controls the smoothness of the screening. The standard scattering theory is formally applicable to the screened
Coulomb potential $w_R$, i. e., the Lippmann-Schwinger equation yields
the two-particle transition matrix
\begin{equation} \label{eq:tr}
t_R = w_R + w_R g_0 t_R ,
\end{equation}
where $g_0 $ is the two-particle free resolvent. It was proven in Ref. ~\cite{taylor:74 a} that
in the limit of infinite screening radius $R$ the on-shell screened Coulomb
transition matrix (screened Coulomb scattering amplitude)
$\langle \mathbf{p}'| t_R | \mathbf{p} \rangle$ with $p'=p$,
renormalized by an infinitely oscillating phase factor
$z_R^{-1 }(p) = e^{2 i\phi_R(p)}$, approaches the full Coulomb amplitude
$\langle \mathbf{p}'| t_C | \mathbf{p} \rangle$
in general as a distribution. The convergence in the sense of distributions
is sufficient for the description of physical observables in a real
experiment where the incoming beam is not a plane wave but wave packet
and therefore the cross section
is determined not directly by the scattering amplitude but
by the outgoing wave packet, i. e., by the scattering amplitude averaged
over the initial state physical wave packet.
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orshkov:61 }, i. e.,
\begin{equation} \label{eq:gorshkov}
\lim_{R \to \infty} (1 + g_0 t_R) |\mathbf{p} \rangle z_R^{-\frac12 }(p)
= |\psi_C^{(+)}(\mathbf{p}) \rangle. \end{equation}
The screening and renormalization method based on the above relations
can be extended to more complicated systems, albeit with some limitations. We consider the system of three-particles with charges $z_\alpha$
of equal sign interacting via pairwise strong short-range and screened
Coulomb potentials $v_\alpha + w_{\alpha R}$ with $\alpha$ being 1, 2, or 3. The corresponding two-particle transition matrices are calculated with
the full channel interaction
\begin{equation} \label{eq:TR}
T^{(R)}_\alpha = (v_\alpha + w_{\alpha R}) +
(v_\alpha + w_{\alpha R}) G_0 T^{(R)}_\alpha,
\end{equation}
and the multichannel transition operators $U^{(R)}_{\beta \alpha}$
for elastic and rearrangement scattering are solutions
of the AGS equation
\begin{equation}
U^{(R)}_{\beta \alpha} = \bar{\delta}_{\beta \alpha} G_0 ^{-1 }
+ \sum_{\gamma=1 }^3 \bar{\delta}_{\beta \gamma}
T^{(R)}_\gamma G_0 U^{(R)}_{\gamma \alpha} ;
\label{eq:Uba}
\end{equation}
all operators depend parametrically on the Coulomb screening radius $R$. In order to isolate the screened Coulomb contributions to the transition
amplitude that diverge in the infinite $R$ limit
we introduce an auxiliary screened Coulomb potential $W^{\mathrm{c\. \:\!. m\. \:\!. }}_{\alpha R}$
between the particle $\alpha$
and the center of mass (c. m. ) of the remaining pair. The same screening function has to be used for both Coulomb potentials
$w_{\alpha R}$ and $W^{\mathrm{c\. \:\!. m\. \:\!. }}_{\alpha R}$. The corresponding transition matrix
\begin{equation} \label{eq:Tcm}
T^{\mathrm{c\. \:\!. m\. \:\!. }}_{\alpha R} = W^{\mathrm{c\. \:\!. m\. \:\!. }}_{\alpha R} +
W^{\mathrm{c\. \:\!. m\. \:\!. }}_{\alpha R} G^{(R)}_{\alpha} T^{\mathrm{c\. \:\!. m\. \:\!. }}_{\alpha R} ,
\end{equation}
with $ G^{(R)}_{\alpha} = (E+i0 -H_0 -v_\alpha - w_{\alpha R})^{-1 }$
is a two-body-like operator and therefore its on-shell and half-shell
behavior in the limit $R \to \infty$
is given by Eqs. ~(\ref{eq:taylor2 }) and (\ref{eq:gorshkov}). As derived in Ref. ~\cite{deltuva:05 a}, the
three-particle transition operators may be decomposed as
\begin{eqnarray}
U^{(R)}_{\beta \alpha} &=& \delta_{\beta\alpha} T^{\mathrm{c\. \:\!. m\. \:\!. }}_{\alpha R}
+ [1 + T^{\mathrm{c\. \:\!. m\. \:\!. }}_{\beta R} G^{(R)}_{\beta}]
\tilde{U}^{(R)}_{\beta\alpha}
[1 + G^{(R)}_{\alpha} T^{\mathrm{c\. \:\!. m\. \:\!. }}_{\alpha R}] \quad
\label{eq:U-T} \\
&=& \delta_{\beta\alpha} T^{\mathrm{c\. \:\!. m\. \:\!. }}_{\alpha R}
+ (U^{(R)}_{\beta \alpha} - \delta_{\beta\alpha} T^{\mathrm{c\. \:\!. m\. \:\!. }}_{\alpha R}). \label{eq:U-T2 }
\end{eqnarray}
where the auxiliary operator $\tilde{U}^{(R)}_{\beta\alpha}$ is of
short range when calculated between on-shell screened Coulomb states. Thus, the three-particle transition operator $U^{(R)}_{\beta \alpha}$
has a long-range part $\delta_{\beta\alpha} T^{\mathrm{c\. \:\!. m\. \:\!. }}_{\alpha R}$ whereas the
remainder $U^{(R)}_{\beta \alpha} - \delta_{\beta\alpha} T^{\mathrm{c\. \:\!. m\. \:\!. }}_{\alpha R}$
is a short-range operator that is externally distorted
due to the screened Coulomb waves generated by
$[1 + G^{(R)}_{\alpha} T^{\mathrm{c\. \:\!. m\. \:\!. }}_{\alpha R}]$. On-shell, both parts do not have a proper limit as $R \to \infty$ but
the limit exists after renormalization by an appropriate phase factor,
yielding the transition amplitude for full Coulomb
\begin{eqnarray} \nonumber
&& \langle b_\beta \mathbf{q}'_\beta | U^{(C)}_{\beta \alpha}
|b_\alpha \mathbf{q}_\alpha\rangle =
\delta_{\beta \alpha}
\langle b_\alpha \mathbf{q}'_\beta |T^{\mathrm{c\. \:\!. m\. \:\!. }}_{\alpha C}
|b_\alpha \mathbf{q}_\alpha \rangle \\ & &
+ \lim_{R \to \infty} [ Z^{-\frac{1 }{2 }}_{\beta R}(q'_\beta)
\langle b_\beta \mathbf{q}'_\beta |
( U^{(R)}_{\beta \alpha}
- \delta_{\beta\alpha} T^{\mathrm{c\. \:\!. m\. \:\!. }}_{\alpha R})
|b_\alpha \mathbf{q}_\alpha \rangle
Z^{-\frac{1 }{2 }}_{\alpha R}(q_\alpha) ]. \quad
\label{eq:UC2 }
\end{eqnarray}
The first term on the right-hand side of Eq. ~(\ref{eq:UC2 }) is known
analytically~\cite{taylor:74 a}; it corresponds to the particle-pair $\alpha$
full Coulomb transition amplitude
that results from the implicit renormalization of $T^{\mathrm{c\. \:\!. m\. \:\!. }}_{\alpha R}$
according to Eq. ~(\ref{eq:taylor2 }). The $R \to \infty$ limit for the remaining part
$( U^{(R)}_{\beta \alpha} - \delta_{\beta\alpha} T^{\mathrm{c\. \:\!. m\. \:\!. }}_{\alpha R})$
of the multichannel transition matrix is performed numerically;
due to the short-range nature of this term
the convergence with the increasing screening radius $R$
is fast and the limit is reached with sufficient accuracy at
finite $R$; furthermore, it can be calculated using the partial-wave expansion. We emphasize that Eq. ~(\ref{eq:UC2 }) is by no means an approximation
since it is based on the obviously exact identity (\ref{eq:U-T2 })
where the $R \to \infty$ limit for each term exists and
is calculated separately. The renormalization factor for $R \to \infty $ is a diverging phase factor
\begin{equation}
Z_{\alpha R}(q_\alpha) = e^{-2 i \Phi_{\alpha R}(q_\alpha)},
\end{equation}
where $\Phi_{\alpha R}(q_\alpha)$, though independent of the particle-pair
relative angular momentum $l_\alpha$ in the infinite $R$ limit,
may be realized by
\begin{equation} \label{eq:phiRl}
\Phi_{\alpha R}(q_\alpha) = \sigma_{l_\alpha}^{\alpha}(q_\alpha) -
\eta_{l_\alpha R}^{\alpha}(q_\alpha),
\end{equation}
with the diverging screened Coulomb phase shift
$\eta_{l_\alpha R}^{\alpha}(q_\alpha)$
corresponding to standard boundary conditions
and the proper Coulomb one $\sigma_{l_\alpha}^{\alpha}(q_\alpha)$ referring to the
logarithmically distorted proper Coulomb boundary conditions. For the screened Coulomb potential of Eq. ~(\ref{eq:wr})
the infinite $R$ limit of $\Phi_{\alpha R}(q_\alpha)$ is known analytically,
\begin{equation} \label{eq:phiRlln}
\Phi_{\alpha R}(q_\alpha)=\mathcal{K}_{\alpha}(q_\alpha)[\ln{(2 q_\alpha R)} - C/n],
\end{equation}
where $C \approx 0.5772156649 $ is the Euler number and
$\mathcal{K}_{\alpha}(q_\alpha) = \alpha_{e. m. }z_\alpha \sum_\gamma
\bar{\delta}_{\gamma\alpha} z_\gamma M_\alpha/q_\alpha$
is the Coulomb parameter with $\alpha_{e. m. } \approx 1 /137 $. The form of the renormalization phase $\Phi_{\alpha R}(q_\alpha)$ to be used
in the actual calculations with finite screening radii $R$ is not unique,
but the converged results show independence of
the chosen form of $\Phi_{\alpha R}(q_\alpha)$. For breakup reactions we follow a similar strategy. However,
the proper three-body Coulomb wave function and its relation to the
three-body screened Coulomb wave function is, in general, unknown. This prevents the application of the screening and renormalization method to the
reactions involving three free charged particles (nucleons or nuclei)
in the final state. However, in the system of two charged particles and a neutral one
with $z_\rho = 0 $, the final-state Coulomb distortion becomes again
a two-body problem with the screened Coulomb transition matrix
\begin{equation}
T_{\rho R} = w_{\rho R} + w_{\rho R} G_0 T_{\rho R}. \end{equation}
This makes the channel $\rho$, corresponding to the correlated pair
of charged particles, the most convenient choice for the
description of the final breakup state. As shown in Ref. ~\cite{deltuva:05 d}, the AGS
breakup operator
\begin{equation}\label{eq:U0 a}
U^{(R)}_{0 \alpha} = {}
G_0 ^{-1 } + \sum_{\gamma=1 }^3 T^{(R)}_{\gamma} G_0 U^{(R)}_{\gamma \alpha} ,
\end{equation}
can be decomposed as
\begin{equation}\label{eq:U0 t}
U^{(R)}_{0 \alpha} = {} (1 + T_{\rho R} G_{0 })
\tilde{U}^{(R)}_{0 \alpha} (1 + G^{(R)}_{\alpha} T^{\mathrm{c\. \:\!. m\. \:\!. }}_{\alpha R}),
\end{equation}
where the reduced operator
$\tilde{U}^{(R)}_{0 \alpha}(Z)$ calculated between screened Coulomb
distorted initial and final states is of finite range. In the full breakup operator $U^{(R)}_{0 \alpha}(Z)$
the external distortions show up in screened
Coulomb waves generated by $(1 + G^{(R)}_{\alpha} T^{\mathrm{c\. \:\!. m\. \:\!. }}_{\alpha R})$
in the initial state and by $(1 + T_{\rho R} G_{0 })$ in the final
state; both wave functions do not have proper limits as $R \to \infty$. Therefore the full breakup transition amplitude in the case of
the unscreened Coulomb potential is obtained via the renormalization
of the on-shell breakup transition matrix $ U^{(R)}_{0 \alpha}$
in the infinite $R$ limit
\begin{equation}
\langle \mathbf{p}'_\rho \mathbf{q}'_\rho | U^{(C)}_{0 \alpha}
|b_\alpha \mathbf{q}_\alpha \rangle =
\lim_{R \to \infty} [ z^{-\frac{1 }{2 }}_{\rho R}(p'_\rho)
\langle \mathbf{p}'_\rho \mathbf{q}'_\rho | U^{(R)}_{0 \alpha}
|b_\alpha \mathbf{q}_\alpha \rangle Z_{\alpha R}^{-\frac{1 }{2 }}(q_\alpha )],
\label{eq:UC1 a}
\end{equation}
where $\mathbf{p}'_\rho$ is the relative momentum between the charged
particles in the final state,
$\mathbf{q}'_\rho$ the corresponding particle-pair relative momentum, and
\begin{equation} \label{eq:phiRp}
z_{\rho R}(p'_\rho) = e^{-2 i\kappa_\rho(p'_\rho)[\ln{(2 p'_\rho R)} - C/n]} ,
\end{equation}
the final-state renormalization factor with the Coulomb parameter
$\kappa_\rho(p'_\rho)$ for the pair $\rho$. The limit in Eq. ~(\ref{eq:UC1 a}) has to be performed numerically,
but, due to the short-range nature of the breakup operator,
the convergence with increasing screening radius $R$
is fast and the limit is reached with sufficient accuracy at
finite $R$. Thus, to include the Coulomb interaction via the screening and renormalization
method one only needs to solve standard scattering theory equations. \subsection{Practical realization}
We calculate the short-range part of the elastic, rearrangement, and breakup
scattering amplitudes (\ref{eq:UC2 }) and (\ref{eq:UC1 a}) by
solving standard scattering equations (\ref{eq:Uba}), (\ref{eq:Tcm}),
and (\ref{eq:U0 a}) with a finite Coulomb screening radius $R$. We work in the momentum-space partial-wave basis~\cite{deltuva:phd},
i. e., we use three sets \\
$|p_\alpha q_\alpha \nu_\alpha \rangle \equiv
|p_\alpha q_\alpha (l_\alpha \{ [L_\alpha(s_\beta s_\gamma)S_\alpha]
I_\alpha s_\alpha \} K_\alpha) { J M} \rangle$
with $(\alpha, \beta, \gamma)$ being cyclic permutations of (1,2,3 ). Here $s_\alpha$ is the spin of particle $\alpha$, $L_\alpha$ and $l_\alpha$ are
the orbital angular momenta associated with $p_\alpha$ and $q_\alpha$ respectively,
whereas $S_\alpha$, $I_\alpha$, and $K_\alpha$ are intermediate angular momenta
that are coupled to a total angular momentum $J$ with projection $M$. All discrete quantum numbers are abbreviated by $\nu_\alpha$. The integration over the momentum variables is discretized using
Gaussian quadrature rules thereby converting a system of integral equations
for each $J$ and parity $\Pi = (-)^{L_\alpha +l_\alpha}$
into a very large system of linear algebraic equations. Due to the huge dimension those linear systems cannot be solved directly. Instead we expand the AGS transition operators (\ref{eq:Uba})
into the corresponding Neumann series
\begin{equation} \label{eq:neumann}
U^{(R)}_{\beta \alpha} = \bar{\delta}_{ \beta \alpha } G_0 ^{-1 } +
\sum_{\gamma=1 }^3 \bar{\delta}_{ \beta \gamma } T^{(R)}_\gamma
\bar{\delta}_{ \gamma \alpha }
+ \sum_{\gamma=1 }^3 \bar{\delta}_{ \beta \gamma } T^{(R)}_\gamma G_0
\sum_{\sigma=1 }^3 \bar{\delta}_{\gamma \sigma } T^{(R)}_\sigma
\bar{\delta}_{\sigma \alpha}
+ \cdots ,
\end{equation}
that are summed up by the iterative Pade method~\cite{chmielewski:03 a};
it yields an accurate solution of Eq. ~(\ref{eq:Uba}) even when the
Neumann series (\ref{eq:neumann}) diverges.
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and renormalization method depends
very much on the choice of the screening function, in our case on the power
$n$ in Eq. ~(\ref{eq:wr}). We want to ensure that the
screened Coulomb potential $w_R$ approximates well the true Coulomb one
$w_C$ for distances $r<R$ and simultaneously vanishes rapidly for $r>R$,
providing a comparatively fast convergence of the partial-wave expansion. As shown in Ref. ~\cite{deltuva:05 a}, this is not the case for simple
exponential screening $(n =1 )$ whereas the sharp cutoff $(n \to \infty)$
yields slow oscillating convergence with the screening radius $R$. However,
we found that values of $3 \le n \le 8 $ provide a sufficiently smooth and
rapid screening around $r=R$. The screening functions
for different $n$ values are compared in Ref. ~\cite{deltuva:05 a}
together with the results demonstrating the superiority of our
optimal choice: using $3 \le n \le 8 $ the convergence with the
screening radius $R$, at which the short range part of the amplitudes was
calculated, is fast enough such that the convergence of the
partial-wave expansion, though being slower than for the nuclear interaction
alone, can be achieved and there is no need to work in a plane-wave basis. Here we use $n=4 $ and show in Figs. ~\ref{fig:Rad}and \ref{fig:Radb}
few examples for the $R$-convergence of the $\alpha$-deuteron scattering
observables calculated in a three-body model $(\alpha, p, n)$;
the nuclear interaction is taken from Ref. ~\cite{deltuva:06 b}. The convergence with $R$ is impressively fast for both
$\alpha$-deuteron elastic scattering and breakup. In addition we note
that the Coulomb effect is very large and clearly improves the description
of the experimental data, especially for the
differential cross section in $\alpha$-deuteron breakup reaction. This is due to the shift of
the $\alpha p$ $P$-wave resonance position when the $\alpha p$ Coulomb
repulsion is included that leads to
the corresponding changes in the structure of the observables. \begin{figure}[t]
\sidecaption[t]
\includegraphics[scale=.55 ]{Rd48 c. eps}
\caption{
Differential cross section and deuteron vector analyzing power $iT_{11 }$
of the $\alpha d$ elastic scattering at 4.81 ~MeV deuteron lab energy
as functions of the c. m. scattering angle. Convergence with the screening radius $R$ used to calculate the
short-range part of the amplitudes is studied:
$R= 5 $~fm (dotted curves), $R= 10 $~fm (dash-dotted curves), and
$R= 15 $~fm (solid curves). Results without Coulomb
are given by dashed curves. The experimental data are from Refs. ~\cite{bruno:80, gruebler:70 a}. }
\label{fig:Rad}
\end{figure}
\begin{figure}[t]
\sidecaption[t]
\includegraphics[scale=.55 ]{Ra15. eps}
\caption{
Fivefold differential cross section of the $\alpha d$ breakup reaction
at 15 ~MeV $\alpha$ lab energy for several combinations of
$\alpha$ and proton scattering angles as function of the final-state
energy variable $S$ with $dS = (dE_\alpha^2 + dE_p^2 )^{1 /2 }$. Convergence with the screening radius $R$ is studied:
$R= 10 $~fm (dotted curves), $R= 15 $~fm (dash-dotted curves), and
$R= 20 $~fm (solid curves). Results without Coulomb
are given by dashed curves. The experimental data are from Ref. ~\cite{koersner:77 }. }
\label{fig:Radb}
\end{figure}
In addition to the internal reliability criterion of the screening and
renormalization method --- the convergence with $R$ --- we note that our
results for proton-deuteron elastic scattering~\cite{deltuva:05 b}
agree well over a
broad energy range with those of Ref. ~\cite{kievsky:01 a} obtained from
the variational configuration-space solution
of the three-nucleon Schr\"odinger equation with unscreened Coulomb potential
and imposing the proper Coulomb boundary conditions explicitly. \section{Configuration space}\label{sec:r}
In contrast to the momentum-space representation,
the Coulomb interaction has a trivial expression
in configuration space and thus may seem to be easier to handle. However the
major obstacle for configuration-space treatment of the scattering problem
is related with the complexity of the wave function asymptotic structure,
which strongly complicates once three-particle breakup is available. Although for
short range interactions the analytical behavior of the breakup asymptote
of the configuration space wave function is well established, this is not a
case once long range interactions (like Coulomb) are present. Therefore a
method which enables the scattering problem to be solved without explicit use
of the wave function asymptotic form is of great importance. The
complex scaling method has been proposed~\cite{Nuttal_csm, CSM_71 } and successfully
applied to calculate the resonance positions~\cite{Moiseyev} by
using bound state boundary conditions. As has been demonstrated recently this method can be extended also for the
scattering problem~\cite{CSM_Curdy_04, Elander_CSM}. We
demonstrate here that this method may be also
successfully applied to solve
three-particle scattering problems which include the long-range Coulomb interaction together
with short range optical potentials. \subsection{Faddeev-Merkuriev equations}
Like in the momentum space formalism described above Jacobi coordinates are
also used in configuration space to separate the
center of mass of the three-particle system. One has three equivalent sets
of three-particle Jacobi coordinates
\begin{eqnarray}
\mathbf{x}_{\alpha } &=&\sqrt{\frac{2 m_{\beta }m_{\gamma }}{(m_{\beta
}+m_{\gamma })m}}(\mathbf{r}_{\gamma }-\mathbf{r}_{\beta }) , \\
\mathbf{y}_{\alpha } &=&\sqrt{\frac{2 m_{\beta }(m_{\beta }+m_{\gamma })}{%
(m_{\alpha }+m_{\beta }+m_{\gamma })m}}(\mathbf{r}_{\alpha }-\frac{m_{\beta }%
\mathbf{r}_{\beta }+m_{\gamma }\mathbf{r}_{\gamma }}{m_{\beta }+m_{\gamma }}) ,
\nonumber
\end{eqnarray}%
here $r_{\alpha }$ and $m_{\alpha }$ are individual particle position
vectors and masses, respectively. The choice of a mass scale $m$ is
arbitrary. The three-particle problem is formulated here using Faddeev-Merkuriev
(FM) equations~\cite{Merkuriev_80 }:
\begin{eqnarray}
(E-H_{0 }-\sum_{\kappa =1 }^{3 }w_{i}^{l})\psi _{\alpha }=(v_{\alpha
}+w_{\alpha }^{s})(\psi _{\alpha }+\psi _{\beta }+\psi _{\gamma })\nonumber , \\
(E-H_{0 }-\sum_{\kappa =1 }^{3 }w_{i}^{l})\psi _{\beta }=(v_{\beta }+w_{\beta
}^{s})(\psi _{\alpha }+\psi _{\beta }+\psi _{\gamma }) , \\
(E-H_{0 }-\sum_{\kappa =1 }^{3 }w_{i}^{l})\psi _{\gamma }=(v_{\gamma
}+w_{\gamma }^{s})(\psi _{\alpha }+\psi _{\beta }+\psi _{\gamma }) , \nonumber%
\end{eqnarray}%
where the Coulomb interaction is split in two parts (short and long range), $%
w_{\alpha }=w_{\alpha }^{s}+w_{\alpha }^{l}$, by means of some arbitrary
cut-off function $\chi _{\alpha }(x_{\alpha }, y_{\alpha })$:
\begin{equation}
w_{\alpha }^{s}(x_{\alpha }, y_{\alpha })=w_{\alpha }(x_{\alpha })\chi
_{\alpha }(x_{\alpha }, y_{\alpha })\qquad w_{\alpha }^{l}(x_{\alpha
}, y_{\alpha })=w_{\alpha }(x_{\alpha })[1 -\chi _{\alpha }(x_{\alpha
}, y_{\alpha })]
\end{equation}%
This cut-off function intends to shift the full Coulomb interaction
in the $w_{\alpha }^{s}$ term if $%
x_{\alpha }$ is small, whereas the $w_{\alpha }^{l}$ term acquires the full Coulomb
interaction if $x_{\alpha }$ becomes large and $y_{\alpha }<x_{\alpha }$. The practical choice of function $\chi _{\alpha }(x_{\alpha }, y_{\alpha })$
has been proposed in~\cite{Merkuriev_80 }:
\begin{equation}
\chi _{\alpha }(x_{\alpha }, y_{\alpha })=\frac{2 }{[1 +exp{(\frac{[x_{\alpha }/x_0 ]^\mu}{1 +y_{\alpha }/y_0 })}]} ,
\end{equation}%
with free parameters $x_{0 }, y_{0 }$ having size comparable with the
charge radii of the respective binary systems; the value of parameter
$\mu$ must be larger than 1 and is usually set $\mu\approx2 $. In such a way the so-called
Faddeev amplitude $\psi _{\alpha }$ intends to acquire full asymptotic
behavior of the binary $\alpha -(\beta \gamma )$ channels, i. e:
\begin{eqnarray}
\psi _{\alpha }(\mathbf{x}_{\alpha }, \mathbf{y}_{\alpha }\rightarrow \infty
)=\delta _{\kappa , \alpha }\psi _{\alpha }^{i_{\kappa }}(\mathbf{x}_{\alpha
})\phi _{\alpha }^{i_{\kappa }, in}(\mathbf{y}_{\alpha })&+&\sum_{j_{\alpha
}}f_{j_{\alpha i_{\kappa }}}(\mathbf{x}_{\alpha }. \mathbf{y}_{\alpha })\psi
_{\alpha }^{j_{\alpha }}(\mathbf{x}_{\alpha })\phi _{\alpha }^{j_{\alpha
}, out}(\mathbf{y}_{\alpha })\nonumber \\
&+&A_{i_{\kappa }}(\mathbf{x}_{\alpha }, \mathbf{y}%
_{\alpha })\Phi _{i_{\kappa }}^{out}(\mathbf{\rho }) ,
\end{eqnarray}
where the hyperradius is $\rho =\sqrt{x_{\alpha }^{2 }+y_{\alpha }^{2 }}$. An
expression $\varphi _{\alpha }^{i_{\alpha }}(\mathbf{x}_{\alpha })\phi
_{\alpha }^{i_{\kappa }, in}(\mathbf{y}_{\alpha })$ represents the incoming wave for particle $\alpha $ on pair $(\beta \gamma )$ in
the bound state $i_{\alpha }$, with $\varphi _{\alpha }^{i_{\alpha }}(%
\mathbf{x}_{\alpha })$ representing the normalized wave
function of bound state $i_{\alpha }$. This wave
function is a solution of the $(E-H_{0 }-w_{\alpha }-v_{\alpha
}-W_{\alpha }^{c. m. })$ two-body Hamiltonian. The $\phi _{\alpha }^{j_{\alpha }, out}(%
\mathbf{y}_{\alpha })$ and $\Phi _{i_{\kappa }}^{out}(\mathbf{\rho }_{\alpha
})$ represent outgoing waves for binary and three-particle breakup channels
respectively. In the asymptote, one has the following behavior:
\begin{eqnarray}
\varphi _{\alpha }^{i_{\alpha }}(x_{\alpha } &\rightarrow &\infty )\propto
\exp (-k_{i_{\alpha }}x_{\alpha }) , \nonumber \\
\phi _{\alpha }^{i_{\alpha }, out}(y_{\alpha } &\rightarrow &\infty )\propto
\exp (iq_{i_{\alpha }}y_{\alpha }) , \\
\Phi _{i_{\alpha }}^{out}(\rho &\rightarrow &\infty )\propto \exp (iK\rho ) ,
\label{eq:assf}
\end{eqnarray}
with $k_{i_{\alpha }}=\sqrt{-\varepsilon _{_{i_{\alpha }}}m}$ representing
momentum of 2 -body bound state $i_{\alpha }$ with a negative binding energy $%
\varepsilon _{_{i_{\alpha }}}$; $q_{i_{\alpha }}=\sqrt{(E-\varepsilon
_{_{i_{\alpha }}})m}$ is relative scattering momentum for the $\alpha -(\beta
\gamma )$ binary channel, whereas $K=\sqrt{mE}$ is a three-particle breakup
momentum (three-particle breakup is possible only if energy value $E$ is
positive). When considering particle's $\alpha $ scattering on the bound state $i_{\alpha
}$ of the pair $(\beta \gamma )$, it is
convenient to separate readily
incoming wave $\psi _{\alpha }^{i_{\alpha }, in}=\psi _{\alpha }^{i_{\alpha }}(\mathbf{x}_{\alpha })\phi _{\alpha }^{i_{\alpha }, in}(\mathbf{y}_{\alpha
})$, by introducing:
\begin{eqnarray}
\psi _{\alpha }^{i_{\alpha }, out} &=&\psi _{\alpha }^{i_{\alpha }}-\psi
_{\alpha }^{i_{\alpha }}(\mathbf{x}_{\alpha })\phi _{\alpha }^{i_{\alpha
}, in}(\mathbf{y}_{\alpha }) , \\
\psi _{\beta }^{i_{\alpha }, out} &=&\psi _{\beta }^{i_{\alpha }}\qquad \beta
\neq \alpha , \nonumber
\end{eqnarray}
Then Faddeev-Merkuriev equations might be rewritten in a so-called driven form:
\begin{eqnarray}
(E-H_{0 }-\sum_{\kappa =1 }^{3 }w_{\kappa }^{l})\psi _{\alpha
}^{out}&=&(v_{\alpha }+w_{\alpha }^{s})(\psi _{\alpha }^{out}+\psi _{\beta
}^{out}+\psi _{\gamma }^{out})+\left[ \sum_{\kappa =1 }^{3 }w_{\kappa
}^{l}-w_{\alpha }-W_{\alpha }^{c. m. }\right] \psi _{\alpha }^{in} ,
\nonumber \\
(E-H_{0 }-\sum_{\kappa =1 }^{3 }w_{\kappa }^{l})\psi _{\beta }^{out}&=&(v_{\beta
}+w_{\beta }^{s})(\psi _{\alpha }^{out}+\psi _{\beta }^{out}+\psi _{\gamma
}^{out}+\psi _{\alpha }^{in}) , \\
(E-H_{0 }-\sum_{\kappa =1 }^{3 }w_{\kappa }^{l})\psi _{\gamma
}^{out}&=&(v_{\gamma }+w_{\gamma }^{s})(\psi _{\alpha }^{out}+\psi _{\beta
}^{out}+\psi _{\gamma }^{out}+\psi _{\alpha }^{in}) . \nonumber
\label{eq:drive_FM}
\end{eqnarray}
In this expression index of the incoming state $i_{\alpha }$ has been omitted in
all Faddeev component expressions $\psi _{\alpha }^{in}$ and $\psi _{\alpha }^{out}$. \subsection{Complex scaling}
Next step is to perform the complex scaling operations i. e. scale all the distances $%
x $ and $y$ by a constant complex factor $e^{i\theta }, $ so that both $%
Re(e^{i\theta })$ and $Im(e^{i\theta })$ are positive (angle $\theta $ must
be chosen in the first quartet in order to satisfy this condition). The complex scaling
operation, in particular, implies that the analytical continuation of the
interaction potentials is performed: $v_{\alpha }(x_{\alpha }e^{i\theta })$
and $w_{\alpha }(x_{\alpha }e^{i\theta })$. Therefore the complex scaling method
may be used only if these potentials are analytic.
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\rho \rightarrow \infty )\right] ^{CS}
&\propto &\exp (-K\rho \sin \theta ) . \nonumber
\end{eqnarray}
Nevertheless an incoming wave diverges in $y_{\alpha }$ after the complex
scaling:
\begin{equation}
\left[ \phi _{\alpha }^{i_{\alpha }, out}(y_{\alpha }\rightarrow \infty )%
\right] ^{CS}\propto \exp (+q_{i_{\alpha }}y_{\alpha }\sin \theta ) . \end{equation}
However these terms appear only on the right hand sides of the driven Faddeev-Merkuriev
equation~(\ref{eq:drive_FM})
being pre-multiplied with the potential terms and under certain conditions they may vanish
outside of some finite (resolution) domain $x_{\alpha }\in \lbrack 0, x^{\max
}]$ and $y_{\alpha }\in \lbrack 0, y^{\max }]$. Let us consider the long range
behavior of the term $\left[ (v_{\beta }+w_{\beta }^{s})\psi _{\alpha }^{in}%
\right] ^{CS}$. Since the interaction terms $v_{\beta }$ and $w_{\beta }^{s}$
are of short range, the only region the former term might not converge is
along $y_{\beta }$ axis in $(x_{\beta }, y_{\beta })$ plane, i. e. for $%
x_{\beta }\ll y_{\beta }$. On the other hand $x_{\alpha }(%
\mathbf{x}_{\beta }\mathbf{, y}_{\beta })\approx \sqrt{m_{\gamma }/(m_{\gamma
}+m_{\beta })}\sqrt{M/(m_{\gamma }+m_{\alpha })}y_{\beta }$ and $y_{\alpha }(%
\mathbf{x}_{\beta }\mathbf{, y}_{\beta })\approx \sqrt{m_{\beta }/(m_{\gamma
}+m_{\beta })}\sqrt{m_{\alpha }/(m_{\gamma }+m_{\alpha })}y_{\beta }$ under
condition $x_{\beta }\ll y_{\beta }$. Then one has:
\begin{equation}
\small{
\left[ (v_{\beta }+w_{\beta }^{s})\psi _{\alpha }^{i_{\alpha }, in}\right]
^{CS}_{x_{\beta }\ll y_{\beta }}\propto \exp\left(-k_{i_{\alpha }}\sqrt{\frac{%
m_{\gamma }M}{(m_{\gamma }+m_{\beta })(m_{\gamma }+m_{\alpha })}}y_{\beta
}\cos \theta +q_{i_{\alpha }}\sqrt{\frac{m_{\alpha }m_{\beta }}{%
(m_{\gamma }+m_{\beta })(m_{\gamma }+m_{\alpha })}}y_{\beta }\sin \theta \right)} . \end{equation}
This term becomes bound to finite domain in $(x_{\beta }, y_{\beta }) $
plane, if condition:
\begin{equation}
\tan \theta <\sqrt{\frac{m_{\gamma }M}{m_{\alpha }m_{\beta }}}\frac{%
k_{i_{\alpha }}}{q_{i_{\alpha }}}=\sqrt{\frac{m_{\gamma }M}{m_{\alpha
}m_{\beta }}}\sqrt{\frac{\left\vert B_{_{i_{\alpha }}}\right\vert }{%
E+\left\vert B_{_{i_{\alpha }}}\right\vert }} ,
\label{max_theta}
\end{equation}
is satisfied. This implies that for rather large scattering energies $E$,
above the break-up threshold, one is obliged to use rather small complex
scaling parameter $\theta $ values. The term $\left[\sum_{\kappa =1 }^{3 }w_{\kappa }^{l}-w_{\alpha }-W_{\alpha
}^{c. m. }\right] \psi _{\alpha }^{i_{\alpha }, in}$, in principle, is not
exponentially bound after the complex scaling. It represents the higher
order corrections to the residual Coulomb interaction between particle $%
\alpha $ and bound pair $(\beta \gamma )$. These corrections are weak $%
o(1 /y^{2 })$ and might be neglected by suppressing this term close to the
border of the resolution domain. Alternative possibility might be to use
incoming wave functions, which account not only for the bare $\alpha -(\beta
\gamma )$ Coulomb interaction but also takes into account higher order
polarization corrections. \bigskip
Extraction of the scattering observables is realized by employing Greens
theorem. One might demonstrate that strong interaction amplitude for $%
\alpha -(\beta \gamma )$ collision is:
\begin{equation}
f_{j_{\alpha i_{\kappa }}}(\mathbf{x}_{\alpha }. \mathbf{y}_{\alpha })=-\frac{m}{q_{j_{\alpha}}}\int
\int \left[ (\psi _{\alpha }^{j_{\alpha }, in})^*\right] ^{CS}(\overline{v}%
_{\alpha }+\overline{w}_{\alpha }-W_{\alpha }^{c. m. })^{CS}\left[ \Psi
_{i_{\kappa }}\right] ^{CS}e^{6 i\theta }d^{3 }\mathbf{x}_{i}d^{3 }\mathbf{y}%
_{i} \label{3 b_amp_nc} ,
\end{equation}%
with $\left[ \Psi _{i_{\kappa }}\right] ^{CS}=\left[ \psi _{\alpha
}^{i_{\kappa }, out}+\psi _{\beta }^{i_{\kappa }, out}+\psi _{\gamma
}^{i_{\kappa }, out}+\psi _{\alpha }^{i_{\kappa }, in}\right] ^{CS}$ being the
total wave function of the three-body system. In the last expression the
term containing product of two incoming waves is slowest to converge. Even
stronger constraint than eq. (\ref{max_theta}) should be implied on complex scaling angle in
order to make this term integrable on the finite domain. Nevertheless this term
contains only the product of two-body wave functions and might be evaluated
without using complex scaling prior to three-body solution. Then the appropriate
form of the integral~(\ref{3 b_amp_nc}) to be used becomes:
\begin{eqnarray}
f_{j_{\alpha i_{\kappa }}}(\mathbf{x}_{\alpha }. \mathbf{y}_{\alpha })
&=&-\frac{m}{q_{j_{\alpha}}}\int \int \left[ (\psi _{\alpha }^{j_{\alpha }, in})^*\right] ^{CS}(%
\overline{v}_{\alpha }+\overline{w}_{\alpha }-W_{\alpha }^{c. m. })^{CS}\left[
\Psi _{i_{\kappa }}-\psi _{\alpha }^{j_{\alpha }, in}\right] ^{CS}e^{6 i\theta
}d^{3 }\mathbf{x}_{i}d^{3 }\mathbf{y}_{i} \nonumber\\
&&-\frac{m}{q_{j_{\alpha}}}\int \int (\psi _{\alpha }^{j_{\alpha }, in})^*(\overline{v}_{\alpha }+%
\overline{w}_{\alpha }-W_{\alpha }^{c. m. })\psi _{\alpha }^{j_{\alpha
}, in}d^{3 }\mathbf{x}_{i}d^{3 }\mathbf{y} . \end{eqnarray}
\bigskip
\bigskip
\section{Application to three-body nuclear reactions}
The two methods presented in sections~\ref{sec:p} and~\ref{sec:r} were first
applied to the proton-deuteron elastic scattering and
breakup~\cite{deltuva:05 a, deltuva:05 d, deltuva:09 e, lazauskas:11 a}. The three-nucleon system is the
only nuclear three-particle system that may be considered realistic in the sense that the interactions
are given by high precision potentials valid over a broad energy range. Nevertheless, in the same way one considers the nucleon as a single particle by neglecting
its inner quark structure, in a further approximation one can
consider a cluster of nucleons (composite nucleus)
to be a single particle that interacts with other nucleons or nuclei via
effective potentials whose parameters are determined from the two-body data. A classical example is the $\alpha$ particle, a tightly
bound four-nucleon cluster. As shown in Figs. ~\ref{fig:Rad} and \ref{fig:Radb}
and in Ref. ~\cite{deltuva:06 b}, the description of the
$(\alpha, p, n)$ three-particle system with real potentials
is quite successful at low
energies but becomes less reliable with increasing energy where the
inner structure of the $\alpha$ particle cannot be neglected anymore. At higher energies the nucleon-nucleus or nucleus-nucleus interactions
are modeled by optical potentials (OP) that provide quite an accurate
description of the considered two-body system
in a given narrow energy range; these potentials are complex to account for the inelastic
excitations not explicitly included in the model space. The methods based on
Faddeev/AGS equations can be applied also in this case, however,
the potentials within the pairs that are bound in the initial or final
channel must remain real. The comparison of the two methods based
on the AGS and FM equations will be performed in section~\ref{sec:compare}
for such an interaction model with OP. In the past the description of three-body-like nuclear reactions involved
a number of approximate methods that have been developed. Well-known examples are
the distorted-wave Born approximation (DWBA), various adiabatic approaches
\cite{johnson:70 a}, and
continuum-discretized coupled-channels (CDCC) method~\cite{austern:87 }. Compared to them the present methods based on exact Faddeev or AGS equations,
being more technically and numerically involved,
have some disadvantages. Namely,
their application in the present technical realization is so far limited
to a system made of two nucleons and one heavier cluster. The reason is
that the interaction between two heavier cluster involves
very many angular momentum states and the partial-wave convergence
cannot be achieved. The comparison between traditional nuclear
reaction approaches and momentum-space Faddeev/AGS methods
for various neutron + proton + nucleus systems
are summarized in section~\ref{sec:cdcc}. On the other hand, the Faddeev and AGS methods may be more flexible
with respect to dynamic input and thereby allows to test novel aspects
of the nuclear interaction not accessible with the traditional approaches. Few examples will be presented in section \ref{sec:nonloc}. \subsection{Numerical comparison of AGS and FM methods} \label{sec:compare}
As an example we consider the $n+p+^{12 }C$ system. For the $n$-$p$ interaction
we use a realistic AV18 model~\cite{wiringa:95 a} that accurately reproduces the
available two-nucleon scattering data and deuteron binding energy. To study not only the $d+{}^{12 }$C but also $p+{}^{13 }$C
scattering and transfer reactions we use
a $n$-$^{12 }$C potential that is real in the $^2 P_\frac{1 }{2 }$
partial wave and supports the ground state of $^{13 }C$
with 4.946 MeV binding energy; the parameters are taken from
Ref. ~\cite{nunes:11 b}. In all other partial waves we
use the $n$-$^{12 }$C optical potential from Ref. ~\cite{CH89 }
taken at half the deuteron energy in the $d+{}^{12 }$C channel. The $p$-$^{12 }$C optical potential is also taken from Ref. ~\cite{CH89 }, however, at the proton energy in the $p+{}^{13 }$C channel. We admit that, depending on the reaction of interest,
other choices of energies for OP may be more appropriate,
however, the aim of the present study is comparison of the
methods and not the description of the experimental data although the latter
are also included in the plots. We consider $d+{}^{12 }$C scattering at 30 MeV deuteron lab energy
and $p+{}^{13 }$C scattering at 30.6 MeV proton lab energy;
they correspond to the same energy in c. m. system. First we perform calculations by neglecting the $p$-$^{12 }$C Coulomb repulsion. One observes a perfect agreement between the AGS and FM methods. Indeed, the calculated S-matrix elements in each three-particle
channel considered (calculations have been performed for total three-particle
angular momentum states up to $J=13 $) agree within
three digits. Scattering observables converge quite slowly with $J$
as different angular momentum state contributions cancel each other
at large angles. Nevertheless, the results of the two methods are practically
indistinguishable as demonstrated in Fig. ~\ref{fig:dC-noC}
for $d+{}^{12 }$C elastic scattering and transfer to $p+{}^{13 }$C. Next we perform the full calculation including the $p$-$^{12 }$C
Coulomb repulsion; we note that inside the nucleus the Coulomb potential
is taken as the one of a uniformly charged sphere~\cite{deltuva:06 b}. Once again we obtain good agreement between the AGS and FM methods. However, this time small variations up to the order of 1 \% are observed when
analyzing separate $S$-matrix elements, mostly in high angular momentum states. This leads to small differences in some scattering observables, e. g.,
differential cross sections for $d+{}^{12 }$C elastic scattering
(at large angles where the differential cross section is very small)
and for the deuteron stripping reaction $d+{}^{12 }$C $ \to p+{}^{13 }$C
shown in Fig. ~\ref{fig:dC}. The $p+{}^{13 }$C elastic scattering observables
presented in Fig. ~\ref{fig:pC} converge faster with $J$. As a consequence, the results of the two calculations are indistinguishable
for the $p+{}^{13 }$C elastic cross section
and only tiny differences can be seen for the proton analyzing power
at large angles. In any case, the agreement between the AGS and FM methods
exceeds both the accuracy of the data and the existing discrepancies
between theoretical predictions and experimental data. \begin{figure
\sidecaption[t]
\includegraphics[scale=.56 ]{dC30 noC. eps}
\caption{
Comparison of momentum- (solid curves) and configuration-space
(dashed-dotted curves) results for the
deuteron-${}^{12 }$C scattering at 30 MeV deuteron lab energy. Differential cross sections for elastic scattering and stripping
are shown neglecting the Coulomb interaction. }
\label{fig:dC-noC}
\end{figure}
\begin{figure
\sidecaption[t]
\includegraphics[scale=.56 ]{dC30. eps}
\caption{
Comparison of momentum- (solid curves) and configuration-space
(dashed-dotted curves) results for the
deuteron-${}^{12 }$C scattering at 30 MeV deuteron lab energy. Differential cross sections for elastic scattering and stripping
are shown, the former in ratio to the Rutherford cross section
$d\sigma_R/d\Omega$. The experimental data are from Refs. ~\cite{perrin:77, dC30 p}. }
\label{fig:dC}
\end{figure}
\begin{figure
\sidecaption[t]
\includegraphics[scale=.56 ]{pC30. eps}
\caption{
Comparison of momentum- (solid curves) and configuration-space
(dashed-dotted curves) results for the
proton-${}^{13 }$C elastic scattering at 30.6 MeV proton lab energy. Differential cross section divided by the Rutherford cross section
and proton analyzing power are shown. The experimental data are from Ref. ~\cite{pC30 }. }
\label{fig:pC}
\end{figure}
\subsection{Comparison with traditional nuclear reaction approaches} \label{sec:cdcc}
The method based on the momentum-space AGS equations has already been
used to test the accuracy of the traditional nuclear reaction approaches;
limitations of their validity in energy and kinematic range have
been estalished. The distorted-wave impulse approximation for breakup of a one-neutron
halo nucleus ${}^{11 }$Be on a proton target has been tested in
Ref.
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accuracy as shown in Ref. ~\cite{deltuva:07 d}.
The semi-inclusive differential cross section for the
breakup reaction $p + {}^{11 }$Be $\to p + n + {}^{10 }$Be
was calculated also using two CDCC versions where the full scattering wave function
was expanded into the eigenstates of either the $n + {}^{10 }$Be (CDCC-BU)
or the $p+n$ (CDCC-TR) pair. Neither of them agrees well with AGS
over the whole angular regime as shown in Fig. ~\ref{fig:cdcc}.
It turns out that, depending on the
${}^{10 }$Be scattering angle, the semi-inclusive breakup cross section
is dominated by different mechanisms: at small angles it is the
proton-neutron quasifree scattering whereas at intermediate and large angles
it is the neutron-${}^{10 }$Be $D$-wave resonance. However, a proper treatment
of proton-neutron interaction in CDCC-BU and of neutron-${}^{10 }$Be
interaction in CDCC-TR is very hard to achieve since the wave function
expansion uses eigenstates of a different pair.
No such problem exists in the AGS method that uses simultaneously
three sets of basis states and each pair is treated in its proper basis.
\begin{figure
\includegraphics[scale=.45 ]{ags-cdcc. eps}
\caption{Semi-inclusive differential cross section for the
breakup reaction $p + {}^{11 }$Be $\to p + n + {}^{10 }$Be
at lab energy of 38.4 MeV/nucleon. Results obtained with
AGS and CDCC methods are compared. }
\label{fig:cdcc}
\end{figure}
\subsection{Beyond standard dynamic models} \label{sec:nonloc}
The standard nucleon-nucleus optical potentials employed
in three-body calculations have central and, eventually, spin-orbit parts
that are local.
This local approximation yields a tremendous simplification in the
practical realization of DWBA, CDCC and other traditional approaches
that are based on configuration-space representations where
the use of nonlocal optical potentials was never attempted.
However, nonlocal optical potentials do not yield any serious technical
difficulties in the momentum-space representation. Thus, they
can be included quite easily in the AGS framework employed by us.
There are very few nonlocal parametrizations of the optical potentials
available. We take the one from Refs. ~\cite{giannini, giannini2 } defined
in the configuration space as
\begin{equation} \label{eq:vnl}
v_{\gamma}(\vec{r}', \vec{r}) = H_c(x)[V_c(y) + iW_c(y)] +
2 \vec{S_\gamma}\cdot \vec{L_\gamma} H_s(x) V_s(y) ,
\end{equation}
with $x = |\vec{r}'-\vec{r}|$ and $y=|\vec{r}'+\vec{r}|/2 $.
The central part has real volume and imaginary surface parts,
whereas the spin-orbit part is real; all of them are expressed in the standard
way by Woods-Saxon functions. Some of their strength parameters were
readjusted in Ref. ~\cite{deltuva:09 b}
to improve the description of the experimental nucleon-nucleus scattering
data. The range of the nonlocality is determined by the functions
$H_i(x) = (\pi \beta_i^2 )^{-3 /2 } \exp{(-x^2 /\beta_i^2 )}$
with the parameters $\beta_i$ being of the order of 1 fm.
A detailed study of nonlocal optical potentials in three-body reactions
involving stable as well as weakly bound nuclei, ranging from
${}^{10 }$Be to ${}^{40 }$Ca,
is carried out in Ref. ~\cite{deltuva:09 b}.
In order to isolate the nonlocality effect
we also performed calculations with a local optical potential that
provides approximately equivalent description
of the nucleon-nucleus scattering at the considered energy.
The nonlocality effect turns out to be very small in the elastic proton
scattering from the bound neutron-nucleus system and of moderate
size in the deuteron-nucleus scattering. However, the
effect of nonlocal proton-nucleus optical potential becomes
significant in deuteron stripping and pickup reactions $(d, p)$ and $(p, d)$;
in most cases it considerably improves agreement with the experimental data.
Examples for $(d, p)$ reactions leading to ground and excited states of
the stable nucleus ${}^{17 }$O and one-neutron halo nucleus ${}^{15 }$C
are presented in Figs. ~\ref{fig:Odp} and \ref{fig:Cdp}.
We note that in these transfer reactions the proton-nucleus potential
is taken at proton lab energy in the proton channel while
the neutron-nucleus potential has to be real in order to support the
respective bound states.
\begin{figure
\sidecaption[t]
\includegraphics[scale=.56 ]{dOp36 nl. eps}
\caption{
Differential cross section for $(d, p)$ reaction on ${}^{16 }$O
at 36 MeV deuteron lab energy
leading to ${}^{17 }$O nucleus in the ground state $5 /2 ^+$ (top)
and first excited state $1 /2 ^+$ (bottom).
Predictions of nonlocal (solid curve) and local
(dashed curve) optical potentials (OP)
are compared with the
experimental data from Ref. ~\cite{dO25 -63 }. }
\label{fig:Odp}
\end{figure}
\begin{figure
\sidecaption[t]
\includegraphics[scale=.56 ]{dCp14. eps}
\caption{
Differential cross section for $(d, p)$ reaction on ${}^{14 }$C
at 14 MeV deuteron lab energy
leading to one-neutron halo nucleus
${}^{15 }$C in the ground state $1 /2 ^+$ (top)
and first excited state $5 /2 ^+$ (bottom).
Curves as in Fig. ~\ref{fig:Odp} and the
experimental data are from Ref. ~\cite{d14 C14 p}. }
\label{fig:Cdp}
\end{figure}
Another extension beyond the standard dynamic models includes
the AGS method using energy-dependent optical potentials
Although such calculations don't correspond
to a rigorous Hamiltonian theory, they may shed some light on the
shortcomings of the traditional nuclear interaction models.
A detailed discussion of the calculations with
energy-dependent optical potentials is given in Ref. ~\cite{deltuva:09 a}.
\section{Summary}
We have presented the results of three-body Faddeev-type calculations for systems of three particles, two of which are charged, interacting through short-range nuclear plus the long-range Coulomb potentials. Realistic applications of three-body theory to three-cluster nuclear reactions --- such as scattering of deuterons on a nuclear target or one-neutron halo nucleus impinging on a proton target --- only became possible to address in recent years
when a reliable and practical momentum-space treatment of the Coulomb
interaction has been developed.
After the extensive and very complete study of $p$-$d$ elastic scattering and breakup, the natural extension of these calculations was the application to complex reactions such as $d$-${}^{4 }$He, $p$-${}^{17 }$O, ${}^{11 }$Be-$p$, $d$-${}^{58 }$Ni and many others using a realistic interaction such as AV18 between nucleons, and optical potentials chosen at the appropriate energy for the nucleon-nucleus interactions. The advantage of three-body calculations vis-\`{a}-vis traditional approximate reaction methods is that elastic, transfer, and breakup channels are treated on the same footing once the interaction Hamiltonian has been chosen. Another advantage of the three-body Faddeev-AGS approach is the possibility to include
nonlocal optical potentials instead of local ones as commonly used in the standard nuclear reaction methods; as demonstrated, this leads to an improvement in the description of transfer reactions in a very consistent way across different energies and mass numbers for the core nucleus.
Although most three-body calculations have been performed in momentum space over a broad range of nuclei from ${}^{4 }$He to ${}^{58 }$Ni and have encompassed studies of cross sections and polarizations for elastic, transfer, charge exchange,
and breakup reactions, coordinate space calculations above breakup threshold are coming to age using the complex scaling method. We have demonstrated here that both calculations agree to within a few percent for all the reactions we have calculated. This is a very promising development that may bring new light to the study of nuclear reactions given that the reduction of the many-body problem to an effective three-body one may be better implemented and understood by the community in coordinate space rather than in momentum space. On the other hand,
compared to DWBA, adiabatic approaches, or CDCC, the
Faddeev-type three-body methods are computationally more demanding and require greater technical expertise rendering them less attractive to analyze the data.
Nevertheless, when benchmark calculations have been performed comparing the Faddeev-AGS results with those obtained using CDCC or adiabatic approaches, some discrepancies were found in transfer and breakup cross sections depending on the specific kinematic conditions. Therefore the Faddeev-AGS approach is imminent in order to calibrate and validate approximate nuclear reaction methods wherever a comparison is possible.
\begin{acknowledgement}
The work of A. D. and A. C. F. was partially supported by the
FCT grant PTDC/FIS/65736 /2006.
The work of R. L. was granted access to the HPC resources of IDRIS under the allocation 2009 -i2009056006
made by GENCI (Grand Equipement National de Calcul Intensif). We thank the staff members of the IDRIS for their constant help.
\end{acknowledgement}
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\section{Introduction}
\label{sec:intro}
The gravitational scattering of classical objects at large impact parameter $b$ is relevant for the study of the inspiral phase of black-hole binaries since it can be used to determine the parameters of the Effective-One-Body description (see~\cite{Damour:2016 gwp} and references therein). For this reason, gravitational scattering has been at the centre of renewed attention and has been recently investigated using a variety of techniques, including the use of quantum field theory (QFT) amplitudes to extract the relevant classical physics~\cite{Goldberger:2004 jt, Melville:2013 qca, Goldberger:2016 iau, Luna:2016 idw, Luna:2017 dtq, Bjerrum-Bohr:2018 xdl, Cheung:2018 wkq, Kosower:2018 adc, Bern:2019 nnu, KoemansCollado:2019 ggb, Cristofoli:2019 neg, Bern:2019 crd, Kalin:2019 rwq, Bjerrum-Bohr:2019 kec, Kalin:2019 inp, Damour:2019 lcq, Cristofoli:2020 uzm, Kalin:2020 mvi, Kalin:2020 lmz, Kalin:2020 fhe, Mogull:2020 sak, Huber:2020 xny}. Here we will focus in particular on the eikonal approach~\cite{Amati:1987 wq, Amati:1987 uf, Muzinich:1987 in, Sundborg:1988 tb}, where the classical gravitational dynamics is derived from standard QFT amplitudes by focusing on the terms that exponentiate in the eikonal phase $e^{2 i \delta}$. The Post-Minkowskian (PM) expansions writes $\delta$ as a perturbative series in the Newton constant $G$ at large values of $b$ and the state-of-the-art results determine the real part of the 3 PM ({\em i. e. } 2 -loop) eikonal $\operatorname{Re} 2 \delta_2 $ (or the closely related scattering angle) and to some extent the imaginary part, both in standard GR~\cite{Amati:1990 xe, Bern:2019 nnu, Bern:2019 crd, Cheung:2020 gyp, Kalin:2020 fhe, Damour:2020 tta} and various supersymmetric generalisations~\cite{DiVecchia:2019 kta, Bern:2020 gjj, Parra-Martinez:2020 dzs, DiVecchia:2020 ymx}. In this letter we expand on the approach discussed in~\cite{Amati:1990 xe, DiVecchia:2020 ymx} where the relation between the real and the imaginary part of $\delta_2 $ was used to derive the 3 PM scattering angle in the ultrarelativistic limit and to show that it is a universal feature of all gravitational theories in the two derivative approximation. Furthermore, it was shown in~\cite{DiVecchia:2020 ymx} for ${\cal N}=8 $ supergravity that taking into account the full soft region in the loop integrals was crucial to obtain a smooth interpolation between the behaviour of $\delta_2 $ in the non-relativistic, {\em i. e. } Post-Newtonian (PN), regime and the ultrarelativistic (or massless) one. The additional contributions coming from the full soft region had the feature of contributing half-integer terms in the PN expansion and were therefore interpreted as radiation-reaction (RR) contributions. This connection was further confirmed in~\cite{Damour:2020 tta} by Damour, who used a linear response relation earlier derived in~\cite{Bini:2012 ji} to connect these new RR terms to the loss of angular momentum in the collision. In this way the result of~\cite{DiVecchia:2020 ymx} was extended to the case of General Relativity~\cite{Damour:2020 tta}. In this paper we argue that there is actually a direct relation between the RR and the much studied soft-bremsstrahlung limits. We claim that the real part of the RR eikonal at 3 PM (indicated by ${\rm Re}\, 2 \delta_2 ^{(rr)}$) is simply related to the infrared divergent contribution of its imaginary part $({\rm Im}\, 2 \delta_2 )$. This relation holds at all energies and reads
\begin{equation}
\lim_{\epsilon\to 0 } {\rm Re} \,2 \delta_2 ^{(rr)} = -\lim_{\epsilon\to 0 }\left[ \pi \epsilon ({\rm Im}\,2 \delta_2 )\right]
\;, \label{1.5 }
\end{equation}
where, as usual, $\epsilon = \frac{4 -D}{2 }$ is the dimensional regularisation parameter. On the other hand, there is a simple connection (see e. g. ~\cite{Addazi:2019 mjh}) between the infrared divergent imaginary part of $\delta_2 $ and the so-called zero-frequency limit~\cite{Smarr:1977 fy} of the bremsstrahlung spectrum reading:
\begin{equation}
\lim_{\epsilon\to 0 } \left[ - 2 \epsilon ({\rm Im}\,2 \delta_2 )\right] = \frac{d E^{rad}}{2 \hbar d \omega}(\omega \to 0 ) ~ \Rightarrow~
\lim_{\epsilon\to 0 }{\rm Re} \,2 \delta_2 ^{(rr)} = \frac{\pi}{4 \hbar} \frac{d E^{rad}}{d \omega}(\omega \to 0 )
\;, \label{ZFL}
\end{equation}
so that, in the end, RR gets directly related to soft bremsstrahlung. We stress that all (massless) particles can contribute to the r. h. s. of \eqref{ZFL} and therefore to the RR. This result was first noticed in the ${\cal N}=8 $ supergravity setup of~\cite{Caron-Huot:2018 ape, Parra-Martinez:2020 dzs} by using the results of~\cite{DiVecchia:2020 ymx, DiVecchia:2021 bdo}, see also~\cite{Herrmann:2021 tct}, where the full 3 PM eikonal is derived by a direct computation of the 2 -loop amplitude describing the scattering of two supersymmetric massive particle. Here we give an interpretation of this connection and conjecture its general validity in gravity theories at the 3 PM level (the first non trivial one) by reconstructing the infrared divergent part of ${\rm Im} \, 2 \delta_2 $ from the three-body discontinuity involving the two massive particles and a massless particle. The building block is of course the $2 \rightarrow 3 $ five-point tree-level amplitude where, for our purposes, it is sufficient to keep only the leading classical divergent term in the soft limit (the so-called Weinberg term) of the massless particle. When focusing on pure GR, the only massless particle that can be involved in the three-particle cut mentioned above is the graviton. We will see that, by using Eq. \eqref{1.5 }, we reproduce the deflection angle recently derived in~\cite{Damour:2020 tta} on the basis of a linear-response formula and of a lowest-order calculation of the angular momentum flux. In the massive ${\cal N}=8 $ case, one needs to consider, in addition to the graviton, the contributions of the relevant vectors and scalar fields (including the dilaton). Once all massless particles that can appear in the three-particle cut are taken into account, one obtains~\eqref{3.6 } which, as already mentioned, satisfies~\eqref{1.5 }. The basic idea underlying all cases is that the calculation of ${\rm Im}\, 2 \delta_2 $ from sewing tree-level, on shell, inelastic amplitudes is far simpler than the derivation of the full two-loop elastic amplitude even when focusing on just the classical contributions. Both for GR and for $\mathcal N=8 $, the infrared divergent piece of $\delta_2 $ can be equivalently obtained exploiting the exponentiation of infrared divergences in momentum space for the elastic amplitude itself (details will be presented elsewhere). The arguments supporting~\eqref{1.5 } appear to be valid within a large class of gravitational theories and so this equation provides a direct, general way to calculate the RR contributions at the 3 PM level. It remains to be seen whether this approach can be generalized, and in which form, beyond 3 PM. The paper is organized as follows. In Sect. ~\ref{softm} we introduce our kinematical set-up for the relevant elastic ($2 \rightarrow 2 $) and inelastic ($2 \rightarrow 3 $) processes and discuss the standard soft limit of the latter in momentum space. In Sect. ~\ref{RRIR} we present the empirical connection between $\operatorname{Re} 2 \delta_2 ^{(rr)}$ and the IR divergent part of $\operatorname{Im} 2 \delta_2 $ in the maximally supersymmetric case. Using unitarity and analyticity of the scattering amplitude, we provide arguments in favour of its general validity. We also outline the logic of the calculations that follow. In Sect. ~\ref{softb} we transform the soft-limit results of Sect. ~\ref{softm} to impact-parameter space in the large-$b$ limit. In Sect. ~\ref{probability} we use these to compute the divergent part of $\operatorname{Im} 2 \delta_2 $ and, through our connection, the RR terms in $\operatorname{Re} 2 \delta_2 $. This is first done for the case of ${\cal N}=8 $ supergravity, where we recover the result of~\cite{DiVecchia:2020 ymx}, and then for Einstein's gravity, reproducing the result of~\cite{Damour:2020 tta}, and for Jordan-Brans-Dicke theory. \section{Soft Amplitudes in Momentum Space}
\label{softm}
Let us start by better defining the processes under consideration. We shall be interested in the scattering of two massive scalar particles in $D=4 -2 \epsilon$ dimensions, with or without the additional emission of a soft massless quantum. For GR, we thus consider minimally coupled scalars with masses $m_1 $, $m_2 $ in $4 -2 \epsilon$ dimensions. For $\mathcal N=8 $ supergravity, that can be obtained by compactifying six directions in ten-dimensional type II supergravity, we instead choose incoming Kaluza--Klein (KK) scalars whose $(10 -2 \epsilon)$-dimensional momenta read as follows:
\begin{equation}
\label{eq:kin10 D}
P_1 = (p_1 ;0,0,0,0,0, m_1 )\,, \qquad P_2 = (p_2 ;0,0,0,0, m_2 \sin\phi, m_2 \cos\phi)\;,
\end{equation}
where the last six entries refer to the compact KK directions and provide $p_1 $, $p_2 $ with the desired effective masses $m_1 $, $m_2 $ in $4 -2 \epsilon$ dimensions. The angle $\phi$ thus describes the relative orientation between the KK momenta,
We work in a centre-of-mass frame and for our purposes it is convenient to regard the amplitudes as functions of $\bar{p}$, encoding the classical momentum of the massive particles, the transferred momentum $q$ (which is related to the impact parameter after Fourier transform) and the emitted momentum $k$. We thus parametrise the momenta of the incoming states as follows,
\begin{equation}
\label{eq:kin}
\begin{aligned}
p_1 &= (E_1, \vec{p}\, ) = \bar{p}_1 - a q + c k \,,
&&\bar{p}_1 = (E_1,0, \ldots,0, \bar{p}\, ) \,,
\\
p_2 &= (E_2, -\vec{p}\, ) = \bar{p}_2 + a q - c k \,,
&&\bar{p}_2 = (E_2,0, \ldots,0, -\bar{p}\, ) \,,
\end{aligned}
\end{equation}
while the outgoing\footnote{We treat all vectors as formally ingoing. } states are a soft particle of momentum $k$ and massive states with momenta
\begin{equation}
\label{eq:kin2 }
k_1 = -\bar{p}_1 - (1 -a) q - c k \,, \qquad \ \; \,
k_2 = -\bar{p}_2 + (1 -a) q - (1 -c) k \,. \end{equation}
We singled out the direction of the classical momentum $\bar{p}$, while $q$ is non-trivial only along the $2 -2 \epsilon$ space directions orthogonal to $\bar{p}_i$. In the elastic case of course $k=0 =c$ and we have $a=1 /2 $. For the inelastic amplitudes one can fix $a$ and $c$ by imposing the on-shell conditions and using $\bar{p}_i q=0 $, but we will not need their explicit expression in what follows. We shall now collect the tree-level amplitudes that will enter our calculation of $\operatorname{Im}\, 2 \delta_2 $ via unitarity, focusing for the most part on $\mathcal N=8 $ and commenting along the way on small amendments that are needed to obtain the GR amplitudes. The simplest building block for our analysis of $\mathcal N=8 $ supergravity is the elastic tree-level amplitude
\begin{equation}
A_{tree} \simeq - \frac{32 \pi G m_1 ^2 m_2 ^2 (\sigma - \cos \phi)^2 }{t}\,, \qquad\text{with } \sigma = - \frac{p_1 p_2 }{m_1 m_2 }\,,
\label{2.1 }
\end{equation}
where we retained only the terms with the pole at $t=-q^2 = 0 $, since we restrict our attention to long-range effects. When $\phi=\frac{\pi}{2 }$, the KK momenta are along orthogonal directions, and, in this case, the pole at $t = 0 $ corresponds to the exchange of the graviton and of the dilaton that are coupled universally to all massive states with the following three-point on-shell amplitudes in $D=4 $:
\begin{equation}
A_3 ^{\mu \nu} =-i \kappa ( p_j^\mu k_j^\nu + p_j^\nu k_j^\mu )\,, \qquad
A_3 ^{dil} = -i \kappa \sqrt{2 }\, m_j^2 \;,
\label{2.2 }
\end{equation}
with $j=1,2 $ and $\kappa = \sqrt{8 \pi G}$. Using the vertices~\eqref{2.2 } and standard propagators, the graviton and the dilaton exchanges yield
\begin{equation}
A^{gr}_{tree} \simeq - \frac{16 \pi G m_1 ^2 m_2 ^2 (2 \sigma^2 - 1 )}{t} ~, \quad A^{dil}_{tree} \simeq -\frac{16 \pi G m_1 ^2 m_2 ^2 }{t} ~. \label{2.1 b}
\end{equation}
Their sum reproduces~\eqref{2.1 } for $\phi=\frac{\pi}{2 }$. For generic $\phi$, in addition to the couplings mentioned above, we also need to consider massless vectors and scalars coming from the KK compactification of the ten dimensional graviton.
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_2 $ couples to another vector and another scalar with a strength depending to the other component of the KK momentum,
\begin{equation}
B_3 ^\mu = - i \kappa m_2 \sqrt{2 } (p_2 -k_2 )^\mu \sin\phi\,, \qquad B_3 = - i \kappa 2 m_2 ^2 \sin^2 \phi \,. \label{2.3 b}
\end{equation}
There is also an extra scalar related to the off-diagonal components of the internal metric whose coupling is proportional to $\cos\phi\sin\phi$; here we will not use this coupling as we will mainly focus on the cases $\phi=0 $ and $\phi=\frac{\pi}{2 }$. Let us now move to the inelastic, $2 \to 3 $ amplitude. As stressed in the introduction, we can restrict ourselves to the leading soft term that diverges as $k ^{-1 }$ for $k\to 0 $. It is given by the product of the elastic tree-level amplitude times a soft factor. For instance, the leading term for the emission of a soft graviton is \cite{Weinberg:1965 nx}:
\begin{equation}
\label{eq:grnde}
A_5 ^{\mu\nu} \simeq {\kappa} \left(\frac{p_1 ^\mu p_1 ^\nu}{p_1 k} + \frac{k_1 ^\mu k_1 ^\nu}{k_1 k} + \frac{p_2 ^\mu p_2 ^\nu}{p_2 k} + \frac{k_2 ^\mu k_2 ^\nu}{k_2 k} \right) A_{tree}\,,
\end{equation}
while in the case of the dilaton one finds\footnote{We neglect possible terms proportional to $\delta(\omega)$ which play no role in the present discussion. } \cite{DiVecchia:2015 jaq}
\begin{equation}
\label{eq:dinde}
A_5 ^{dil} \simeq -\frac{\kappa}{\sqrt{2 }} \left(\frac{m_1 ^2 }{p_1 k} + \frac{m_1 ^2 }{k_1 k} + \frac{m_2 ^2 }{p_2 k} + \frac{m_2 ^2 }{k_2 k} \right) A_{tree}\,. \end{equation}
We now use~\eqref{eq:kin} and~\eqref{eq:kin2 } and keep the leading terms in the soft limit $k\to 0 $. By further keeping only the {\em classical} contributions, which are captured by the linear terms in the $q \to 0 $ limit, one obtains
\begin{equation}
A_5 ^{\mu \nu} \simeq \kappa \left[ \left(\frac{\bar{p}_1 ^\mu \bar{p}_1 ^\nu}{(\bar{p}_1 k)^2 } - \frac{\bar{p}_2 ^\mu \bar{p}_2 ^\nu}{(\bar{p}_2 k)^2 } \right) (qk) - \frac{\bar{p}_1 ^\mu q^\nu+ \bar{p}_1 ^\nu q^\mu}{(\bar{p}_1 k) } + \frac{\bar{p}_2 ^\mu q^\nu+ \bar{p}_2 ^\nu q^\mu}{(\bar{p}_2 k) } \right] A_{tree}
\label{2.4 }
\end{equation}
for the graviton
and
\begin{equation}
\label{eq:dfe}
A_5 ^{dil} \simeq -\frac{\kappa}{\sqrt{2 }} \left( \frac{m_1 ^2 (qk)}{(\bar{p}_1 k)^2 } - \frac{m_2 ^2 (qk)}{(\bar{p}_2 k)^2 } \right)A_{tree}
\end{equation}
for the dilaton. From now on we focus for simplicity on the case $\phi=\frac{\pi}{2 }$ and so only the first line of~\eqref{2.3 } is non-trivial; together with the contribution of~\eqref{2.3 b} we need to consider the emission of the two vectors and of the two scalars. For the soft amplitudes we find:
\begin{equation}
A_5 ^\mu \simeq \kappa m_1 \sqrt{2 } \left( \frac{\bar{p}_1 ^\mu (qk)}{(\bar{p}_1 k)^2 } - \frac{q^\mu}{\bar{p}_1 k}\right) A_{tree} \,, \quad B_5 ^\mu \simeq \kappa m_2 \sqrt{2 } \left(- \frac{\bar{p}_2 ^\mu (qk)}{(\bar{p}_2 k)^2 } + \frac{q^\mu}{\bar{p}_2 k}\right)A_{tree} \; ,
\label{2.5 }
\end{equation}
\begin{equation}
A_5 \simeq \kappa m_1 ^2 \frac{(qk)}{(\bar{p}_1 k)^2 } A_{tree} \,, \qquad B_5 \simeq - \kappa m_2 ^2 \frac{(qk)}{(\bar{p}_2 k)^2 } A_{tree} \;. \label{2.6 }
\end{equation}
\section{Radiation Reaction from Infrared Singularities}
\label{RRIR}
In this section we briefly present our arguments for the validity, at two-loop level and for generic gravity theories, of the relation \eqref{1.5 }. We leave a more detailed discussion to a longer paper \cite{DiVecchia:2021 bdo}. Our starting point is an empirical observation made in the context of a recent calculation in ${\cal{N}}=8 $ supergravity \cite{DiVecchia:2020 ymx} whose set-up has been recalled in the previous section. An interesting outcome of that calculation (made for $\cos \phi =0 $) was the identification of a radiation-reaction contribution to the real part of the (two loop) eikonal phase, given by
\begin{equation}
{\rm Re}\,2 \delta_2 ^{(rr)} = \frac{16 G^3 m_1 ^2 m_2 ^2 \sigma^4 }{\hbar b^2 (\sigma^2 -1 )^2 } \Bigg[ \sigma^2 + \frac{\sigma (\sigma^2 -2 )}{(\sigma^2 -1 )^{\frac{1 }{2 }}}\cosh^{-1 } (\sigma)\Bigg] + {\cal O}(\epsilon)\; . \label{Redel2 }
\end{equation}
This contribution emerges from the inclusion of radiation modes in the loop integrals and gives rise to half-integer-PN corrections to the deflection angle. Considering the full massive ${\cal N}=8 $ result~\cite{DiVecchia:2021 bdo}, we then noticed a simple relation between the contribution in eq. ~\eqref{Redel2 } and two terms appearing in the imaginary part of the same eikonal phase so that, in the full expression for $\delta_2 ^{(rr)}$, there are three terms that appear in the following combination:
\begin{equation}
\left[
1 + \frac{i}{\pi}
\left(
- \frac{1 }{\epsilon} + \log(\sigma^2 -1 )
\right)
\right]
{\rm Re}\,2 \delta_2 ^{(rr)}. \label{comb2 }
\end{equation}
The two imaginary contributions to $2 \delta_2 ^{(rr)}$ that appear in \eqref{comb2 } are an IR-singular term, which captures the full contribution proportional to $\epsilon^{-1 }$, and a $\log(\sigma^2 -1 )$ term, which captures the branch cuts starting at $\sigma=\pm1 $. Let us now examine whether this feature is to be regarded as an accident of the maximally supersymmetric theory or rather as a more general fact. As we shall discuss below and will explain in more detail in \cite{DiVecchia:2021 bdo}, the precise combination of the two imaginary terms in the round bracket of \eqref{comb2 } is dictated by the three-particle unitarity cut, where the phase space integration over the soft momentum of the massless quantum is responsible for the infrared singularity in ${\rm Im}\,2 \delta_2 $ (let us recall that ${\rm Im}\,2 \delta_2 $ contains just the inelastic contribution to the cut \cite{Amati:1990 xe}). Furthermore, using real-analyticity of the amplitude forces the $\log (\sigma^2 -1 )$ to appear in $\delta_2 $ as $\log (1 -\sigma^2 ) = \log (\sigma^2 -1 ) - i \pi$ yielding precisely the analytic structure of \eqref{comb2 }. Combining these two observations, which are based purely on unitarity, analyticity and crossing symmetry, we are led to conjecture the validity of \eqref{1.5 } independently of the specific theory under consideration. As anticipated, this relation opens the way to a much simpler calculation of RR effects since it trades the computation of $\operatorname{Re}2 \delta_2 ^{(rr)}$ to that of the IR-divergent part of ${\rm Im}\,2 \delta_2 $. In the following sections we will carry out this calculation both for the supersymmetric case at hand, for pure gravity where we shall recover a recent result by Damour~\cite{Damour:2020 tta}, and for the scalar-tensor theory of Jordan-Brans-Dicke. For the purpose of computing the IR-divergent piece in ${\rm Im}\,2 \delta_2 $, one can focus on the leading ${\cal O}(k^{-1 })$ term in the soft expansion of the inelastic amplitudes given in Sect~\ref{softm}. This allows us to factor out, for each specific theory, the corresponding elastic amplitude. Next, and in this order, one has to take the leading term in a small-$q$ expansion so as to get the sought-for classical contribution. In terms of the impact parameter $b$ which will be introduced in~\eqref{eq:ft}, the small-$q$ limit is equivalent to an expansion for large values of $b$. Since the soft factor is linear in $q$ (it goes to zero at zero scattering angle), and the tree amplitude has a $q^{-2 }$ singularity, the result for the inelastic amplitude is (modulo $\epsilon$ dependence) of $\mathcal O(b^{-1 })$ and thus of the desired $\mathcal O(b^{-2 })$ in ${\rm Im}\, 2 \delta_2 $. \section{Soft Amplitudes in $b$-space}
\label{softb}
We now start from the momentum space soft amplitudes given in Sect. ~2 and go to impact parameter space using for a generic amplitude the notation
\begin{equation}\label{eq:ft}
\tilde A(b) = \int \frac{d^{2 -2 \epsilon} q}{(2 \pi)^{2 -2 \epsilon}}\frac{A(q)}{4 m_1 m_2 \sqrt{\sigma^2 -1 }}\, e^{i b \cdot q}\;. \end{equation}
We can now simply replace the factors of $q_j$ in the numerators of the various amplitudes $A_5 $ by the derivative $- i \frac{\partial}{\partial b^j}$ and then perform the Fourier transform where the $q$-dependence appears only in $A_{tree}$. Starting from the ${\cal N}=8 $ elastic tree-level amplitude with $\phi=\frac{\pi}{2 }$, given, up to analytic terms as $q^2 \to0 $, by
\begin{equation}
A_{tree} = 8 \pi \beta(\sigma) \frac{m_1 m_2 }{q^2 } \,,
\qquad
\beta(\sigma) = 4 G m_1 m_2 \sigma^2 \,,
\label{A0 }
\end{equation}
the leading eikonal takes the form
\begin{equation}
2 \delta_0 = -\beta(\sigma) \frac{\Gamma(1 -\epsilon) (\pi b^2 )^{\epsilon}}{2 \epsilon \hbar \sqrt{\sigma^2 -1 }} ~ \Rightarrow - i \frac{\partial}{\partial b^j} 2 \delta_0 = \frac{ i\, \Gamma(1 -\epsilon)\, b^j (\pi b^2 )^{\epsilon}}{ b^2 \hbar \sqrt{\sigma^2 -1 }} \beta(\sigma)\;. \label{delta0 }
\end{equation}
As clear from~\eqref{2.1 b}, one can move from $\mathcal N=8 $ to the case of pure GR simply by replacing the prefactor $\beta(\sigma)$ by
\begin{equation}
\label{eq:tAd}
\beta^{GR}(\sigma) = 2 G m_1 m_2 (2 \sigma^2 -1 )\;. \end{equation}
We then obtain the following result for the classical part of the soft graviton and soft dilaton amplitudes in impact parameter space
\begin{align}
\label{2.81 l}
\begin{split}
{\tilde{A}}_5 ^{\mu \nu} (\sigma, b, k) &\simeq {i \frac{\kappa \beta(\sigma)(\pi b^2 )^{\epsilon}}{b^2 \sqrt{\sigma^2 -1 }}} \\
&\times \left[ (k b)\left( \frac{\bar{p}_1 ^\mu \bar{p}_1 ^\nu}{(\bar{p}_1 k)^2 } - \frac{\bar{p}_2 ^\mu \bar{p}_2 ^\nu}{(\bar{p}_2 k)^2 }\right)- \frac{\bar{p}_1 ^\mu {b}^\nu+ \bar{p}_1 ^\nu {b}^\mu}{ (\bar{p}_1 k) } + \frac{\bar{p}_2 ^\mu {b}^\nu+ \bar{p}_2 ^\nu {b}^\mu}{ (\bar{p}_2 k) } \right]\,,
\end{split}
\\
{\tilde{A}}_5 ^{dil} (\sigma, b, k) &\simeq - {i \frac{\kappa \beta(\sigma)(\pi b^2 )^{\epsilon}}{ \sqrt{2 (\sigma^2 -1 )}}} {\frac{(k b)}{b^2 } } \left[ \frac{m_1 ^2 }{(\bar{p}_1 k)^2 } - \frac{m_2 ^2 }{(\bar{p}_2 k)^2 } \right] \,,
\label{2.82 l}
\end{align}
where we approximated the factor of $\Gamma(1 -\epsilon)$ in~\eqref{delta0 } to 1 as we are interested in the $D\to 4 $ case, but we continue to keep track of the dimensionful factor of $b^{2 \epsilon}$. Having obtained Eqs. ~\eqref{2.81 l}, \eqref{2.82 l} with the appropriate normalization, we follow the same procedure to go over to $b$-space for the other fields relevant to the ${\cal N}=8 $ analysis.
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\mu}{\bar{p}_2 k}\right],
\label{2.10 }
\end{align}
while for the two scalars we get
\begin{eqnarray}
{\tilde{A}}_5 \simeq{i \frac{\kappa m_1 ^2 \beta(\sigma)(\pi b^2 )^{\epsilon}}{ b^2 \sqrt{\sigma^2 -1 }} } \frac{(k b)}{(\bar{p}_1 k)^2 } \,, \qquad
{\tilde{B}}_5 \simeq - {i \frac{\kappa m_2 ^2 \beta(\sigma)(\pi b^2 )^{\epsilon}}{ b^2 \sqrt{\sigma^2 -1 }} } \frac{(k b)}{(\bar{p}_2 k)^2 }\,. \label{2.11 }
\end{eqnarray}
Note that all our soft amplitudes are homogeneous functions of $\omega$ and $b$ of degree $-1 $ and $-1 + 2 \epsilon$, respectively. \section{IR Divergence of the 3 PM Eikonal}
\label{probability}
Motivated by the discussion of Sect. ~\ref{RRIR} and armed with the results of the Sect. ~\ref{softb}, we now turn to the calculation of the infrared divergent part of ${\rm Im} \,2 \delta_2 $ from the three-particle unitarity cut. Indeed the unitarity convolution in momentum space diagonalizes in impact parameter space giving (see e. g. \cite{Amati:2007 ak})
\begin{equation}
2 {\rm Im} \,2 \delta_2 = \sum_i \int \frac{d^{D-1 } \vec k}{2 |\vec{k}| (2 \pi)^{D-1 }} | \tilde{A}_{5 i} |^2 \,,
\label{3.1 }
\end{equation}
where the sum is over each massless state in the theory under consideration. For spin-one and spin-two particles this also includes a sum over helicities. Instead of separating different helicity contributions, we use the fact that all the $2 \to 3 $ amplitudes we use are gauge invariant/transverse and simply insert the corresponding on shell Feynman and de Donder propagators, {\em i. e. } $\eta^{\mu\nu}$ for the vectors and $\frac{1 }{2 } \left(\eta^{\mu\rho} \eta^{\nu\sigma} + \eta^{\mu \sigma} \eta^{\nu \rho} - \eta^{\mu\nu} \eta^{\rho\sigma} \right)$ for the graviton. Equation \eqref{3.1 } implies that $\beta^2 (\sigma)$ always factors out of the integral over $\vec k$. In spherical coordinates the latter splits into an integral over the modulus $|\vec{k}|$ and one over the angles defined by the following parametrisation of the vector $\vec{k}$:
\begin{equation}
\vec{k} = |\vec{k}| ( \sin \theta \cos \varphi, \, \sin \theta \sin \varphi, \, \cos \theta)\,, \qquad
(k b) = - |\vec{k}| b \sin \theta \cos \varphi\, ,
\label{3.1 a}
\end{equation}
that implies
\begin{equation}
\label{eq:pkang}
(\bar{p}_1 k) = |\vec{k}| (E_1 - \bar{p} \cos\theta) \,, \qquad
(\bar{p}_2 k) = |\vec{k}| (E_2 + \bar{p} \cos\theta)\, ,
\end{equation}
where we have taken $b$ in \eqref{3.1 a} along the $x$ axis. It is clear that the integral over $|\vec{k}| = \hbar \omega$ in~\eqref{3.1 } factorises together with an $\epsilon$-dependent power of $b$ to give\footnote{We need to keep $D=4 -2 \epsilon$ only for the integral over $|\vec{k}|$ while the integration over the angular variables can be done for $\epsilon=0 $, so that effectively $d^{D-1 } \vec k = |\vec{k}|^{2 -2 \epsilon} d|\vec{k}|\, \sin \theta \, d \theta\, d \varphi$. }
\begin{equation}
2 {\rm Im} \,2 \delta_2 \sim \int \frac{d \omega}{\omega} \omega^{- 2 \epsilon} (b^2 )^{-1 +2 \epsilon} \sim (b^2 )^{-1 + 3 \epsilon} \int \frac{d \omega}{\omega} (\omega b)^{- 2 \epsilon}
\label{eq:of}
\end{equation}
where the factor $ (b^2 )^{-1 + 3 \epsilon}$ is precisely the one expected (also on dimensional grounds) to appear in $\delta_2 $. On the other hand, the integral over $\omega$ produces a $\frac{1 }{\epsilon}$ divergence in the particular combination:
\begin{equation}
\int \frac{d \omega}{\omega} (\omega b)^{- 2 \epsilon} = - \frac{1 }{2 \epsilon} (\, \overline{\omega b}\, )^{-2 \epsilon} = -\frac{1 }{2 \epsilon} + \log \overline{\omega b}+ {\cal O}(\epsilon)
\label{comb}
\end{equation}
where $\overline{\omega b}$ is an appropriate upper limit on the classical dimensionless quantity $\omega b$. To determine $\overline{\omega b}$ one can argue as follows. By energy conservation:
\begin{equation}
\label{hom}
\hbar \omega = \Delta E_1 + \Delta E_2
\end{equation}
where $\Delta E_i$ is the energy loss for the $i^{\rm th}$ particle. On the other hand, in order for the spatial components of the momentum transfers $q_i = -(p_i + k_i)$ to provide a classical contribution, they should be of order $\hbar/b \ll |\vec{p}_i|$. But then we can estimate \eqref {hom} by using (for on-shell particles):
\begin{equation}
\label{onshell}
\Delta E_i \lesssim\frac{ |\vec{p}_i|}{E_i} |\Delta \vec{p}_i| \qquad (i = 1,2 )\,. \end{equation}
Combining \eqref{hom} and \eqref{onshell} we arrive at
\begin{equation}
\omega b \lesssim \frac{ |\vec{p}_1 |}{E_1 }+ \frac{ |\vec{p}_2 |}{E_2 }\,. \end{equation}
Using now the following (centre-of-mass) expressions,
\begin{equation}
\label{eq:rws}
\begin{gathered}
\bar{p} \; \simeq \; |\vec{p}\, | = \frac{m_1 m_ 2 \sqrt{\sigma^2 -1 }}{\sqrt{m_1 ^2 +m_2 ^2 +2 m_1 m_2 \sigma}}\;, \\
E_1 = m_1 \frac{m_1 + \sigma m_2 }{\sqrt{m_1 ^2 +m_2 ^2 +2 m_1 m_2 \sigma}}\;, \quad
E_2 = m_2 \frac{m_2 + \sigma m_1 }{\sqrt{m_1 ^2 +m_2 ^2 +2 m_1 m_2 \sigma}}\;,
\end{gathered}
\end{equation}
we find:
\begin{equation}
\overline{\omega b} \sim \sqrt{\sigma^2 -1 } \left( \frac{m_2 }{m_1 + \sigma m_2 } + \frac{m_1 }{m_2 + \sigma m_1 } \right) = \sqrt{\sigma^2 -1 }( 1 + {\cal O}(\sigma -1 ))\;. \label{baromb}
\end{equation}
Therefore, inserting this result in \eqref{comb} and using the real-analyticity argument mentioned in Sect. ~3, precisely the combination appearing in \eqref{comb2 } is indeed recovered. This is the essence of our argument for conjecturing \eqref{comb2 } as a general connection between RR and soft limits. The rest of this section provides examples and non trivial tests of such a connection. \subsection{Massive ${\cal{N}}=8 $ Supergravity}
We evaluate separately the ${\cal O}(\epsilon^{-1 })$ contribution to~\eqref{3.1 } for each massless state: the graviton, the dilaton, two vectors and two scalars coupling to the particle of mass $m_1 $ and other two vectors and two scalars coupling to the particle of mass $m_2 $. We first start from the dilaton contribution. By using~\eqref{2.82 l} in~\eqref{3.1 } we obtain
\begin{equation}
\label{eq:dp1 }
({\rm Im} \,2 \delta_2 )_{dil} \simeq \frac{\kappa^2 {\beta}^2 (\sigma)}{4 b^2 (\sigma^2 -1 )} \int \frac{d |\vec{k}| |\vec{k}|^{-2 \epsilon-1 } }{2 (2 \pi)^{3 }} \. \int_{-1 }^1 \. \. \. dx\, \pi (1 -x^2 )\. \left[ \frac{m_1 ^2 }{({E_1 } -\bar{p} x)^2 } - \frac{m_2 ^2 }{({E_2 } +\bar{p} x)^2 } \right]^2 \. \. \!,
\end{equation}
where $x=\cos\theta$. The extra factor of $\pi\sin^2 \theta=\pi (1 -x^2 ) $ in the integrand follows from the integration over the angle $\varphi$. As already mentioned the integral over $|\vec{k}|$ factorises out of the whole integral and provides the sought for $\epsilon^{-1 }$ factor. Finally, by using~\eqref{eq:rws}, we express everything in terms of $\sigma$ introduced in \eqref{2.1 }. Then, using~\eqref{eq:rws} in~\eqref{eq:dp1 } and performing the integral over $x$, we obtain
\begin{eqnarray}
({\rm Im} \,2 \delta_2 )_{dil} (\sigma, b) \simeq -\frac{1 }{2 \epsilon } \frac{G {\beta}^2 (\sigma)}{\pi \hbar b^2 (\sigma^2 -1 )^2 }
\left[ \frac{\sigma^2 +2 }{3 } - \frac{\sigma}{ (\sigma^2 -1 )^{\frac{1 }{2 }} }
\cosh^{-1 } (\sigma) \right] . \label{3.3 }
\end{eqnarray}
Note that the final result depends on the masses only through $\sigma$ even if the integrand depends on $m_1, m_2 $ and $\sigma$ separately. The term with the factor of $\cosh^{-1 }(\sigma)$ emerges from the cross-product of the square in~\eqref{eq:dp1 }, while the other terms yield only rational contributions in $\sigma$. For the graviton's contribution, using~\eqref{2.81 l} in~\eqref{3.1 }, we obtain %
\begin{align}
({\rm Im} \, & 2 \delta_2 )_{gr} (\sigma, b) \simeq - \frac{\kappa^2 {\beta}^2 (\sigma)}{2 b^2 (\sigma^2 -1 )} \left(-\frac{1 }{2 \epsilon} \, \frac{1 }{2 (2 \pi)^{3 }} \right)
\pi \int_{-1 }^1 \. \. dx \nonumber \\
\times & \Bigg\{4 \left[ \frac{m_1 ^2 }{(E_1 -\bar{p}x)^2 } + \frac{m_2 ^2 }{(E_2 +\bar{p} x)^2 } - \frac{2 m_1 m_2 \sigma}{(E_1 -\bar{p} x) (E_2 + \bar{p} x)} \right] \\ \nonumber
& - \frac{1 -x^2 }{2 } \left[ \frac{m_1 ^4 }{(E_1 - \bar{p} x)^4 } + \frac{m_2 ^4 }{(E_2 + \bar{p}x)^4 } - \frac{2 m_1 ^2 m_2 ^2 (2 \sigma^2 -1 )}{(E_1 -\bar{p} x)^2 (E_2 +\bar{p} x)^2 }\right] \Bigg\}\,. \end{align}
The integral over $x$ is again elementary\footnote{Surprisingly, it turns out to be the same as the integral appearing in Eq. ~(4.4 ) of~\cite{Damour:2020 tta} and thus reproduces exactly the function ${\cal I}$ in~(4.7 ) of that reference. }. In terms of the variable $\sigma$ we obtain:
\begin{equation}
({\rm Im} \,2 \delta_2 )_{gr} (\sigma, b) \simeq -\frac{1 }{2 \epsilon } \frac{G {\beta}^2 (\sigma)}{\pi \hbar b^2 (\sigma^2 -1 )^2 }
\left[ \frac{ 8 -5 \sigma^2 }{3 } - \frac{\sigma(3 -2 \sigma^2 )}{(\sigma^2 -1 )^{\frac{1 }{2 }}} \cosh^{-1 } (\sigma) \right] . \label{3.2 }
\end{equation}
Following the same procedure for the contribution of the two vectors in~\eqref{2.5 } we get
\begin{eqnarray}
({\rm Im} \,2 \delta_2 )_{vec} (\sigma, b) \simeq -\frac{1 }{2 \epsilon} \frac{G {\beta}^2 (\sigma)}{\pi \hbar b^2 (\sigma^2 -1 )^2 } \left[\frac{8 }{3 } (\sigma^2 -1 )\right]
\label{3.4 }
\end{eqnarray}
and for the sum of the two scalars in~\eqref{2.6 } we obtain
\begin{eqnarray}
({\rm Im} \,2 \delta_2 )_{sca} (\sigma, b)\simeq -\frac{1 }{2 \epsilon} \frac{G {\beta}^2 (\sigma)}{\pi \hbar b^2 (\sigma^2 -1 )^2 } \left[\frac{2 }{3 } (\sigma^2 -1 )\right]. \label{3.5 }
\end{eqnarray}
In the last two types of contributions the soft particles are attached to the same massive state, so there are no terms in the integrand with the structure appearing in the cross term of~\eqref{eq:dp1 } and hence no factors of $\cosh^{-1 }(\sigma)$ in the final result. Also the graviton and the dilaton results contain contributions of this type corresponding to the terms in the integrands which depend only on $E_1 $ or $E_2 $. In the ${\cal N}=8 $ setup these contributions cancel when summing over all soft particles. Notice also that the static limit $\sigma\to 1 $ of~\eqref{3.4 } and~\eqref{3.5 } is qualitatively different from that of the full graviton and dilaton contributions as it starts one order earlier. Then thanks to~\eqref{1.5 } also the leading term of the PN expansion of the ${\cal N}=8 $ eikonal or deflection angle is due to the vectors and the scalars in~\eqref{2.3 } and~\eqref{2.3 b}.
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in pure GR follows exactly the same steps with only the contribution of the graviton and yields again the result in Eq. ~\eqref{3.2 } just with the prefactor $({\beta}^{GR}(\sigma))^2 $ in place of ${\beta}^2 (\sigma)$. Then, assuming that Eq. ~\eqref{1.5 } is also valid in GR, we get
\begin{equation}
({\rm Re} \,2 \delta^{(rr)}_2 )_{GR} (\sigma, b) = \frac{G ({\beta}^{GR}(\sigma))^2 }{2 \hbar b^2 (\sigma^2 -1 )^2 }
\left[ \frac{ 8 -5 \sigma^2 }{3 } - \frac{\sigma(3 -2 \sigma^2 )}{(\sigma^2 -1 )^{\frac{1 }{2 }}} \cosh^{-1 } (\sigma) \right]
\label{3.2 b}
\end{equation}
and, from it, we obtain the deflection angle
\begin{equation}
\label{eq:6.6 }
(\chi^{(rr)}_3 )_{GR} = - \frac{\hbar}{|\vec{p}|} \frac{\partial {\rm Re} 2 \delta^{(rr)}_2 }{\partial b} = \frac{G ({\beta}^{GR}(\sigma))^2 }{|\vec{p}| b^3 (\sigma^2 -1 )^2 }
\left[ \frac{ 8 -5 \sigma^2 }{3 } - \frac{\sigma(3 -2 \sigma^2 )}{(\sigma^2 -1 )^{\frac{1 }{2 }}} \cosh^{-1 } (\sigma) \right]
\end{equation}
which reproduces the one given in Eq. (6.6 ) of~\cite{Damour:2020 tta}. At the moment, the physical reason for this agreement is unclear.
The results obtained so far allow one to derive in a straightforward way the zero-frequency limit (ZFL) of the energy spectrum $\frac{dE^{rad}}{d\omega}$. Indeed, the energy spectrum is just the integrand of~\eqref{3.1 } for the graviton multiplied by an extra factor of $ \hbar \omega$ (see also~\cite{Goldberger:2016 iau}) so that,
\begin{equation}
\label{eq:EradG3 }
E^{\rm rad} = \int \frac{d^{D-1 } k}{2 (2 \pi)^{D-1 }} \tilde{A}^{*\, \mu\nu}_{5 }\left(\eta_{\mu\rho} \eta_{\nu\sigma} - \frac{1 }{2 } \eta_{\mu\nu} \eta_{\rho\sigma}\right) \tilde{A}_5 ^{\rho\sigma} \equiv \int_0 ^\infty\! \! \! d\omega \, \frac{d E^{rad}}{d\omega}\;.
\end{equation}
Since we computed only the $k\to 0 $ limit of this integrand, we can reliably extract just the ZFL
\begin{equation}
\frac{d E^{rad}}{d\omega} ( \omega\to 0 ) = \lim_{\epsilon\to 0 } \left[-4 \hbar \epsilon ({\rm Im} 2 \delta_2 )\right] \,.
\label{3.7 }
\end{equation}
In the case of GR we can use~\eqref{3.2 } with $({\beta}^{GR}(\sigma))^2 $ in place of ${\beta}^2 (\sigma)$ and reproduce Eq. ~(2.11 ) of~\cite{Kovacs:1978 eu} (taken from~\cite{Ruffini:1970 sp}) by taking the static limit $\sigma \to 1 $
\begin{equation}
\frac{dE}{d \omega} ( \omega \to 0 ) = \frac{32 G^3 m_1 ^2 m_2 ^2 }{5 \pi b^2 }\;.
\label{3.12 }
\end{equation}
Our result \eqref{3.7 } should hold true\footnote{T. Damour kindly informed us that he has carried out the explicit check. } at all values of $\sigma$, extending Smarr's original result \cite{Smarr:1977 fy} to arbitrary kinematics (see~\cite{Kovacs:1978 eu}). Possibly, our approach can be extended to compute the energy spectrum to sub and sub-sub leading order in $\omega$ and to reproduce, in particular cases, the results of~\cite{Sahoo:2018 lxl}, \cite{Addazi:2019 mjh} and~\cite{Saha:2019 tub}.
On the other hand, our method looks inadequate to deal with the full spectrum and with the total energy loss\footnote{Such a calculation has been recently tackled by a different approach in \cite{Herrmann:2021 lqe}. }. For instance, extrapolating the ZFL result \eqref{3.7 } to the upper limit given in \eqref{baromb} would reproduce, at large $\sigma$, the qualitative behaviour of Eq. (5.10 ) of \cite{Kovacs:1978 eu}. But, as anticipated to be the case in \cite{Kovacs:1978 eu}, and discussed in \cite{Gruzinov:2014 moa} and \cite{Ciafaloni:2018 uwe}, such a result needs to be amended, as in the ultra-relativistic/massless limit, at fixed $G$, it would violate energy conservation.
Our connection between RR and soft limits readily applies to Jordan-Brans-Dicke (JBD) scalar-tensor theory.
The coupling of the massless scalar to massive particles is very much like that of the dilaton except for a rescaling of the coupling by a function of the JBD parameter $\omega_J$
(the coefficient of the JBD kinetic term):
\begin{equation}
g_{JBD} = \frac{1 }{\sqrt{2 \omega_J + 3 }}\, g_{dil}
\label{JBDcoup}\,.
\end{equation}
The string dilaton case is recovered for $\omega_J = -1 $.
It is then straightforward to calculate the RR in JBD theory. It amounts to inserting in~\eqref{3.2 } and in~\eqref{3.3 } the JBD $\beta(\sigma)$ factor,
\begin{equation}
\beta^{JBD}(\sigma) = 4 G m_1 m_2 \left(\sigma^2 - \frac{\omega_J +1 }{2 \omega_J + 3 }\right),
\label{JBDbeta}
\end{equation}
and to further multiplying the dilaton's contribution of \eqref{3.3 } by a factor $(2 \omega_J + 3 )^{-2 }$. Thus the contribution to the radiation reaction part of the eikonal from the JBD scalar reads
\begin{equation}
\frac{G {(\beta^{JBD})}^2 (2 \omega_J + 3 )^{-2 }}{2 \hbar b^2 (\sigma^2 -1 )^2 }
\left[ \frac{\sigma^2 +2 }{3 } - \frac{\sigma}{ (\sigma^2 -1 )^{\frac{1 }{2 }} }
\cosh^{-1 } (\sigma) \right] .
\label{JBDdelta}
\end{equation}
In the limit $\omega_J \to \infty$, this result vanishes leaving just the contribution of the graviton and thus reproducing the GR result. Since the present lower limit on $\omega_J$ is about $4 \times 10 ^4 $ the effect is unfortunately unobservable.
\subsection*{Acknowledgements}
We thank Enrico Herrmann, Julio Parra-Martinez, Michael Ruf and Mao Zeng for sharing with us a first draft of their paper \cite{Herrmann:2021 lqe} and for useful comments on ours.
We also thank Zvi Bern, Emil Bjerrum-Bohr, Poul Henrik Damgaard, Thibault Damour, Henrik Johansson, Rafael Porto and Ashoke Sen for valuable observations on a preliminary version of this letter.
The research of RR is partially supported by the UK Science and Technology Facilities Council (STFC) Consolidated Grant ST/P000754 /1 ``String theory, gauge theory and duality''. The research of CH (PDV) is fully (partially) supported by the Knut and Alice Wallenberg Foundation under grant KAW 2018.0116.
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\section{Introduction}
\subsection{Innermost stable circular orbits in General Relativity}
Let us consider a classical test non-spinning particle moving at stable circular orbit around a central massive body. In Newtonian theory, this orbit can have arbitrary radius. This follows from the fact that the effective potential of particle always has minimum, for any value of particle angular momentum, and if angular momentum tends to zero the radius of stable circular orbit goes to zero too \cite{Zeld-Novikov1971 }, see Fig. \ref{fig-U-Newt}. All circular orbits are stable till zero radius, and in Newtonian theory there is no minimum radius of stable circular orbit \cite{Kaplan}. \begin{figure}
\centerline{\hbox{\includegraphics[width=0.45 \textwidth]{U-Newt. eps}}} \caption{Newtonian effective potential $U = - M/r + L^2 /2 r^2 $ for a test particle moving in the gravitational field of a central body with mass $M$, for different values of $L$ ($L$ is the angular momentum per unit mass, $G=1 $). Solid circles show positions of minima corresponding to stable circular orbits. } \label{fig-U-Newt}
\end{figure}
In General Relativity (GR) the situation is different. The effective potential has a more complicated form, depending on the particle angular momentum, see Fig. \ref{fig-U-Schw} for the potential in the Schwarzschild metric. For large values of the angular momentum the effective potential has two extrema: maximum which corresponds to unstable circular orbit, and minimum which corresponds to stable circular orbit. With decreasing of angular momentum, radii of unstable and stable circular orbits become closer to each other. When angular momentum reach a boundary value, two extrema of effective potential merge into one inflection point. This point corresponds to minimal possible radius of stable circular orbit. Such orbit is called the innermost stable circular orbit (ISCO). Further decreasing of angular momentum leads to potential without extrema. For these values of angular momentum neither type of finite motion is possible. \begin{figure}
\centerline{\hbox{\includegraphics[width=0.45 \textwidth]{U-Schw1. eps}}} \caption{Effective potential (per unit particle rest mass) for motion in Schwarzschild metric $U_{Schw}= \sqrt{\left( 1 -\frac{2 M}{r} \right) \left( 1 +\frac{L^2 }{r^2 } \right)}$ for different values of $L$ ($L$ is the angular momentum per unit particle rest mass, $G=1 $). Maxima of potential are shown by circles, and minima are shown by solid circles. } \label{fig-U-Schw}
\end{figure}
Radius and other values of the ISCO parameters (total angular momentum, energy, orbital angular frequency) are different in different metrics. For the Schwarzschild background the radius of ISCO equals to $6 M$\footnote{In this paper we use the system of units where $G=c=1 $, the Schwarzschild radius $R_S=2 M$, and other physical quantities which will be introduced further have the following dimensionalities: $[L]=[M]$, $[J]=[M]$, $[E]=1 $, $[a]=[M]$, $[s]=[M]$. } , it was found by Kaplan \cite{Kaplan}, see also \cite{LL2 }. In the Kerr space-time circular motion is possible only in the equatorial plane of BH and the radius of ISCO depends on the direction of motion of the particle in comparison with the direction of BH rotation, whether they co-rotate or counter-rotate. Co-rotation and counter-rotation cases correspond to parallel and antiparallel orientation of vectors of the orbital angular momentum of the particle and the BH angular momentum. In a case of orbital co-rotation the ISCO radius becomes smaller than $6 M$, in case of counter-rotation -- bigger, see Fig. \ref{fig-schw-kerr}. For the case of the extreme Kerr background the difference between these two variants is quite considerable: we have $9 M$ for the antiparallel and $M$ for the parallel orientation. The parameters of ISCO in the Kerr space-time for a non-spinning particle were obtained in works of Ruffini \& Wheeler \cite{Ruffini1971 } and Bardeen, Press \& Teukolsky \cite{Bardeen1972 }. This problem is described at length, for example, in the textbook by Hobson \textit{et al. } \cite{Hobson}. \begin{figure}
\centerline{\hbox{\includegraphics[width=0.45 \textwidth]{SchwKerr. eps}}} \caption{Innermost stable circular orbits in Schwarzschild and Kerr metric. In Kerr metric radius of ISCO depends on direction of orbital motion of test particle. } \label{fig-schw-kerr}
\end{figure}
\subsection{Spinning particles in General Relativity}
In GR a rotation of the central gravitating body influences a motion of the particle orbiting it. Due to this reason the orbits of test particles in Kerr metric differ from orbits in Schwarzschild metric. When, in turn, a test particle has spin as well, it will also influence the particle's orbit. In particular, the motion of a spinning particle will differ from the non-spinning one even in the Schwarzschild background. The problem of the motion of a classical spinning test body in GR was considered in papers of Mathisson \cite{Mathisson1937 }, Papapetrou \cite{Papapetrou1951 a} and Dixon \cite{Dixon1970 a, Dixon1970 b, Dixon1978 }, using different techniques. The equations of motion of a spinning test particle in a given gravitational field were derived in different forms; they are now referred to as Mathisson-Papapetrou-Dixon equations. From these equations it follows that the motion of the centre of mass and the particle rotation are connected to each other, and when the particle has spin the orbits will differ from geodesics of a spinless massive particle. \subsection{The ISCO of spinning particles}
Influence of a spin on the orbits in the Schwarzschild metric was investigated in the paper of Corinaldesi and Papapetrou \cite{Papapetrou1951 b}, and in the paper of Micoulaut \cite{Micoulaut1967 }. A motion of a spinning test particle in Kerr metric was considered by Rasband \cite{Rasband1973 } and Tod, de Felice \& Calvani \cite{Tod1976 }, in particular ISCO radius was calculated numerically, see also papers of Abramowicz \& Calvani \cite{Abramowicz1979 } and Calvani \cite{Calvani1980 }. Subsequently the number of works on this subject were published \cite{Hojman1977, Suzuki1997, Suzuki1998, SaijoMaeda1998, TanakaMino1996, Apostolatos1996, Semerak1999, Semerak2007, Plyatsko2012 a, Plyatsko2012 b, Plyatsko2013, Bini2004 a, Bini2004 b, Bini2011 a, Bini2011 b, BiniDamour2014, Damour2008, Faye2006, Favata2011, Steinhoff2011, Steinhoff2012, Hackmann2014, Kunst2015, Putten1999 }. The detailed derivation of the equations of motion of spinning particle in Kerr space-time is presented in the work of Saijo \textit{et al. } \cite{SaijoMaeda1998 }. Method of calculation of ISCO parameters of spinning particle moving in Kerr metric is presented in details \cite{Favata2011 }. Linear corrections in spin for the ISCO parameters in Schwarzschild metric have been found by Favata \cite{Favata2011 }. In paper of Jefremov, Tsupko \& Bisnovatyi-Kogan \cite{Jefremov2015 } we have analytically obtained the small spin corrections for the ISCO parameters for the Kerr metric at arbitrary value of Kerr parameter $a$. The cases of Schwarzschild, slowly rotating and extreme Kerr black hole were considered in details. For a slowly rotating black hole the ISCO parameters are obtained up to quadratic in $a$ and particle's spin $s$ terms. For the extreme $a=M$ and almost extreme $a=(1 -\delta)M$ Kerr BH we succeeded to find the exact analytical solution for the ISCO parameters for arbitrary spin, with only restrictions connected with applicability of Mathisson-Papapetrou-Dixon equations. It has been shown that the limiting values of ISCO radius and frequency for $a=M$ do not depend on the particle's spin while values of energy and total angular momentum do depend on it. In this work we review some results of our recent research of innermost stable circular orbits (ISCO) \cite{Jefremov2015 } and present some new calculations. ISCO radius, total angular momentum, energy, orbital angular frequency are considered. We calculate the ISCO parameters numerically for different values of Kerr parameter $a$ and investigate their dependence on both black hole and test particle spins. Then we describe in details how to calculate analytically small-spin corrections to ISCO parameters for arbitrary values of $a$, presenting our formulae in different forms. \section{The motion of a spinning test body in the equatorial plane of a Kerr black hole} \label{section-MPD}
In the present treatment of the problem of spinning body motion in GR we use the so-called "pole-dipole" approximation \cite{Papapetrou1951 a}, in the frame of which the motion is described by the Mathisson-Papapetrou-Dixon (MPD) equations \cite{Papapetrou1951 a, Mathisson1937, Dixon1970 a, Dixon1970 b, Dixon1978, SaijoMaeda1998 }:
\begin{equation}
\begin{split}
&\frac{Dp^\mu}{D\tau}=-\frac{1 }{2 }R^{\mu}{}_{\nu \rho \sigma}v^{\nu}S^{\rho \sigma} , \\
&\frac{DS^{\mu \nu}}{D\tau}=p^\mu v^\nu - p^\nu v^\mu. \label{MPD}
\end{split}
\end{equation}
Here $D/D \tau$ is a covariant derivative along the particle trajectory, $\tau$ is an affine parameter of the orbit \cite{SaijoMaeda1998 }, $R^{\mu}{}_{\nu \rho \sigma}$ is the Riemannian tensor, $p^\mu$ and $v^\mu$ are 4 -momentum and 4 -velocity of a test body, $S^{\rho \sigma}$ is its spin-tensor. The equations were derived under the assumption that characteristic radius of the spinning particle is much smaller than the curvature scale of a background spacetime \cite{SaijoMaeda1998 } (see also \cite{Rasband1973 }, \cite{Apostolatos1996 }) and the mass of a spinning body is much less than that of BH. It is known, however, that these equations are incomplete, because they do not define which point on the test body is used for spin and trajectory measurements. Therefore we need some extra condition (`spin supplementary condition') to do that and to close the system of equations \cite{Papapetrou1951 b}. We use the condition of Tulczyjew \cite{Tulczyjew1959 } given by
\begin{equation}
p_\mu S^{\mu \nu}=0. \label{SSC}
\end{equation}
The system of equations (\ref{MPD}) with (\ref{SSC}) in a general space-time admits two conserved quantities: particle's mass $m^2 = -p^{\mu}p_{\mu}$ and the magnitude of its specific spin $s^2 = S^{\mu \nu}S_{\mu \nu}/(2 m^2 )$, see \cite{SaijoMaeda1998 }. We will consider the motion of a spinning particle in the equatorial plane of Kerr metric ($\theta=\pi/2 $),
which is given in Boyer-Lindquist coordinates by \cite{LL2, Hobson}
\begin{equation}
\begin{split}
ds^2 & = - \left( 1 -\frac{2 M r}{\Sigma} \right) dt^2 - \\
&- \frac{4 M a r \sin^2 \theta}{\Sigma} dt \ d\varphi + \frac{\Sigma}{\Delta}dr^2 + \Sigma \ d\theta^2 + \\
&+ \left( r^2 +a^2 +\frac{2 M ra^2 \sin^2 \theta}{\Sigma} \right)\sin^2 \theta \ d\varphi^2,
\end{split}
\label{}
\end{equation}
where $a$ is the specific angular momentum of a black hole, $\Sigma \equiv r^2 + a^2 \cos^2 \theta$, $\Delta \equiv r^2 - 2 Mr + a^2 $. In this case there are two additional conserved quantities: total energy of the particle and the projection of its total angular momentum onto $z$-axis. In case of the motion of a spinning particle in the equatorial plane, the angular momentum of a spinning particle is always perpendicular to the equatorial plane \cite{SaijoMaeda1998 }. Therefore we can describe the test particle spin by only one constant $s$ which is the specific spin angular momentum of the particle. Value $|s|$ indicates the magnitude of the spin and $s$ itself is its projection on the $z$-axis. It is more obvious to think of the spin in terms of the particle's spin angular momentum $\mathbf{S_1 }=sm\mathbf{\hat{z}}$ which is parallel to the BH spin angular momentum $\mathbf{S_2 }=aM\mathbf{\hat{z}}$, when $s>0 $, and antiparallel, when $s<0 $. Here $\mathbf{\hat{z}}$ is a unit vector in the direction of the $z$-axis and $m$ is a mass of the particle \cite{SaijoMaeda1998 }, \cite{Favata2011 }. Saijo \textit{et al} \cite{SaijoMaeda1998 } have derived the equations of motion of a spinning test particle for the equatorial plane of Kerr BH. The equations of motion for the variables $r$, $t$, $\varphi$ in this case have the form \cite{SaijoMaeda1998 }
\begin{equation}
\begin{split}
& (\Sigma_s \Lambda_s \dot r)^2 =R_s, \\
& \Sigma_s \Lambda_s \dot t =a\left( 1 +\frac{3 Ms^2 }{r \Sigma_s}\right)\left[ J -(a+s)E \right] +\\
& +\frac{r^2 +a^2 }{\Delta}P_s, \\
& \Sigma_s \Lambda_s \dot \varphi =\left( 1 +\frac{3 Ms^2 }{r \Sigma_s}\right)\left[ J -(a+s)E \right] +\frac{a}{\Delta}P_s. \\
\end{split}
\label{spin-eqs}
\end{equation}
where
\begin{equation}
\begin{split}
& \Sigma_s= r^2 \left(1 -\frac{Ms^2 }{r^3 }\right), \\
& \Lambda_s= 1 - \frac{3 M s^2 r [-(a + s) E +J]^2 }{\Sigma_s^3 }, \\
&R_s = P_s^2 - \Delta \left\{ \frac{\Sigma_s^2 }{r^2 } + [-(a + s) E +J]^2 \right\}, \\
&P_s = \left[r^2 + a^2 + \frac{a s (r + M)}{r} \right] E - \left(a + \frac{M s}{r} \right) J.
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}, \cite{Rasband1973 }
\begin{equation}
(\Sigma_s \Lambda_s \dot r)^2 =\alpha_s E^2 -2 \beta_s E +\gamma_s,
\label{rad-motion}
\end{equation}
where
\begin{equation}
\begin{split}
&\alpha_s = \left[r^2 + a^2 + \frac{a s (r + M)}{r} \right]^2 - \Delta (a + s)^2, \\
&\beta_s = \left[ \left(a + \frac{M s}{r} \right) \left(r^2 + a^2 + \frac{a s (r + M)}{r} \right) - \right. \\
&\left. - \Delta (a + s) \right]J, \\
&\gamma_s = \left(a + \frac{M s}{r} \right)^2 J^2 - \Delta \left[r^2 \left(1 - \frac{M s^2 }{r^3 }\right)^2 + J^2 \right]. \end{split} \label{abc-spin}
\end{equation}
We can consider the whole right-hand side of (\ref{rad-motion}) as an effective potential. For the matter of convenience, let us further divide it by $r^4 $ and define the effective potential as
\begin{equation}
V_s(r;J, E)= \frac{1 }{r^4 } (\alpha_s E^2 -2 \beta_s E +\gamma_s). \label{V-spin}
\end{equation}
\section{Numerical calculation of ISCO parameters}
The equations which define circular orbits are given by a system
\begin{equation}
\left\{
\begin{aligned}
V_s &= 0 \, , \\
\frac{dV_s}{dr} &=0 \, . \\
\end{aligned}
\right. \label{eff12 }
\end{equation}
In order to find the last stable orbit (ISCO) we need to demand additionally that the second derivative of the effective potential vanishes:
\begin{equation}
\begin{aligned}
&\frac{d^2 V_s}{dr^2 } = 0. \\
&\\
\end{aligned}
\label{eff3 }
\end{equation}
For the sake of convenience, we change variables and work not with $r$ and $J$ but with $u=1 /r$ and $x=J-aE$, so the function $V_s(u; x, E)$ will be used. In new variables (see \cite{Jefremov2015 } for details), system of equations determining ISCO have the form
\begin{equation}
\left\{
\begin{aligned}
V_s &= 0 \, , \\
\frac{dV_s}{du} &=0 \, , \\
\frac{d^2 V_s}{du^2 } &= 0 \, . \\
\end{aligned}
\right. \label{n}
\end{equation}
The explicit form of these equations is \cite{Jefremov2015 }:
\begin{equation}
\begin{aligned}
&(1 + 2 a s u^2 - s^2 u^2 + 2 M s^2 u^3 ) E^2 + \\
&+(-2 a u^2 x + 2 s u^2 x - 6 M s u^3 x - 2 a M s^2 u^5 x) E -\\ &-1 + 2 M u - a^2 u^2 + 2 M s^2 u^3 -\\
&- 4 M^2 s^2 u^4 + 2 a^2 M s^2 u^5 - M^2 s^4 u^6 +\\
&+ 2 M^3 s^4 u^7 - a^2 M^2 s^4 u^8 - u^2 x^2 + 2 M u^3 x^2 +\\
&+ 2 a M s u^5 x^2 + M^2 s^2 u^6 x^2 = 0 \, ;\\
&(4 a s u - 2 s^2 u + 6 M s^2 u^2 ) E^2 + \\
&+ (-4 a u x + 4 s u x - 18 M s u^2 x - 10 a M s^2 u^4 x) E +\\
&+2 M - 2 a^2 u + 6 M s^2 u^2 - 16 M^2 s^2 u^3 +\\
& + 10 a^2 M s^2 u^4 - 6 M^2 s^4 u^5 + 14 M^3 s^4 u^6 -\\
&- 8 a^2 M^2 s^4 u^7 - 2 u x^2 + 6 M u^2 x^2 +\\
&+ 10 a M s u^4 x^2 + 6 M^2 s^2 u^5 x^2 =0 \, ;\\
&(4 a s - 2 s^2 + 12 M s^2 u) E^2 + \\
&+ (-4 a x + 4 s x - 36 M s u x - 40 a M s^2 u^3 x)E -\\
&- 2 (a^2 - 6 M s^2 u + 24 M^2 s^2 u^2 - 20 a^2 M s^2 u^3 +\\
&+ 15 M^2 s^4 u^4 - 42 M^3 s^4 u^5 +28 a^2 M^2 s^4 u^6 + x^2 - \\
& - 6 M u x^2 - 20 a M s u^3 x^2 - 15 M^2 s^2 u^4 x^2 ) =0 \, . \end{aligned}
\label{system}
\end{equation}
These three equations form a closed system for three parameters of ISCO $E$, $x$ and $u$, which are dependent only on the Kerr parameter $a$ and particle's spin $s$. This system can be used for numerical calculation of $E$, $x$, $u$ of ISCO at given $a$ and $s$. Then, values of $r$ and $J$ can be found numerically by using $r=1 /u$ and $J=x+aE$. Another important characteristics of the particle circular motion is its angular velocity. The orbital angular frequency of the particle at the ISCO, as seen from an observer at infinity, is defined as
\begin{equation}
\Omega \equiv \frac{d \varphi / d\tau}{ dt / d\tau}. \label{Omega-definition}
\end{equation}
The values $d \varphi / d\tau$ and $dt / d\tau$ are found from the second and the third equations in (\ref{spin-eqs}), where we should substitute values of $r$, $E$ and $J$ at a given orbit. To find the ISCO frequency $\Omega_{\mathrm{\, ISCO}}$ we need to use the ISCO values of $r$, $E$ and $J$, see \cite{Favata2011 }. At given $a$ and $s$ the system lead to solutions both for co-rotating and counter-rotating case. Corotation means parallel orientation of particle's angular momentum $\mathbf{J}$ and BH spin $\mathbf{a}$, $J>0 $; counter-rotation means antiparallel orientation, $J<0 $. The $z$-axis is chosen to be parallel to BH spin $\mathbf{a}$, so $a$ is positive or equal to zero. Spin $s$ is the projection of spin on the $z$-axis and can be positive (spins of particle and BH are parallel) or negative (antiparallel). Specifying $a$ and $s$, we can find $r_{\mathrm{\, ISCO}}$, $E_{\mathrm{\, ISCO}}$ and $J_{\mathrm{\, ISCO}}$ numerically by solving system (\ref{system}), all parameters are in units of $M$. Results for the radius calculation are presented in Figures \ref{fig-r-co} and \ref{fig-r-counter}. For $a=0 $ and non-spinning particle ($s=0 $) radius equals to $6 M$. Increase of $a$ leads to decrease of $r_{\mathrm{\, ISCO}}$ for co-rotating case and to increase of $r_{\mathrm{\, ISCO}}$ for counter-rotating case. \begin{figure}
\centerline{\hbox{\includegraphics[width=0.45 \textwidth]{r-co. eps}}} \caption{Radius of ISCO for the case of corotation: angular momentum of black hole and total angular momentum of particle are parallel, $J>0 $. All values are in units of $M$. See also Figure 3 in paper of Suzuki and Maeda \cite{Suzuki1998 }. } \label{fig-r-co}
\end{figure}
\begin{figure}
\centerline{\hbox{\includegraphics[width=0.45 \textwidth]{r-counter. eps}}} \caption{Radius of ISCO for the case of counterrotation: angular momentum of black hole and total angular momentum of particle are antiparallel, $J<0 $. All values are in units of $M$. } \label{fig-r-counter}
\end{figure}
Calculations of ISCO energy are shown in Figures \ref{fig-E-co} and \ref{fig-E-counter}. For $a=0 $ and $s=0 $ the energy equals to $2 \sqrt{2 }/3 $. Calculations of ISCO total angular momentum are presented in Figures \ref{fig-J-co} and \ref{fig-J-counter}. For $a=0 $ and $s=0 $ the magnitude of angular momentum equals to $2 \sqrt{3 }M$. Calculations of ISCO angular frequency are shown in Figures \ref{fig-Omega-co} and \ref{fig-Omega-counter}. For $a=0 $ and $s=0 $ the magnitude of angular frequency equals to $1 /6 \sqrt{6 }M$. \begin{figure}
\centerline{\hbox{\includegraphics[width=0.45 \textwidth]{E-co. eps}}} \caption{Energy of ISCO for the case of corotation. } \label{fig-E-co}
\end{figure}
\begin{figure}
\centerline{\hbox{\includegraphics[width=0.45 \textwidth]{E-counter. eps}}} \caption{Energy of ISCO for the case of counterrotation. } \label{fig-E-counter}
\end{figure}
\begin{figure}
\centerline{\hbox{\includegraphics[width=0.45 \textwidth]{J-co. eps}}} \caption{Total angular momentum of ISCO for the case of corotation. } \label{fig-J-co}
\end{figure}
\begin{figure}
\centerline{\hbox{\includegraphics[width=0.45 \textwidth]{J-counter-modulus. eps}}} \caption{Total angular momentum (absolute value) of ISCO for the case of counterrotation. } \label{fig-J-counter}
\end{figure}
\begin{figure}
\centerline{\hbox{\includegraphics[width=0.45 \textwidth]{Omega-co. eps}}} \caption{Orbital angular frequency of ISCO for the case of corotation. } \label{fig-Omega-co}
\end{figure}
\begin{figure}
\centerline{\hbox{\includegraphics[width=0.45 \textwidth]{Omega-counter-modulus. eps}}} \caption{Orbital angular frequency (absolute value) of ISCO for the case of counterrotation. } \label{fig-Omega-counter}
\end{figure}
\section{Analytical calculation of ISCO parameters, small-spin corrections}
In paper \cite{Jefremov2015 } we have derived linear small-spin corrections for ISCO parameters for arbitrary $a$. Parameters are written there with using variable $u_0 =1 /r_0 $ which is inverse radius of ISCO for non-spinning particle. The scheme of calculation of all parameters are presented there, see text after formula (61 ) in \cite{Jefremov2015 }. Here, we rewrite all formulas in terms of $r_0 $. The scheme of calculation of ISCO parameters with linear small-spin corrections:
(i) Radius of ISCO. We need to solve equation for ISCO radius $r_0 $ of non-spinning particle
\begin{equation}
r_0 ^2 - 6 Mr_0 - 3 a^2 \mp 8 a \sqrt{Mr_0 } =0. \label{eq-r0 }
\end{equation}
and find $r_0 $. In this equation and all formulas below the upper sign corresponds to the antiparallel orientation of particle's angular momentum $\mathbf{J}$ and BH spin $\mathbf{a}$ (counter-rotation, $J<0 $), the lower -- to the parallel one (corotation, $J>0 $). Solution of (\ref{eq-r0 }) can be found analytically, see \cite{Bardeen1972 }. To avoid large formulas, we write all other unknowns not as explicit functions of $a$ but as explicit functions of $a$ and $r_0 $, keeping in mind that $r_0 $ can be found from Eq. (\ref{eq-r0 }) at arbitrary $a$. We need to notice that representation of all unknowns via $r_0 $ given below could be rewritten in a different form using (\ref{eq-r0 }), see \cite{Jefremov2015 } for different representations. Linear correction is
\begin{equation}
r_1 = \frac{4 }{r_0 } (a \pm \sqrt{Mr_0 }) . \end{equation}
Finally radius of ISCO for given $a$ is
\begin{equation}
r_{\mathrm{\, ISCO}} = r_0 + s r_1 \, . \end{equation}
(ii) Energy of ISCO is:
\begin{equation}
E_{\mathrm{\, ISCO}} = E_0 + s E_1 ,
\end{equation}
\begin{equation}
E_0 = \sqrt{1 -\frac{2 M}{3 r_0 }} ,
\end{equation}
\begin{equation}
E_1 = \pm \frac{1 }{\sqrt{3 }} \frac{M}{r_0 ^2 } . \end{equation}
(iii) Total angular momentum of ISCO is:
\begin{equation}
J_{\mathrm{\, ISCO}} = J_0 + s J_1 ,
\end{equation}
\begin{equation}
J_0 = \mp \frac{r_0 }{\sqrt{3 }} + a \sqrt{1 -\frac{2 M}{3 r_0 }} ,
\end{equation}
\begin{equation}
J_1 = \frac{2 \sqrt{M} r_0 ^{3 /2 } \pm a(3 r_0 +M) }{\sqrt{3 } r_0 ^2 } .
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to the spin of the test-particle is calculated numerically, see right picture on Fig. 2 in \cite{Favata2011 } and the sixth column in Table I in \cite{Favata2011 }. Our analytical results (with appropriate change of variables) agree with calculations of paper of Favata: value of $\Omega_0 $ for given $a$ (see first column in Table I; note that BH spin can be negative in this Table, it corresponds to case of counterrotation in our work) give numbers in second column of Table I, value of $\Omega_1 /\Omega_0 $ give numbers in six column.
Now let us consider particular cases.
Results for the Schwarzschild case are \cite{Jefremov2015 }:
\begin{equation}
\begin{aligned}
J_{\mathrm{\, ISCO}}&=2 \sqrt{3 } M + \frac{\sqrt{2 }}{3 }s_J , \\
E_{\mathrm{\, ISCO}}&= \frac{2 \sqrt{2 }}{3 } -\frac{1 }{36 \sqrt{3 }}\frac{s_J}{M} , \\
r_{\mathrm{\, ISCO}}&= 6 M -2 \sqrt{\frac{2 }{3 }}s_J , \\
\Omega_{\mathrm{\, ISCO}}&= \frac{1 }{6 \sqrt{6 }M} +\frac{s_J}{48 M^2 } . \\
\end{aligned}
\label{Schw}
\end{equation}
Here instead of $s$, which is the projection on $z$-axis that does not unabiguously correspond to any physical direction in Schwarzschild case, we use $s_J$ which is the projection of particle's spin upon the direction of $\mathbf{J}$ and is positive when the particle's spin is parallel to it and negative when it is antiparallel. Value $J$ is considered as positive in this case. Small-spin corrections for Schwarzschild metric were derived by Favata \cite{Favata2011 }.
For Kerr BH with slow rotation ($a \ll M$) we have obtained the corrections up to quadratic terms \cite{Jefremov2015 }:
\begin{equation}
\begin{aligned}
J_{\mathrm{\, ISCO}}&= \mp 2 \sqrt{3 }M -\frac{2 \sqrt{2 }}{3 }a +\frac{\sqrt{2 }}{3 }s \pm\frac{11 }{36 \sqrt{3 }}\frac{a}{M}s \pm\\&\pm \frac{4 \sqrt{3 } M}{27 }\left( \frac{a}{M} \right)^2 \pm \frac{1 }{4 M \sqrt{3 }}s^2, \\
E_{\mathrm{\, ISCO}}&= \frac{2 \sqrt{2 }}{3 } \pm\frac{1 }{18 \sqrt{3 }}\frac{a}{M} \pm \frac{1 }{36 \sqrt{3 }}\frac{s}{M} -\frac{\sqrt{2 }}{81 }\frac{a}{M}\frac{s}{M} -\\&-\frac{5 }{162 \sqrt{2 }}\left( \frac{a}{M} \right)^2 -\frac{5 }{432 \sqrt{2 }M^2 }s^2 , \\
r_{\mathrm{\, ISCO}}&= 6 M \pm 4 \sqrt{\frac{2 }{3 }}a \pm 2 \sqrt{\frac{2 }{3 }}s + \frac{2 }{9 } \frac{a}{M}s -\\&-\frac{7 M}{18 }\left( \frac{a}{M} \right)^2 -\frac{29 }{72 M}s^2 . \\
\Omega_{\mathrm{\, ISCO}} &= \mp \frac{1 }{6 \sqrt{6 }M} +\frac{11 }{216 M}\frac{a}{M} +\frac{1 }{48 M^2 }s \, \mp\\
&\mp \left( \frac{1 }{18 \sqrt{6 }M}\frac{as}{M^2 } + \frac{59 }{648 \sqrt{6 }M} \frac{a^2 }{M^2 } + \right. \\
&\left. + \frac{97 }{3456 \sqrt{6 }M}\frac{s^2 }{M^2 } \right).
\end{aligned}
\label{Kerr-slow1 }
\end{equation}
For extreme Kerr BH ($a=M$) for counterrotation we have obtained \cite{Jefremov2015 }:
\begin{equation}
\begin{aligned}
J_{\mathrm{\, ISCO}}&= -\frac{22 \sqrt{3 }}{9 }M +\frac{82 \sqrt{3 }}{243 }s, \\
E_{\mathrm{\, ISCO}}&=\frac{5 \sqrt{3 }}{9 } +\frac{\sqrt{3 }}{243 }\frac{s}{M}, \\
r_{\mathrm{\, ISCO}}&= 9 M +\frac{16 }{9 }s, \\
\Omega_{\mathrm{\, ISCO}} &= -\frac{1 }{26 M} +\frac{3 s}{338 M^2 }. \\
\end{aligned}
\label{Kerr-counter}
\end{equation}
For the case of corotation we have considered nearly extreme Kerr BH with $a=(1 -\delta)M$ with $\delta \ll 1 $, and have obtained:
\begin{equation}
\begin{aligned}
J_{\mathrm{\, ISCO}}&=\left( \frac{2 }{\sqrt{3 }} +\frac{2 \times 2 ^{2 /3 } \delta^{1 /3 }}{\sqrt{3 }} \right)M +\\
&+\left( -\frac{2 }{\sqrt{3 }} +\frac{4 \times 2 ^{2 /3 }\delta^{1 /3 }}{\sqrt{3 }} \right)s, \\
E_{\mathrm{\, ISCO}}&=\left( \frac{1 }{\sqrt{3 }} +\frac{2 ^{2 /3 } \delta^{1 /3 }}{\sqrt{3 }} \right) +\\
&+\left( -\frac{1 }{\sqrt{3 }} +\frac{2 \times 2 ^{2 /3 }\delta^{1 /3 }}{\sqrt{3 }} \right) \frac{s}{M}, \\
r_{\mathrm{\, ISCO}}&= \left( 1 +2 ^{2 /3 }\delta^{1 /3 } \right)M -2 \times 2 ^{2 /3 } \delta^{1 /3 }s, \\
\Omega_{\mathrm{\, ISCO}} &= \frac{1 }{2 M} - \frac{3 \times 2 ^{2 /3 } \delta^{1 /3 }}{8 M} + \frac{9 \times 2 ^{2 /3 } \delta^{1 /3 }}{16 M^2 } s . \\
\end{aligned}
\label{Kerr-co}
\end{equation}
We see that in the case of $a=M$ ($\delta=0 $) the corrections, linear in spin, are absent in formulae for ISCO radius and frequency. This was also demonstrated in \cite{Abramowicz1979 }. In the work \cite{TanakaMino1996 } on basis of the numerical calculation, it was noticed that in the extreme Kerr background for the parallel case the magnitude of test-body's spin does not influence the radius of the last stable orbit and it always remains equal to $M$. We have succeeded in proving this analytically. We have obtained the exact (in spin) values of ISCO parameters for nearly extreme Kerr BH in case of corotation \cite{Jefremov2015 }:
\begin{equation}
\begin{aligned}
&J_{\mathrm{\, ISCO}}= 2 M E_{\mathrm{\, ISCO}} \, , \\
&E_{\mathrm{\, ISCO}}= \frac{M^2 -s^2 }{M^2 \sqrt{3 +6 s/M}} +\\
&+ \frac{(M^2 -s^2 )^{1 /3 }(2 M +s)^{2 /3 } Z(M, s)^{2 /3 }}{\sqrt{3 }M^{5 /2 }(M +2 s)^{3 /2 }} \, \delta^{1 /3 }, \\
&r_{\mathrm{\, ISCO}}=M +\frac{M(M^2 -s^2 )^{1 /3 }(2 M +s)^{2 /3 }}{Z(M, s)^{1 /3 }} \, \delta^{1 /3 }, \\
&\Omega_{\mathrm{\, ISCO}} = \frac{1 }{2 M} -\\
&-\frac{3 (M -s)^{1 /3 }(M +2 s)}{4 (2 M +s)^{1 /3 }(M +s)^{2 /3 } Z(M, s)^{1 /3 }}\, \delta^{1 /3 } , \\
&Z(M, s) \equiv M^4 +7 M^3 s +9 M^2 s^2 +11 M s^3 -s^4 . \\
\end{aligned}
\label{Kerr-delta}
\end{equation}
From this solution we see that for $\delta=0 $ ($a=M$) the radius and the angular frequency are independent of the particle's spin $s$ while the values of energy and total angular momentum depend on it.
It can be easily seen from the exact solution (\ref{Kerr-delta}) that for extreme Kerr BH solution for energy and angular momentum diverges with $s \rightarrow -M/2 $. It shows that an approximation of test body does not work with such large values of $s$. Of course, limits of test body application depend on $a$, but we emphasize that all results beyond the approximation $s \ll M$ should be considered with big care, see \cite{SaijoMaeda1998 }, \cite{Jefremov2015 }.
For a spinless particle the conserved quantity is the orbital angular momentum $L_z$, whereas in the case of a spinning particle the conserved quantity is the total angular momentum $J_z$, which includes spin terms \cite{SaijoMaeda1998 }. In this case 'orbital angular momentum' at infinity $L_z$ can also be introduced as $L_z=J_z - s$, see \cite{SaijoMaeda1998 }.
In the paper \cite{Jefremov2015 } we present formulae for a small-spin linear corrections for $E$, $J$ and $\Omega$ at circular orbit with a given radius $r$. It can be seen that for $r \rightarrow \infty$ the total angular momentum equals to $J=J_0 + s$, where $J_0 $ is total angular momentum for non-spinning particle. In spinless case it consists of orbital angular momentum part only, therefore $J_0 =L$. This justifies the introduction of orbital momentum at infinity just as difference between $J$ and $s$.
For circular orbits at finite radius (in particular, ISCO) orbital angular momentum cannot be defined by such simple way \cite{SaijoMaeda1998 }. But using test body approximation $s \ll M$ allows us to use term 'corotation' and 'counterrotation' as terms for orbital motion.
If we use tentatively the orbital angular momentum in the form $L_z=J_z - s$ for ISCO, we will get for the Schwarzschild case:
\begin{equation}
L_{\mathrm{\, ISCO}} = 2 \sqrt{3 } M - \left(1 - \frac{\sqrt{2 }}{3 } \right) s_J .
\label{L-orb}
\end{equation}
We see from (\ref{L-orb}) and (\ref{Schw}) that increasing of positive $s_J$ leads to increasing of $r_{\mathrm{\, ISCO}}$ and decreasing of orbital angular momentum $L_{\mathrm{\, ISCO}}$. At the same time the total angular momentum becomes bigger but it happens only due to increasing of its spin part.
\section{Binding energy in the innermost stable circular orbit}
Let us denote efficiency $\varepsilon$ as the fraction of the rest mass
energy that can be released in making the transition from rest at infinity to
the innermost stable circular orbit \cite{Hobson}, in our units it is given by
\begin{equation}
\varepsilon = 1 - E_{\mathrm{\, ISCO}} = E_{\mathrm{\, bind}}.
\end{equation}
Note that in our notations $E$ is energy per unit particle rest mass. In other words the efficiency is the binding energy $E_{\mathrm{\, bind}}$ at ISCO per unit mass. The efficiency shows how much energy can be released by radiation during the accretion process. For the Schwarzschild black hole and non-spinning test particle the efficiency equals to 0.057, and it reaches maximum for the extreme Kerr black hole -- 0.42 \cite{Hobson}, \cite{MTW}.
In the case of spinning test particles we can easily calculate the efficiency with using of expressions for $E_{\mathrm{\, ISCO}}$ presented in (\ref{Schw}), (\ref{Kerr-delta}), see Fig. \ref{fig-binding-energy}.
\begin{figure}
\centerline{\hbox{\includegraphics[width=0.45 \textwidth]{bind-energy. eps}}} \caption{The efficiency (binding energy) of spinning test particle at ISCO. For case of extreme black hole the binding energy can be smaller or larger than 0.42 depending on spin orientation. } \label{fig-binding-energy}
\end{figure}
For the Schwarzschild black hole (see (\ref{Schw})) the efficiency is:
\begin{equation}
\varepsilon = 1 - \frac{2 \sqrt{2 }}{3 } + \frac{1 }{36 \sqrt{3 }}\frac{s_J}{M} .
\end{equation}
For extreme Kerr black hole in case of orbital corotation (see (\ref{Kerr-co})), the efficiency is:
\begin{equation}
\varepsilon = 1 - \frac{1 }{\sqrt{3 }} + \frac{1 }{\sqrt{3 }} \frac{s}{M} .
\end{equation}
It means that in case when particle spin is parallel to the total angular momentum of particle and the black hole spins, the efficiency can be larger than 42 \%. Note that the ISCO radius and angular frequency do not depend on spin in the case of extreme Kerr BH.
\section*{Acknowledgments}
The work of GSBK and OYuT was partially supported by the Russian Foundation for Basic Research Grant No. 14 -02 -00728 and the Russian Federation President Grant for Support of Leading Scientific Schools, Grant No. NSh-261.2014.2. GSBK acknowledges partial support by by the Russian Foundation for Basic Research Grant No. OFI-M 14 -29 -06045.
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\section{Introduction}
At the meeting of the American Mathematical Society in Hayward, California, in April 1977, Olga Taussky-Todd \cite{TausskyTodd} asked whether one could characterize the values of the group determinant when the entries are all integers. For a prime $p, $ a complete description was obtained for $\mathbb Z_{p}$ and $\mathbb Z_{2 p}$, the cyclic groups of order $p$ and $2 p$, in \cite{Newman1 } and \cite{Laquer}, and for $D_{2 p}$ and $D_{4 p}$ the dihedral groups of order $2 p$ and $4 p$ in \cite{dihedral}. The values for $Q_{4 n}$, the dicyclic group of order $4 n$ were explored in \cite{dicyclic}
with a near complete description for $Q_{4 p}$. In general though this quickly becomes a hard problem,
with only partial results known even for $\mathbb Z_{p^2 }$ once $p\geq 7 $ (see \cite{Newman2 } and \cite{Mike}). The remaining groups of order less than 15 were tackled in \cite{smallgps} and $\mathbb Z_{15 }$ in \cite{bishnu1 }. The integer group determinants have been determined for all five abelian groups of order 16 ($\mathbb Z_2 \times \mathbb Z_8 $, $\mathbb Z_{16 }$, $\mathbb Z_2 ^4 $, $\mathbb Z_4 ^2 $, $\mathbb Z_2 ^2 \times\mathbb Z_4 $ in \cite{Yamaguchi1, Yamaguchi2, Yamaguchi3, Yamaguchi4, Yamaguchi5 }), and for three of the non-abelian groups
($D_{16 }$, $\mathbb Z_2 \times D_8 $, $\mathbb Z_2 \times Q_8 $ in \cite{dihedral, ZnxH}). Here we determine the the group determinants for $Q_{16 }$, the dicyclic or generalized quaternion group of order 16. $$ Q_{16 }=\langle X, Y \; | \; X^8 =1, \; Y^2 =X^4, \; XY=YX^{-1 }\rangle. $$
This leaves five unresolved non-abelian groups of order 16
\begin{theorem} The even integer group determinants for $Q_{16 }$ are exactly the multiples of $2 ^{10 }$. The odd integer group determinants are all the integers $n\equiv 1 $ mod 8 plus those $n\equiv 5 $ mod 8 of the form
$n=mp^2 $ where $m\equiv 5 $ mod 8 and $p\equiv 7 $ mod $8 $ is prime. \end{theorem}
We shall think here of the group determinant as being defined on elements of the group ring $\mathbb Z [G]$
$$ \mathcal{D}_G\left( \sum_{g\in G} a_g g \right)=\det\left( a_{gh^{-1 }}\right) . $$
\begin{comment}
We observe the multiplicative property
\begin{equation} \label{mult} \mathcal{D}_G(xy)= \mathcal{D}_G(x)\mathcal{D}_G(y), \end{equation}
using that
$$ x=\sum_{g \in G} a_g g, \;\;\; y=\sum_{g \in G} b_g g \; \Rightarrow \; xy=\sum_{g\in G} \left(\sum_{hk=g}a_hb_k\right) g. $$
\end{comment}
Frobenius \cite{Frob} observed that the group determinant can be factored using the groups representations (see for example \cite{Conrad} or \cite{book})
and an explicit expression for a dicyclic group determinant was given in \cite{smallgps}. For $Q_{16 }$, arranging the
16 coefficients into two polynomials of degree 7
$$ f(x)=\sum_{j=0 }^7 a_j x^j, \;\; g(x)=\sum_{j=0 }^7 b_jx^j, $$
and writing the primitive 8 th root of unity $\omega:=e^{2 \pi i/8 }=\frac{\sqrt{2 }}{2 }(1 +i)$, this becomes
\begin{equation} \label{form}\mathcal{D}_G\left( \sum_{j=0 }^7 a_j X^j + \sum_{j=0 }^7 b_j YX^j\right) =ABC^2 D^2 \end{equation}
with integers $A, B, C, D$ from
\begin{align*}
A=& f(1 )^2 - g(1 )^2 \\
B=& f(-1 )^2 -g(-1 )^2 \\
C=& |f(i)|^2 -|g(i)|^2 \\
D=& \left(|f(\omega)|^2 +|g(\omega)|^2 \right)\left(|f(\omega^3 )|^2 +|g(\omega^3 )|^2 \right). \end{align*}
From \cite[Lemma 5.2 ]{dicyclic} we know that the even values must be multiples of $2 ^{10 }$. The odd values must be
1 mod 4 (plainly $f(1 )$ and $g(1 )$ must be of opposite parity and $A\equiv B\equiv \pm 1 $ mod 4 with $(CD)^2 \equiv 1 $ mod 4 ). \section{Achieving the values $n\not \equiv 5 $ mod 8 }
We can achieve all the multiples of $2 ^{10 }$. Writing $h(x):=(x+1 )(x^2 +1 )(x^4 +1 ), $ we achieve the $2 ^{10 }(-3 +4 m)$ from
$$
f(x) = (1 -m)h(x), \quad
g(x)=1 +x^2 +x^3 +x^4 -mh(x), $$
the $2 ^{10 }(-1 +4 m)$ from
$$ f(x)= 1 +x+x^4 +x^5 -mh(x), \;\;\;\;
g(x)= 1 +x-x^3 -x^7 -mh(x), $$
the $2 ^{11 }(-1 +2 m)$ from
$$ f(x)= 1 +x+x^2 +x^3 +x^4 +x^5 -mh(x), \;\;\quad
g(x)=1 +x^4 -mh(x), $$
and the $2 ^{12 }m$ from
$$ f(x)= 1 +x+x^4 +x^5 -x^6 -x^7 -mh(x), \;\;
g(x)= 1 +x-x^3 +x^4 +x^5 -x^7 +mh(x). $$
We can achieve all the $n\equiv 1 $ mod 8 ; the $1 +16 m$ from
$$ f(1 )=1 +mh(x), \;\; g(x)=mh(x), $$
and the $-7 +16 m$ from
$$f(x)= 1 -x+x^2 +x^3 +x^7 - mh(x), \;\;
g(x)= 1 +x^3 +x^4 +x^7 -mh(x). $$
\section{ The form of the $n\equiv 5 $ mod 8 }
This leaves the $n\equiv 5 $ mod 8. Since $(CD)^2 \equiv 1 $ mod 8 we must have $AB\equiv 5 $ mod 8. Switching $f$ and $g$ as necessary we assume that $f(1 ), f(-1 )$ are odd and $g(1 ), g(-1 )$ even. Replacing $x$ by $-x$ if needed we can assume that $g(1 )^2 \equiv 4 $ mod 8 and $g(-1 )^2 \equiv 0 $ mod 8. We write
$$ F(x)=f(x)f(x^{-1 })= \sum_{j=0 }^7 c_j (x+x^{-1 })^j, \quad G(x)=g(x)g(x^{-1 })= \sum_{j=0 }^7 d_j (x+x^{-1 })^j, $$
with the $c_j, d_j$ in $\mathbb Z$. From $F(1 ), F(-1 )\equiv 1 $ mod 8 we have
$$ c_0 +2 c_1 +4 c_2 \equiv 1 \text{ mod }8, \quad c_0 -2 c_1 +4 c_2 \equiv 1 \text{ mod }8, $$
and $c_0 $ is odd and $c_1 $ even. From $G(1 )\equiv 4 $, $G(-1 )\equiv 0 $ mod 8 we have
$$ d_0 +2 d_1 +4 d_2 \equiv 4 \text{ mod 8 }, \quad d_0 -2 d_1 +4 d_2 \equiv 0 \text{ mod } 8, $$
and $d_0 $ is even and $d_1 $ is odd. Since $\omega+\omega^{-1 }=\sqrt{2 }$ we get
\begin{align*} F(\omega) & = (c_0 +2 c_2 +4 c_4 +\ldots ) + \sqrt{2 }(c_1 +2 c_3 +4 c_5 +\cdots), \\
G(\omega) & = (d_0 +2 d_2 +4 d_4 +\ldots ) + \sqrt{2 }(d_1 +2 d_3 +4 d_5 +\cdots),
\end{align*}
and
$$|f(\omega)|^2 +|g(\omega)|^2 = F(\omega)+G(\omega) = X+ \sqrt{2 } Y>0, \;\; \quad X, Y \text{odd, } $$
with $ |f(\omega^3 )|^2 +|g(\omega^3 )|^2 =F(\omega^3 )+G(\omega^3 ) = X- \sqrt{2 } Y>0 $. Hence the positive integer $D=X^2 -2 Y^2 \equiv -1 $ mod 8. Notice that primes 3 and 5 mod 8 do not split in $\mathbb Z[\sqrt{2 }]$ so only their squares can occur in $D$. Hence
$D$ must contain at least one prime $p\equiv 7 $ mod 8, giving the claimed form of the values 5 mod 8. \section{Achieving the specified values 5 mod 8 }
Suppose that $p\equiv 7 $ mod 8 and $m\equiv 5 $ mod 8. We need to achieve $mp^2 $. Since $p\equiv 7 $ mod 8 we know that $\left(\frac{2 }{p}\right)=1 $ and $p$ splits in $\mathbb Z[\sqrt{2 }]. $ Since $\mathbb Z[\sqrt{2 }]$ is
a UFD, a generator for the prime factor gives a solution to
$$ X^2 -2 Y^2 =p, \;\; X, Y\in \mathbb N. $$
Plainly $X, Y$ must both be odd and $X+\sqrt{2 }Y$ and $X-\sqrt{2 }Y$ both positive. Since $(X+\sqrt{2 }Y)(3 +2 \sqrt{2 })=(3 X+4 Y)+\sqrt{2 }(2 X+3 Y)$ there will be $X, Y$ with $X\equiv 1 $ mod 4 and with
$X\equiv -1 $ mod 4. Cohn \cite{Cohn} showed that $a+b\sqrt{2 }$ in $\mathbb Z[\sqrt{2 }]$ is a sum of four squares in $\mathbb Z[\sqrt{2 }]$ if and only if $2 \mid b$. Hence we can write
$$ 2 (X+\sqrt{2 }Y)= \sum_{j=1 }^4 (\alpha_j + \beta_j\sqrt{2 })^2, \;\;\alpha_j, \beta_j\in \mathbb Z. $$
That is,
$$ 2 X=\sum_{j=1 }^4 \alpha_j^2 + 2 \sum_{j=0 }^4 \beta_j^2, \;\;\quad Y=\sum_{j=1 }^4 \alpha_j\beta_j. $$
Since $Y$ is odd we must have at least one pair, $\alpha_1 $, $\beta_1 $ say, both odd. Since $2 X$ is even we must have two
or four of the $\alpha_i$ odd. Suppose that $\alpha_1 $, $\alpha_2 $ are odd and $\alpha_3, \alpha_4 $ have the same parity. We get
\begin{align*} X+\sqrt{2 }Y & = \left( \frac{\alpha_1 +\alpha_2 }{2 } + \frac{\sqrt{2 }}{2 }(\beta_1 +\beta_2 )\right)^2 + \left( \frac{\alpha_1 -\alpha_2 }{2 } + \frac{\sqrt{2 }}{2 }(\beta_1 -\beta_2 )\right)^2 \\
& \quad + \left( \frac{\alpha_3 +\alpha_4 }{2 } + \frac{\sqrt{2 }}{2 }(\beta_3 +\beta_4 )\right)^2 + \left( \frac{\alpha_3 -\alpha_4 }{2 } + \frac{\sqrt{2 }}{2 }(\beta_3 -\beta_4 )\right)^2. \end{align*}
Writing
$$ f(\omega)=a_0 +a_1 \omega+a_2 \omega^2 +a_3 \omega^3 =a_0 + \frac{\sqrt{2 }}{2 }(1 +i)a_1 +a_2 i+ \frac{\sqrt{2 }}{2 }(-1 +i)a_3, $$
we have
$$ \abs{f(\omega)}^2 =\left(a_0 + \frac{\sqrt{2 }}{2 }(a_1 -a_3 )\right)^2 + \left(a_2 + \frac{\sqrt{2 }}{2 }(a_1 +a_3 )\right)^2 $$
and can make
$$ |f(\omega)|^2 +|g(\omega)|^2 = X + \sqrt{2 }Y $$
with the selection of integer coefficients for $f(x)=\sum_{j=0 }^3 a_jx^j$ and $g(x)=\sum_{j=0 }^3 b_jx^j$
\begin{align*} a_0 =&\frac{1 }{2 }(\alpha_1 -\alpha_2 ), \quad a_1 =\beta_1, \quad a_2 =\frac{1 }{2 }(\alpha_1 +\alpha_2 ), \quad a_3 =\beta_2, \\
b_0 =& \frac{1 }{2 }(\alpha_3 -\alpha_4 ), \quad b_1 =\beta_3, \quad b_2 =\frac{1 }{2 }(\alpha_3 +\alpha_4 ), \quad b_3 =\beta_4. \end{align*}
These $f(x)$, $g(x)$ will then give $D=p$ in \eqref{form}. We can also determine the parity of the coefficients. \vskip0.1 in
\noindent
{\bf Case 1 }: the $\alpha_i$ are all odd. Notice that $a_0 $ and $a_2 $ have opposite parity, as do $b_0 $ and $b_2 $. Since $Y$ is odd we must have one or three of the
$\beta_i$ odd. If $\beta_1 $ is odd and $\beta_2, \beta_3, \beta_4 $ all even, then $2 X\equiv 6 $ mod 8 and $X\equiv -1 $ mod 4. Then $a_0, a_1, a_2, a_3 $ are either odd, odd, even, even or even, odd, odd, even and $f(x)=u(x)+2 k(x)$
with $u(x)=1 +x$ or $x(1 +x)$.
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gives
$A=(5 -16 m)$, $B=1 $, $C=-1 $, $D=p$ achieving $(5 -16 m)p^2 $.
\vskip0.1 in
\noindent
{\bf Case 2 }: $\alpha_1 $, $\alpha_2 $ are odd, $\alpha_3 $, $\alpha_4 $ are even.
In this case $a_0 $, $a_2 $ will have opposite parity and $b_0 $, $b_2 $ the same parity.
Since $Y$ is odd we must have $\beta_1 $ odd, $\beta_2 $ even. Since $2 X\equiv 2 $ mod 4 we must have one more odd $\beta_i$, say $\beta_3 $ odd and $\beta_4 $ even.
If $\alpha_3 \equiv \alpha_4 $ mod 4 then $2 X\equiv 6 $ mod 8 and $X\equiv -1 $ mod 4. Hence
$a_0, a_1, a_2, a_3 $ are either odd, odd, even, even or even, odd, odd, even, that is $u(x)=1 +x$ or $x(1 +x)$ and $b_0, b_1, b_2, b_3 $ are even, odd, even, even and $v(x)=x^2 $ and again \eqref{shift} gives $(16 m-3 )p^2 $.
If $\alpha_3 \not\equiv \alpha_4 $ mod 4 then $2 X\equiv 10 $ mod 8 and $X\equiv 1 $ mod 4. In this case
$a_0, a_1, a_2, a_3 $ are either odd, odd, even, even or even, odd, odd, even, that is $u(x)=1 +x$ or $x(1 +x)$ and $b_0, b_1, b_2, b_3 $ are odd, odd, odd, even and $v(x)=1 +x+x^2 $ and again \eqref{shift} gives $(5 -16 m)p^2 $.
Hence, in either case, starting with an $X\equiv 1 $ mod 4 gives the $mp^2 $ with $m\equiv 5 $ mod 16 and
an $X\equiv -1 $ mod 4 the $mp^2 $ with $m\equiv -3 $ mod 16.
\section*{Acknowledgement}
\noindent
We thank Craig Spencer for directing us to Cohn's four squares theorem in $\mathbb Z[\sqrt{2 }]$.
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\section{Introduction}
Inspired by Shannon’s classic information theory \cite{shannon1948 mathematical}, Weaver and Shannon proposed a more general definition of a communication system involving three different levels of problems, namely, (i) transmission of bits (the technical problem); (ii) semantic exchange of transmitted bits (the semantic problem); and (iii) effect of semantic information exchange (the effectiveness problem). The first level of communication, which is the transmission of bits, has been well studied and realized in conventional communication systems based on Shannon’s bit-oriented technical framework. However, with the massive deployment of emerging devices, including Extended Reality (XR) and Unmanned Aerial Vehicles (UAVs), diverse tasks with stringent requirements pose critical challenges to traditional bit-oriented communications, which are already approaching the Shannon physical capacity limit. This imposes the Sixth Generation (6 G) network towards a communication paradigm shift to semantic level and effectiveness level by exploiting the context of data and its importance to the task. Initial works on ``semantic communications'' have mainly focused on identifying the content of the traditional text and speech \cite{luo2022, shi2021 semantic}, and the information freshness, i. e., AoI \cite{NowAoI} as a semantic metric that captures the timeliness of the information. However, these cannot capture the data importance sufficiently of achieving a specific task. In \cite{kountouris2021 semantics}, a joint design of information generation, transmission, and reconstruction was proposed. Although the authors explored the benefits of including the effectiveness level in \cite{strinati20216 g, popovski2020 semantic}, an explicit and systematic communication framework incorporating both semantic level and effectiveness level has not been proposed yet. There is an urgent need for a unified communication framework aiming at task-oriented performances for diverse data types. %
Motivated by this, in this paper, we propose a generic task-oriented and semantics-aware (TOSA) communication framework, which jointly considers the semantic level information about the data context, and effectiveness level performance metric that determines data importance, for different tasks with various data types. The main contributions of this paper are:
\begin{enumerate}
\item We first present unique characteristics of the traditional text, speech, image, video data types, and emerging $360 ^\circ$ video, sensor, haptic, and machine learning models. For each data type, we summarize the semantic information definition and extraction methods in Section~II. \item We then propose a generic TOSA communication framework for typical time critical and non-critical tasks, where semantic level and effectiveness level are jointly considered. Specifically, by exploiting the unique characteristics of different tasks, we present TOSA information, their extraction and recovery methods, and effectiveness level performance metrics to guarantee the task requirements in Section~III. \item To demonstrate the effectiveness of our proposed TOSA communication framework, we present the TOSA solution tailored for interactive Virtual Reality (VR) data
with the aim to maximize the long-term quality of experience (QoE) within the VR interaction latency constraints and analyze the results in Section IV. \item Finally, we conclude the paper in Section V. \end{enumerate}
\section{Semantic Information Extraction}
In this section, we focus on analyzing the characteristics of all data types, and summarizing the semantic information definition with corresponding extraction methods as shown in Table~\ref{feature_summarization}. \begin{table}[. h]
\caption{Semantic information extraction of different data types}
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
\textbf{Data Type} &Semantic Information &\makecell[c]{Semantic Information\\ Extraction Method}\\
\hline
Text &Embedding&BERT \\
\hline
Speech &Embedding&BERT \\
\hline
Image &Edge, Corner, Blob, Ridge&SIFT, CNN \\
\hline
Video &Temporal Correlation&\makecell[c]{CNN} \\
\hline
\makecell[c]{$360 ^\circ$ Video }&FoV&\makecell[c]{Biological Information\\ Compression}\\
\hline
Haptic Data &JND&Web's Law\\
\hline
\makecell[c]{Sensor and\\ Control Data}&Freshness& AoI\\
\hline
\end{tabular}
\label{feature_summarization}
\end{center}
\end{table}
\subsection{{{Speech and Text}}}
For a one-dimensional speech signal, the speech-to-text conversion can be first performed by speech recognition. With the extracted text information,
various approaches developed by Natural Language Processing (NLP) community can be applied to extract embedding as typical semantic information, which represents the words, phrases, or text as a low-dimensional vector. The most famous embedding extraction method is Bidirectional Encoder Representations from Transformers (BERT) proposed by Google, which can be pre-trained and fine-tuned via one additional output layer for different text tasks. However, during the speech-to-text conversion process, the timbre and emotion conveyed in the speech may lose. \subsection{{Image and Video}}
The image is a two-dimensional data type, where the image geometric structures, including edges, corners, blobs, and ridges, can be identified as typical semantic information. Although various traditional signal processing methods have been developed to extract image geometric structures such as SIFT (Scale-Invariant Feature Transform), the convolutional Neural Network (CNN) has shown stronger capability to extract complex geometric structures with its matrix kernel. Video is a typical three-dimensional data type as the combination of two-dimensional images with an extra time dimension. Therefore, the temporal correlation between adjacent frames can be identified as important semantic information, where the static background can be ignored during transmission. To extract temporal correlation information from video, different CNN structures have been utilized. \subsection{{$360 ^\circ$ Video}}
The $360 ^\circ$ rendered video is a new data type in emerging XR applications. The most important semantic information is identified as human field-of-view (FoV), which occupies around one-third of the $360 ^\circ$ video and only has the highest resolution requirement at the center \cite{Liu_vr}. In this case, biological information, such as retinal foveation and ballistic saccadic eye movements can be leveraged for semantic information extraction. Therefore, biological information compression methods have been utilized to extract the semantic information, where retinal foveation and ballistic saccadic eye movements are jointly considered to optimize the semantic information extraction process. \subsection{{Haptic Data}}
Haptic data consists of two submodalities, which are tactile information and kinesthetic \cite{Antonakoglou2018 }. For tactile information,
five major dimensions can be identified, which are friction, hardness perception, warmth conductivity, macroscopic roughness, and microscopic roughness. Kinesthetic information refers to the position/orientation of human body parts and external forces/torques applied to them. To reduce the redundant raw haptic data, Just Noticeable
Difference (JND) is identified as valuable semantic information to filter the haptic signal that cannot be perceived by the human, where the Weber's law serves as an important semantic information extraction criterion. \subsection{Sensor and Control data}
Sensors are usually deployed to monitor the physical characteristics of the environment (e. g., temperature, humidity, or traffic) in a geographical area. The acquisition of data is transformed to status updates that are transmitted through a network to the destination nodes. Then these data are processed in order to extract useful information, such as control commands or remote source reconstruction, that can be further utilized in the prediction of the evolution of the initial source. The accuracy of the reconstructed data either in control commands or to predict the evolution is directly related to the relevance or the semantic value of the data measurements. Thus, one important aspect is the generation of traffic and how it can be affected in order to filter only the most important samples so the redundant or less useful data will be eliminated to reduce potential congestion inside the network. The AoI has also a critical role in dynamic control systems, since it was shown that non-linear AoI and VoI are paradigm shifts and they can improve the performance of such systems. Furthermore, we have seen in early studies that the semantics of information (beyond timeliness) can provide further gains, by reducing the amount of information that is generated and transmitted without degrading the performance. \begin{figure*}[. h]
\centerline{\includegraphics[scale=0.7 ]{framework_2. pdf}}
\caption{TOSA communication framework for different tasks with diverse data types. }
\label{TOSA_framework_fig}
\end{figure*}
\subsection{{Machine Learning Model}}
With the massive deployment of machine learning algorithms, machine learning-related model has been regarded as another important data type. \begin{itemize}
\item \textbf{Federated Learning (FL) Model:} The FL framework has been considered as a promising approach to preserve data privacy, where each participating device uploads the model gradients or model weights to the server and receives the global model from the server. \item \textbf{Split Learning (SL) Model:} Due to the limited computation capability of devices and heavy computation burden, SL has been proposed to split the neural network model between the server and devices, where the device executes the model up to the cut layer and sends the smashed data to the central server to execute the remaining layers. Then the gradient of the smashed data is transmitted back from the server to update the local model. \end{itemize}
However, it is noted that explicit semantic information definition for machine learning model has not been proposed yet. \begin{table*}[. h]
\caption{TOSA Communication Framework Summarization of different tasks}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
\textbf{Data Type} & Task & Communication Entity& Recovery& Latency Type & \makecell[c]{Effectiveness Level\\ Perfromance Metrics}\\
\hline
\multirow{1 }{*}{\makecell[l]{Speech}}&\makecell[c]{Speech Recognition} & Human-Machine &Yes/No&Non-critical &\makecell[l]{$\bullet$F-measure\\$\bullet$Accuracy\\$\bullet$BLEU\\$\bullet$Perplexity} \\
\hline
\multirow{5 }{*}{\makecell[l]{Image}}& \makecell[c]{Face Detection} & Machine-Machine &No&Non-critical & \makecell[l]{$\bullet$IoU\\$\bullet$mAP\\$\bullet$F-measure\\$\bullet$MAE} \\
\cline{2 -5 }
& \makecell[c]{Road Segmentation} & Machine-Machine&Yes & Critical &\makecell[l]{$\bullet$IoU\\$\bullet$Pixel Accuracy\\$\bullet$MPA\\$\bullet$Latency}\\
\hline
\multirow{5 }{*}{\makecell[c]{$360 ^\circ$ Video}} & \makecell[c]{Display in AR}& Machine-Human& Yes& Non-Critical &\makecell[l]{$\bullet$PSNR\\$\bullet$SSIM\\$\bullet$Alignment Accuracy} \\
\cline{2 -5 }
& \makecell[c]{Display in VR} & Machine-Human& No& Non-Critical & \makecell[l]{$\bullet$PSNR\\$\bullet$SSIM\\$\bullet$Timing Accuracy\\$\bullet$Position Accuracy}\\
\hline
\makecell[c]{Haptic Data} & Grasping and Manipulation & Machine-Human&No & Critical& \makecell[l]{$\bullet$SNR\\$\bullet$SSIM}\\
\hline
Sensor& Networked control systems & Machine-Machine& No& Critical& \makecell[l]{$\bullet$LGQ}\\
\hline
\multirow{5 }{*}{\makecell[c]{ML\\Model}}& Federated Learning & Machine-Machine &\rule[0 pt]{1 cm}{0.1 em}& Critical/Non-Critical& \makecell[l]{$\bullet$Latency\\$\bullet$Reliability\\ $\bullet$Convergence Speed\\$\bullet$Accuracy} \\
\cline{2 -6 }
& Split Learning & Machine-Machine &\rule[0 pt]{1 cm}{0.1 em}& Critical/Non-Critical & \makecell[l]{$\bullet$Latency\\ $\bullet$Reliability \\$\bullet$Convergence Speed \\$\bullet$Accuracy}\\
\hline
\end{tabular}
\label{Tasks_summarization}
\end{center}
\end{table*}
\section{TOSA Communication Framework}
\label{different_task}
In this section, we propose a generic TOSA communication framework incorporating both semantic level and effectiveness level, for typical time critical and non-critical tasks as shown in Fig. ~\ref{TOSA_framework_fig}, where the TOSA information, its extraction and recovery methods, and effectiveness level performance metrics are presented in detail. \subsection{One-hop Task}
We consider one-hop tasks with a single link transmission in this section, where each communication entity can be either human or machine as summarized in Table~\ref{Tasks_summarization}. \subsubsection{{Speech Recognition}} In a speech recognition task, the human speech needs to be transmitted to the server, and the speech recognition task can be further divided into conversation-type task (e. g., human inquiry) and command-type task (e. g., smart home control) depending on the speech content. The conversation-type task focuses on understanding the intent, language, and sentiment to provide human with free-flow conversations. The command-type task focuses on parsing the specific command over the transmitted speech and then controlling the target device/robot. In the conversation-type task, the TOSA information can be keywords and emotions. The device can obtain the TOSA information by transforming the speech signal into text and extracting keywords and emotions via BERT. Then the server recovers the text via transformer decoder. In the command-type task, the TOSA information can be the binary command, the device can directly parse the speech signal and obtain the binary command signal for transmission, where no recovery is needed at the receiver side. The effectiveness level performance metrics include F-measure, accuracy, bilingual evaluation understudy (BLEU), and perplexity, where user satisfaction should also be considered. \subsubsection{{Face Detection and Road Segmentation}}
Face detection and road segmentation are two emerging image processing tasks \cite{Zhao2019, Minaee2022 }, where the captured images are required to be transmitted to the central server for processing. However, the road segmentation task in autonomous driving applications imposes stringent latency and reliability requirements due to road safety issues. This is because the vehicles need to instantaneously react to the rapidly changing environment. For the time non-critical face detection task, TOSA information can be the face feature that is extracted via CNN. After being transmitted to the central server, the regions with CNN features (R-CNN) can be applied to perform face detection. For the time critical road segmentation task, one possible solution is to identify the region of interest (ROI) features, i. e., road, as the TOSA information, and crop the images via region proposal algorithms. Then, the central server can perform image segmentation via mask R-CNN. Both tasks can be evaluated via effectiveness level performance metrics, including Intersection over Union (IoU), mean average precision (mAP), F-measure, and mean absolute error (MAE). It is noted that road segmentation in the autonomous driving application can be evaluated via pixel accuracy, and mean pixel accuracy (MPA). However, the trade-off between accuracy and latency remains to be an important challenge to solve.
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average precision (mAP), F-measure, and mean absolute error (MAE). It is noted that road segmentation in the autonomous driving application can be evaluated via pixel accuracy, and mean pixel accuracy (MPA). However, the trade-off between accuracy and latency remains to be an important challenge to solve.
\subsubsection{{Display in Extended Reality}}
Based on the Milgram \& Kishino’s Reality–Virtuality Continuum, the XR can be classified as Augmented Reality (AR), Mixed Reality (MR), and VR, where MR is defined as a superset of AR. Therefore, we focus on the AR and VR display tasks in the following.
In the AR display task, the central server transmits the rendered 3 D model of a specific virtual object to the user. It is noted that the virtual object identification and its pose information related to the real world is the key to achieving alignment between virtual and physical objects. Therefore, the virtual object identification and pose information can be extracted as TOSA information to reduce the data size. Then, by sharing the same 3 D model library, the receiver can locally reconstruct the 3 D virtual object model based on the received TOSA information. To evaluate the 3 D model transmission, Peak Signal-to-Noise Ratio (PSNR) and Structural Similarity (SSIM) can be adopted as effectiveness level performance metrics. However, how to quantify the alignment accuracy among the virtual objects and physical objects as a performance metric remains to solve.
In the VR display task, the central server transmits virtual $360 ^\circ$ video streaming to the user. To avoid the transmission of the whole $360 ^\circ$ video, the central server can predict the eye movements of the user and extract the corresponding FoV as TOSA information. Apart from the PSNR and SSIM mentioned in AR, timing accuracy and position accuracy are also important effectiveness level performance metrics to avoid cybersickness including: 1 ) initial delay: time difference between the start of head motion and that of the corresponding feedback; 2 ) settling delay: time difference between the stop of head motion and that of the corresponding feedback; 3 ) precision: angular positioning consistency between physical movement and visual feedback in terms of degrees; and 4 ) sensitivity: capability of inertial sensors to perceive subtle motions and subsequently provide feedback to users.
\subsubsection{{Grasping and Manipulation}}
Haptic communication has been incorporated by industries to perform grasping and manipulation for efficient manufacturing and profitable
production rates, where the robot transmits the haptic data to the manipulator. The shape and weight of the objects to be held are measured using cutaneous feedback derived from the fingertip contact pressure and kinesthetic feedback of finger positions, which should be transmitted within stringent latency requirement to guarantee industrial operation safety.
Due to the difficulty in supporting massive haptic data with stringent latency requirement, JND can be identified as an important TOSA information to ignore the haptic signal that cannot be perceived by the manipulator. Two effectiveness level performance metrics including SNR and SSIM have been verified to be applicable to vibrotactile quality assessment.
\subsubsection{Control}
In networked control systems (NCS), typically, multiple sensors measure the system state of their control processes and transmit the generated data over a resource-limited shared wireless network. These data usually are en-queued and then transmitted over unreliable channels that cause excessive delays resulting in outdated or even obsolete for decision-making based on less reliable information. Therefore, data freshness and importance are extracted as TOSA information via AoI and value of information (VoI) to guarantee the timing requirement, respectively \cite{TimingProcIEEE}. A typical effectiveness level performance metric that is used to minimize is the Linear Quadratic Gaussian (LQG) cost function, and usually the lower the value of the LQG function the higher the quality of control (QoC).
\subsubsection{{Machine Learning}}
In the following, we focus on task-oriented communications for two distributed ML models, which are FL and SL.
\paragraph{Federated Learning} For the time non-critical tasks, such as NLP and image classification, the goal of FL is to guarantee a high learning accuracy without latency constraints. Traditional loss functions for NLP and image classification, such as mean square error (MSE), MAE, and cross-entropy, can be directly used as effectiveness level performance metrics. However, for time critical tasks, such as object recognition in self-driving cars, the goal of FL is to balance the trade-off between learning accuracy, communication latency, and computation latency. The effectiveness level performance metrics are loss functions with latency constraints. Time critical tasks bring
communication challenges, and communication-efficient FL should be designed to decrease the model size to satisfy latency constraints via federated dropout, federated pruning, and model compression.
Federated dropout is a simple way to prevent the learning
model from overfitting through randomly dropping neurons
and is only used during the training phase, which decreases
communication and computation latencies and slightly improves learning accuracy. However, during the testing phase, the extracted task-oriented information is the whole learning model and transmitted between the server and devices. The extracted task-oriented information is the model with non-dropped weights. Meanwhile, federated dropout does not need model recovery.
Unlike federated dropout only temporarily removing neurons, federated pruning permanently removes neurons in either or both training and testing phases. The extracted task-oriented information in federated pruning is the pruned model. The decision of which parameters to remove is made by considering the importance of each parameter. The pruning ratio should be carefully designed to guarantee learning accuracy and extra computation latency is required to calculate the importance of parameters. Thus, how to design federated pruning methods with low computation complexity needs to be investigated. In addition, federated pruning does not need model recovery.
Model-compression schemes, such as sparsification and quantization decrease the model size. The extracted task-oriented information is the sparse or quantized model. However, these methods slightly decrease the convergence rate and achieve a modest accuracy (about 85 $\%$). Thus, how to design a model compression algorithm with high learning accuracy still needs to be investigated. In addition, compressed FL model needs to be recovered at the receiver.
\paragraph{Split Learning} In SL, the smashed data and its gradient associated with the cut layer are the extracted task-oriented information and transmitted between the server and devices, where no model recovery is required. When multiple devices exist in SL, all devices interact with the edge server in a sequential manner, resulting in high training latency. For time non-critical tasks, such as NLP and image classification, the goal of SL is to achieve high learning accuracy without latency constraints and the effectiveness level performance metrics are the same as that of the FL. However, for time critical tasks, the SL cannot guarantee the requirement of low latency because of its sequential training pattern.
To adapt the SL to time critical tasks, such as real-time object tracking, splitfed learning (SFL) \cite{thapa2022 splitfed} and hybrid split and federated learning (HSFL) \cite{Xiaolan} are proposed, where they combine the primary advantages of FL and SL. The effectiveness level performance metrics of the SFL and HSFL are learning accuracy and training latency. However, SFL and HSFL assume that the model is split at the same cut layer and the server-side model is trained in a synchronous mode. Splitting at the same cut layer leads to asynchronization of device-side model training and smashed data transmission. Thus, how to select an optimal split point and deal with the asynchronization of SL remain important challenges to solve. Also, different split points can result in different smashed data. Thus, how to merge these smashed data in the server-side model should be considered.
\subsection{Chain Task}
In this section, we analyze more complicated but practical chain tasks including XR-aided teleoperation and chain of control, where multiple entities cooperate through communication links to execute the task.
\subsubsection{XR-aided Teleoperation}
XR-aided teleoperation aims to integrate 3 D virtual objects/environment into remote robotic control, which can provide the manipulator with immersive experience and high operation efficiency \cite{xr-aided}. To implement a closed-loop XR-aided teleoperation system, the wireless network is required to support mixed types of data traffic, which includes control and command (C\&C) transmission, haptic information feedback transmission, and rendered $360 ^{\circ}$ video feedback transmission. Stringent communication requirements have been proposed to support XR-aided teleoperation use case, where over 50 $\mathrm{Mb/s}$ bandwidth is needed to support video transmission, and reliability over $1 -10 ^{-6 }$ within millisecond latency is required to support haptic and C\&C transmission.
As XR-aided teleoperation task relies on both parallel and consecutive communication links, how to guarantee the cooperation among these communication links to execute the task is of vital importance. Specifically, the parallel visual and haptic feedback transmissions should be aligned with each other when arriving at the manipulator, and consecutive C\&C and feedback transmissions should be within the motion-to-photon delay constraint. Either violation of alignment in parallel links or latency constraint in consecutive links will lead to BIP and cybersickness. Therefore, both parallel alignment and consecutive latency should be quantified into effectiveness level performance metrics to guarantee the success of XR-aided teleoperation. Moreover, due to the motion-to-photon delay, the control error between the expected trajectory and actual trajectory will accumulate along with the time, which may lead to task failure. Hence, how to alleviate the accumulated error remains an important challenge to solve.
\subsubsection{Chain of control}
In the scenario of a swarm of (autonomous) robots where they need to perform a collaborative task (or a set of tasks) within a deadline over a wireless network, an effective communication protocol that takes into account the peculiarities of such a scenario is needed. Otherwise, the generated and transmitted data will be of very high volume that eventually will face congestion in the network causing large delays and the operated control mechanisms will not be synced causing inefficient or even dangerous operation. Consider the simple case of two robots, let's say Robot A and Robot B that are communicating through a wireless network and they are not collocated. Robot A controls remotely Robot B such that to execute a task and the outcome of that operation will be fed to Robot A for performing a second operation to send the outcome back to Robot B. All this must happen within a strict deadline. The amount of information that is generated, transmitted, processed, and sent back can be very large with the traditional information agnostic approach. On the other hand, if we take into account the semantics of information and the purpose of communication, we change the whole information chain, from its generation point until its utilization. Therefore, defining TOSA metrics for the control loop and communication between a swarm of (autonomous) robots is crucial and it will significantly reduce the amount of information leading to a more efficient operation.
\section{Case Study}
We validate the effectiveness of ML algorithms in optimizing time-critical TOSA communication under one typical case, namely, MEC-enabled and reconfigurable intelligent surface (RIS)-assisted terahertz (THz) VR network.
\begin{figure}[! h]
\centering
\includegraphics[width=3.5 in]{latency_qoe. pdf}
\caption{Average QoE and VR interaction latency of the MEC-enabled THz VR network of each time slot via CDRL with viewpoint prediction, where the VR interaction latency constraint is 20 ms. }
\label{basic_modules}
\end{figure}
The goal of wireless VR networks is to guarantee high QoE under VR interaction latency constraints. Note that the QoE guarantees the seamless, continuous, smooth, and uninterrupted experience of each VR user. However, traditional wireless VR networks transmit whole $360 ^\circ$ videos, which leads to low QoE and high VR interaction latency. In the simulation, we extract the FoV of VR users as TOSA information to decrease the transmission data size. To extract the FoV, viewpoints of VR users are predicted via recurrent neural network (RNN) algorithms. Based on the predicted viewpoints, the corresponding FoVs of VR video frames can be rendered and transmitted in advance. Thus, the MEC does not need to render and transmit the whole $360 ^\circ$ VR video frames, which can substantially decrease the VR interaction latency and improve the QoE of VR users.
Fig. 2 plots the average QoE and VR interaction latency of each time slot via constrained deep reinforcement learning with the viewpoint prediction via RNN. It is observed that the proposed ML algorithms maximize the long-term QoE of VR users under the VR interaction latency threshold.
\section{Conclusion}
In this article, we propose a generic task-oriented and semantics-aware (TOSA) communication framework incorporating both semantic and effectiveness levels for various tasks with diverse data types.
We first identify the the unique characteristics of all existing and new data types in 6 G networks and summarize the semantic information with its extraction methods. To achieve task-oriented communications for various data types, we then present the corresponding TOSA information, their TOSA information extraction and recovery methods, and effectiveness level performance metrics
for both time critical and non-critical tasks.
Importantly, our results demonstrate that our proposed TOSA communication framework can be tailored for VR data to maximize the long-term QoE within VR interaction latency constraints. The paradigm shift from conventional Shannon’s bit-oriented communication design towards the TOSA communication design
will flourish new research on task-driven, context and importance-aware data transmission in 6 G networks.
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\section{Supplementary Material}
\begin{figure}[!h]
\raggedright
\includegraphics[width=0.48\linewidth]{plength}
\includegraphics[width=0.48\linewidth]{pmom}
\includegraphics[width=0.48\linewidth]{p_mlike}
\includegraphics[width=0.48\linewidth]{rtot2}
\includegraphics[width=0.48\linewidth]{ortot}
\includegraphics[width=0.48\linewidth]{aang}
\includegraphics[width=0.48\linewidth]{dist}
\includegraphics[width=0.48\linewidth]{nmue}
\caption{The distribution of the eight input variables into the multi-variate analysis for simulated protons (blue solid line, scaled up x10 for visibility) and non-proton events (red solid line) in the SKI-IV atmospheric neutrino sample after reduction and precuts.
The corresponding data is superimposed on the figures as black points.
The definition of the variables can be found in the main text.}
\label{fig:MVAdist}
\end{figure}
For a given light pattern, compared to muons, protons tend to be reconstructed with a lower momentum due to their larger mass, and the track lengths can be inferred from the sharper ring edges by the hadronic interaction cutoff.
Therefore, the fitted track length, fitted momentum, as well as the proton likelihood over muon assumption provided by the proton fitter~\cite{Super-Kamiokande:2009kfy} are used in the MVA.
Since the kinematic range of protons that can be reconstructed is narrower than muons, the corrected charge within $70^\circ$ of the direction of proton events is typically more peaked.
On the other hand, the corrected charge outside $70^\circ$ of the direction for protons is typically lower since a recoil proton is a clean event, while a low energy muon from atmospheric neutrinos can have $\gamma$ emissions and other processes yielding low intensity Cherenkov light.
These charge distributions, together with the reconstructed Cherenkov angle, are provided by an initial event fitter without PID assumption, and are sent into the MVA as well.
The initial vertex reconstruction at SK utilizes hit time, while ring fitting gives a more precise vertex for muons that is typically close to the initial vertex.
We therefore include the distance between the initial vertex and the ring fitting vertex in the MVA.
Finally, since muons typically produce decay electrons, while protons can only produce decay electrons via secondary particle decays, we employed the number of decay electron to reject muons.
The distributions of the eight input variables described above for the simulated protons, non-proton events mostly composed of muons, and the data after the precuts are shown in Fig.~\ref{fig:MVAdist}.
\begin{figure}
\centering
\includegraphics[width=0.98\linewidth]{skymap.png}
\caption{The skymap of the proton sample. Events passing the zenith angle cut $-\cos\theta_z>0.2$ is marked by red, and other events are in blue. The solid star marker in black marks the Galactic Center and the star marker in grey marks the off-source, which is 180$^\circ$ away in right ascension. The yellow region around the star marker indicates the directional cut of $\cos\theta>0.6$.}
\label{fig:skymap}
\end{figure}
\end{document}
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\section{Introduction}
OODIDA is a modular system for concurrent distributed data analytics for the automotive domain, targeting fleets of reference vehicles~\cite{ulm2019oodida}. Its main purpose is to process telemetry data at its source as opposed to transferring all data over the network and processing it on a central cloud server (cf.~Fig.~\ref{fig:whole_fleet}). A data analyst interacting with this system uses a Python library that assists in creating and validating assignment specifications. Updating this system with new computational methods necessitates terminating and redeploying software. However, we would like to perform updates without terminating ongoing tasks. We have therefore extended our system with the ability to execute user-defined code both on client devices (on-board) and the cloud server (off-board), without having to redeploy any part of it. As a consequence, OODIDA is now highly suited for rapid prototyping. The key aspect of our work is that active-code replacement of Python modules piggybacks on the existing Erlang/OTP infrastructure of OODIDA for sending assignments to clients, leading to a clean design. This paper is a condensed version of a work-in-progress paper~\cite{ulm2019active}, giving an overview of our problem (Sect.~\ref{problem}) and its solution (Sect.~\ref{solution}), followed by an evaluation (Sect.~\ref{eval}) and related work~(Sect.~\ref{related}).
\vspace{-2.3em}
\newsavebox{\tempbox}
\begin{figure}
\sbox{\tempbox}{\includegraphics[width=0.45\textwidth, trim=2cm 2.5cm 2cm 2.5cm, clip]{context.pdf}}
\subfloat[OODIDA in context]{\usebox{\tempbox}\label{fig:context}}%
\qquad
\subfloat[Whole-fleet assignment]{\vbox to \ht\tempbox{%
\vfil
\hbox to 0.45\textwidth{
\includegraphics[width=0.45\textwidth, trim=2cm 2.5cm 2cm 2.5cm, clip]{exec_3.pdf}
}
\vfil}\label{fig:whole_fleet_assignment}}%
\caption{OODIDA overview and details: In (a) user nodes $\boldsymbol{u}$ connect to a central cloud $b$, which connects to clients $\boldsymbol{c}$. The shaded nodes are implemented in Erlang/OTP; the other nodes are external Python applications, i.e.\ the user front-ends $\boldsymbol{f}$, the external server application $a$, and external client applications $\boldsymbol{a}$. In (b) the core of OODIDA is shown with permanent nodes (dark) and temporary handlers (light) in an instance of a whole-fleet assignment. Cloud node $b$ spawned an assignment handler $b'$. After receiving an incoming task, clients $x, y$ and $z$ spawned task handlers $x'$, $y'$, and $z'$ that interact with external applications. Nodes $x$ and $x'$ correspond to $c_1$ in (a) etc.}
\label{fig:whole_fleet}
\end{figure}
\section{Problem}
\label{problem}
OODIDA has been designed for rapid prototyping, which implies that it frequently needs to be extended with new computational methods, both for on-board and off-board data processing. To achieve this goal, Python applications on the cloud and clients have to be updated. Assuming that we update both, the following steps are required: The user front-end $f$ needs to be modified to recognize the new off-board and on-board keywords for the added methods, including checks of assignment parameter values. In addition, the cloud and client applications have to be extended with the new methods. All ongoing assignments need to be terminated and the cloud and clients shut down. Afterwards, we can redeploy and restart the system. This is disruptive, even without taking into account potentially long-winded software development processes in large organizations. On the other hand, the turn-around time for adding custom methods would be much shorter if we could do so at runtime. Active-code replacement targets this particular problem, with the goal of further improving the suitability of OODIDA for rapid prototyping.
\section{Solution}
\label{solution}
With active-code replacement, the user can define a custom Python module for the cloud and for client devices. It is implemented as a special case of an assignment. The front-end $f$ performs static and dynamic checks, attempting to verify correctness of syntax and data types. If these checks succeed, the provided code is turned into a JSON object and ingested by user node $u$ for further processing. Within this JSON object, the user-defined code is stored as an encoded text string. It is forwarded to cloud node $b$, which spawns an assignment handler $b'$ for this particular assignment. Custom code can be used on the cloud and/or clients. Assuming clients have been targeted with active-code replacement, node $b'$ turns the assignment specification into tasks for all clients $\boldsymbol{c}$ specified in the assignment. Afterwards, task specifications are sent to the specified client devices. There, the client process spawns a task handler for the current task, which monitors task completion. The task handler sends the task specification in JSON to an external Python application, which turns the given code into a file, thus recreating the Python module the data analyst initially provided. The resulting files are tied to the ID of the user who provided it. After the task handler is done, it notifies the assignment handler $b'$ and terminates. Similarly, once the assignment handler has received responses from all task handlers, it sends a status message to the cloud node and terminates. The cloud node sends a status message to inform the user that their custom code has been successfully deployed. Deploying custom code to the cloud is similar, the main difference being that $b'$ communicates with the external Python application on the cloud.
If a custom on-board or off-board computation is triggered by a special keyword in an assignment specification, Python loads the user-provided module. The user-specified module is located at a predefined path, which is known to the Python application. The custom function is applied to the available data after the user-specified number of values has been collected. When an assignment uses custom code, external applications reload the custom module with each iteration of an assignment. This leads to greater flexibility: Consider an assignment that runs for an indefinite number of iterations. As external applications can process tasks concurrently, and code replacement is just another task, the data analyst can react to intermediate results of an ongoing assignment by deploying custom code with modified algorithmic parameters while this assignment is ongoing. As custom code is tied to a user ID, there is furthermore no interference due to custom code that was deployed by other users. The description of active-code replacement so far indicates that the user can execute custom code on the cloud server and clients, as long as the correct inputs and outputs are consumed and produced. What may not be immediately obvious, however, is that we can now create \emph{ad hoc} implementations of even the most complex OODIDA use cases in custom code, such as federated learning~\cite{mcmahan2016communication}.
Inconsistent updates are a problem in practice, i.e.~results sent from clients may have been produced with different custom code modules in the same iteration of an assignment. This happens if not all clients receive the updated custom code before the end of the current iteration. To solve this problem, each provided module with custom code is tagged with its md5 hash signature, which is reported together with the results from the clients. The cloud only uses the results tagged with the signature that achieves a majority. Consequently, results are never tainted by using different versions of custom code in the same iteration.
\section{Evaluation}
\label{eval}
The main benefit of active-code replacement is that code for new computational methods can be deployed right away and executed almost instantly, without affecting other ongoing tasks. In contrast, a standard update of the cloud or client installation necessitates redeploying and restarting the respective components of the system. In an idealized test setup, where the various workstations that run the user, cloud and client components of OODIDA are connected via Ethernet, it takes a fraction of a second for a custom on-board or off-board method to be available for the user to call when deployed with active-code replacement, as shown in Table~\ref{tab:comparison}. On the other hand, automated redeployment of the cloud and client installation takes roughly 20 and 40 seconds, respectively. The runtime difference between a standard update and active-code replacement amounts to three orders of magnitude. Of course, real-world deployment via a wireless or 4G connection would be slower as well as error-prone. Yet, the idealized evaluation environment reveals the relative performance difference of both approaches, eliminating potentially unreliable data transmission as a source of error.
This comparison neglects that, compared to a standard update, active-code replacement is less bureaucratic and less intrusive as it does not require interrupting any currently ongoing assignments. Also, in a realistic industry scenario, an update could take days or even weeks due to software development and organizational processes. However, it is not the case that active-code replacement fully sidesteps the need to update the library of computational methods on the cloud or on clients as OODIDA enforces restrictions on custom code. For instance, some parts of the Python standard library are off-limits. Also, the user cannot install external libraries. Yet, for typical algorithmic explorations, which users of our system regularly conduct, active-code replacement is a vital feature that increases user productivity far more than the previous comparison may imply. That being said, due to the limitations of active-code replacement, it is complementary to the standard update procedure rather than a competitive approach.
\begin{table}[]
\centering{
\caption{Runtime comparison of active-code replacement of a moderately long Python module versus regular redeployment in an idealized setting. The former has a significant advantage. Yet, this does not factor in that a standard update is more invasive but can also be more comprehensive. The provided figures are the averages of five runs.\\}
\label{tab:comparison}
\begin{tabular}{@{}lll@{}}
\toprule
& Cloud & Client \\ \toprule
Active-code replacement \phantom{aaaaaaa}
& 20.3 ms \phantom{aaaaaaa} & 45.4 ms \\
Standard redeployment
& 23.6 s & 40.8 s\\
\bottomrule
\end{tabular}
}
\end{table}
\section{Related Work}
\label{related}
The feature described in this paper is an extension of the OODIDA platform~\cite{ulm2019oodida}, which originated from \texttt{ffl-erl}, a framework for federated learning in Erlang/OTP~\cite{ulm2019b}. In terms of descriptions of systems that perform active-code replacement, Polus by Chen et al.~\cite{chen2007polus} deserves mention. A significant difference is that it replaces larger units of code instead of isolated modules. It also operates in a multi-threading environment instead of the highly concurrent message-passing environment of OODIDA. We also noticed a similarity between our approach and Javelus by Gu et al.~\cite{gu2012javelus}. Even though they focus on updating a stand-alone Java application as opposed to a distributed system, their described "lazy update mechanism" likewise only has an effect if a module is indeed used. This mirrors our approach of only loading a custom module when it is needed.
\subsubsection*{Acknowledgements.}
\small{
This research was financially supported by the project On-board/Off-board Distributed Data Analytics (OODIDA) in the funding program FFI: Strategic Vehicle Research and Innovation (DNR 2016-04260), which is administered by VINNOVA, the Swedish Government Agency for Innovation Systems. It took place in the Fraunhofer Cluster of Excellence "Cognitive Internet Technologies." Simon Smith and Adrian Nilsson helped with a supplementary part of the implementation of this feature and carried out the performance evaluation. Ramin Yahyapour (University of G{\"o}ttingen) provided insightful comments during a poster presentation.
}
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\section{Introduction}
Gas dynamical processes are believed to play an important role in
the evolution of astrophysical systems on all length scales.
Smoothed particle hydrodynamics~(SPH) is a powerful gridless
particle method to solve complex fluid-dynamical problems. SPH
has a number of attractive features such as its low numerical
diffusion in comparison to grid based methods. An adequate
scenario for SPH application is the unbound astrophysical
problems, especially on the shock propagation~(see, e.g., Liu \&
Liu 2003). In this way, the basic principles of the SPH is
written in this paper and the simulation of isothermal and
adiabatic shocks are applied to test the ability of this numerical
simulation to produce known analytic solutions.
The program is written in Fortran and is highly portable. This
package supports only calculations for isothermal and adiabatic
shock waves. It is possible to change (to complete) the program
for a wide variety of applications ranging from astrophysics to
fluid mechanics. The program is written in modular form, in the
hope that it will provide a useful tool. I ask only that:
\begin{itemize}
\item If you publish results obtained using some parts of this
program, please consider acknowledging the source of the
package.
\item If you discover any errors in the program or
documentation, please promptly communicate the to the author.
\end{itemize}
\section{Formulation of Shock Waves}
An extremely important problem is the behavior of gases subjected
to compression waves. This happens very often in the cases of
astrophysical interests. For example, a small region of gas
suddenly heated by the liberation of energy will expand into its
surroundings. The surroundings will be pushed and compressed.
Conservation of mass, momentum, and energy across a shock front
is given by the Rankine-Hugoniot conditions~(Dyson \& Williams
1997)
\begin{equation}\label{e:rh1}
\rho_1 v_1=\rho_2 v_2
\end{equation}
\begin{equation}
\rho_1 v_1^2+ K\rho_1^\gamma =\rho_2 v_2^2+ K\rho_2^\gamma
\end{equation}
\begin{equation}\label{e:rh3}
\frac{1}{2}v_1^2 + \frac{\gamma}{\gamma-1} K \rho_1^{\gamma-1}=
\frac{1}{2}v_2^2 + \frac{\gamma}{\gamma-1} K \rho_2^{\gamma-1} +Q
\end{equation}
where the equation of state, $p=K\rho^\gamma$, is used. In
adiabatic case, we have $Q=0$, and for isothermal shocks, we will
set $\gamma=1$.
We would interested to consider the collision of two gas sheets
with velocities $v_0$ in the rest frame of the laboratory. In this
reference frame, the post-shock will be at rest and the pre-shock
velocity is given by $v_1=v_0+v_2$, where $v_2$ is the shock front
velocity. Combining equations (\ref{e:rh1})-(\ref{e:rh3}), we have
\begin{equation}\label{e:v2}
v_2=a_0[-\frac{b}{2}+\sqrt{1+\frac{b^2}{4}+(\gamma-1)
(\frac{M_0^2}{2}-q)}]
\end{equation}
where $a_0\equiv \gamma K\rho_1 ^{\gamma-1}$ is the sound speed,
$M_0\equiv v_0/a_0$ is the Mach number, $b$ and $q$ are defined as
\begin{equation}
b\equiv \frac{3-\gamma}{2}M_0+ \frac{\gamma-1}{M_0}q\quad ;\quad
q\equiv \frac{Q}{a_0^2}.
\end{equation}
Substituting (\ref{e:v2}) into equation (\ref{e:rh1}), density of
the post-shock is given by
\begin{equation}\label{e:den}
\rho_2=\rho_1\{1+\frac{M_0}{[-\frac{b}{2}+\sqrt{1+\frac{b^2}{4}+(\gamma-1)
(\frac{M_0^2}{2}-q)}]}\}.
\end{equation}
\section{SPH Equations}
The smoothed particle hdrodynamics was invented to simulate
nonaxisymmetric phenomena in astrophysics~(Lucy 1977, Gingold \&
Monaghan 1977). In this method, fluid is represented by $N$
discrete but extended/smoothed particles (i.e. Lagrangian sample
points). The particles are overlapping, so that all the physical
quantities involved can be treated as continues functions both in
space and time. Overlapping is represented by the kernel
function, $W_{ab} \equiv W(\textbf{r}_a-\textbf{r}_b,h_{ab})$,
where $h_{ab} \equiv (h_a+h_b)/2$ is the mean smoothing length of
two particles $a$ and $b$. The continuity, momentum and energy
equation of particle $a$ are~(Monaghan 1992)
\begin{equation}
\rho_a=\sum_b m_b W_{ab}
\end{equation}
\begin{equation}
\frac{d\textbf{v}_a}{dt}=-\sum_b m_b (\frac{p_a}{\rho_a}+
\frac{p_b}{\rho_b}+ \Pi_{ab}) \nabla_a W_{ab}
\end{equation}
\begin{equation}
\frac{du_a}{dt}=\frac{1}{2} \sum_b m_b (\frac{p_a}{\rho_a}+
\frac{p_b}{\rho_b}+ \Pi_{ab}) \textbf{v}_{ab} \cdot \nabla_a
W_{ab}
\end{equation}
where $\textbf{v}_{ab}\equiv \textbf{v}_a- \textbf{v}_b$ and
\begin{equation}
\Pi_{ab}=\cases{
\frac{\alpha v_{sig} \mu_{ab}
+\beta \mu_{ab}^2}{\bar{\rho}_{ab}}, &
if $\textbf{v}_{ab}.\textbf{r}_{ab}<0$,\cr
0 , & otherwise,}
\end{equation}
is the artificial viscosity between particles $a$ and $b$, where
$\bar{\rho}_{ab}= \frac{1}{2}(\rho_a+\rho_b)$ is an average
density, $\alpha\sim 1$ and $\beta\sim 2$ are the artificial
coefficients, and $\mu_{ab}$ is defined as its usual form
\begin{equation}
\mu_{ab}=-\frac{\textbf{v}_{ab}
\cdot\textbf{r}_{ab}}{\bar{h}_{ab}}
\frac{1}{r_{ab}^2/\bar{h}_{ab}^2+\eta^2}
\end{equation}
with $\eta\sim 0.1$ and $\bar{h}_{ab}= \frac{1}{2}(h_a+h_b)$. The
signal velocity, $v_{sig}$, is
\begin{equation}
v_{sig}=\frac{1}{2}(c_a+c_b)
\end{equation}
where $c_a$ and $c_b$ are the sound speed of particles. The SPH
equations are integrated using the smallest time-steps
\begin{equation}
\Delta t_a=C_{cour}MIN[ \frac{h_a}{\mid \textbf{v}_a\mid},
(\frac{h_1}{\mid\textbf{a}_1\mid})^{0.5}, \frac{u_a}{\mid du_a/dt
\mid}, \frac{h_a}{\mid dh_a/dt \mid}, \frac{\rho_a}{\mid
d\rho_a/dt \mid}]
\end{equation}
where $C_{cour}\sim 0.25$ is the Courant number.
\section{Results and Prospects}
The chosen physical scales for length and time are $[l]=3.0 \times
10^{16} m$ and $[t]=3.0 \times 10^{13} s$, respectively, so the
velocity unit is approximately $1km.s^{-1}$. The gravity constant
is set $G= 1$ for which the calculated mass unit is $[m]=4.5
\times 10^{32} kg$. There is considered two equal one dimensional
sheets with extension $x= 0.1 [l]$, which have initial uniform
density and temperature of $\sim 4.5\times 10^8 m^{-3}$ and $\sim
10K$, respectively.
Particles with a positive x-coordinate are given an initial
negative velocity of Mach 5, and those with a negative
x-coordinate are given a Mach 5 velocity in the opposite
direction. In adiabatic shock, with $M_0=5$, the post-shock
density must be $2.9$, which is obtained from analytic solution
(\ref{e:den}) with $Q=0$ and $\gamma=2$. The Results of adiabatic
shock are shown in Fig.~1-3. In isothermal shock, with $M_0=5$,
the post-shock density must be $26.9$, which is obtained from
analytic solution Equ.~(\ref{e:den}) with $\gamma=1$. The Results
of isothermal shock are shown in Fig.~4-5. Algorithm of the
program is shown in Fig.~6.
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\section{{\label{sec:intro}} Introduction}
Understanding of the steady state of non-equilibrium systems is the subject of intense research. The typical situation is a solid in contact with two heat baths at different temperature. At the difference of equilibrium systems where the Boltzmann-Gibbs formalism provides an explicit description of the steady state, no equivalent theory is available for non-equilibrium stationary state (NESS). In the last few years, efforts have been concentrated on stochastic lattice gases (\cite{S1 }). For these latter precious informations on the steady state like the typical macroscopic profile of conserved quantities and the form of the Gaussian fluctuations around this profile have been obtained (\cite{S1 }). Recently, Bertini, De Sole, Gabrielli, Jona-Lasinio and Landim proposed a definition of non-equilibrium thermodynamic functionals via a macroscopic fluctuation theory (MFT) which gives for large diffusive systems the probability of atypical profiles (\cite{BL1 }, \cite{BL2 }) in NESS. The method relies on the theory of hydrodynamic limits and can be seen as an infinite-dimensional generalization of the Freidlin-Wentzel theory. The approach of Bertini et al. provides a variational principle from which one can write the equation of the time evolution of the typical profile responsible of a given fluctuation. The resolution of this variational problem is, however, in general very difficult and it has only been carried for two models : the Symmetric Simple Exclusion Process (SSEP) (\cite{BL2 }) and the Kipnis Marchioro Presutti (KMP) model (\cite{BL3 }). Hence, it is of extreme importance to identify simple models where one can test the validity of MFT. The most studied stochastic lattice gas is the Simple Exclusion Process. Particles perform random walks on a lattice but jumps to occupied sites are suppressed. Hence the only interaction is due to the exclusion condition. The only conserved quantity by the bulk dynamics is the number of particles. In this situation, the heat reservoirs are replaced by particles reservoirs which fix the density at the boundaries. The KMP process is a Markov process composed of particles on a lattice. Each particle has an energy and a stochastic mechanism exchange energy between nearest-neigbor particles (\cite{KMP}). The real motivation is to extend MFT for Hamiltonian systems (\cite{B-r}). Unfortunately, for these later, even the derivation of the typical profile of temperature adopted by the system in the steady state is out of range of the actual techniques (\cite{BLR}). The difficulty is to show that the systems behave ergodically, e. g. that the only time invariant measures locally absolutely continuous w. r. t. Lebesgue measure are, for infinitely extended spatial uniform systems, of the Gibbs type. For some stochastic lattice gases it can be proven but it remains a challenging problem for Hamiltonian dynamics. We investigate here the MFT for a system of harmonic oscillators perturbed by a conservative noise (\cite{Ber1 }, \cite{BO}, \cite{BBO}). These stochastic perturbations are here to reproduce (qualitatively) the effective (deterministic) randomness coming from the Hamiltonian dynamics (\cite{OVY}, \cite{LO}, \cite{FFL}). This hybrid system can be considered as a first modest step in the direction of purely Hamiltonian systems. From a more technical point of view, SSEP and KMP are gradient systems and have only one conserved quantity. For gradient systems the microscopic current is a gradient (\cite{KL}) so that the macroscopic diffusive character of the system is trivial. Dealing with non-gradient models, we have to show that microscopically, the current is a gradient up to a small fluctuating term. The decomposition of the current in these two terms is known in the hydrodynamic limit literature as a \textit{fluctuation-dissipation equation} (\cite{EMY}). In general, it is extremely difficult to solve such an equation. Our model has two conserved quantities, energy and deformation, and is non-gradient. But fortunately, an exact fluctuation-dissipation equation can be established. In fact we are not able to apply MFT for the two conserved quantities but only for the temperature field which is a simple, but non-linear, functional of the energy and deformation fields. The paper is organized as follows. In section \ref{SEC:2 }, we define the model. In section \ref{SEC:3 } we establish the fluctuation-dissipation equation and obtain hydrodynamic limits for the system in a diffusive scale. Section \ref{SEC:4 } is devoted to a physical interpretation of the fluctuating term appearing in the fluctuation-dissipation equation. In section \ref{SEC:5 } we compute the covariance of the fluctuation fields in the NESS by a dynamical approach and show the covariance for the energy presents a non-locality we retrieve in the large deviation functional (the quasi-potential). The latter is studied in section \ref{SEC:6 } for the temperature field. \section{{\label{sec:2 }} The Model}
We consider the dynamics of the open system of length $N$. Atoms are labeled by $x \in \{1, \dots, N\}$. Atom $1 $ and $N$ are in contact with two heat reservoirs at different
temperatures $T_{\ell}$ and $T_r$. Momenta of atoms are denoted by $p_1, \dots, p_{N}$ and the distance between particles are denoted by $r_1,
\dots, r_{N-1 }$. The Hamiltonian of the system is given by
\begin{eqnarray*}
{\ensuremath{\mathcal H}}^N = \sum_{x=1 }^{N} {e}_x, \quad {e}_x = \frac{ p_x^2 + r_x^2 }2 \qquad x= 1, \dots, N-1 \\
{e}_{N} = \frac{p_{N}^2 }2. \end{eqnarray*}
We consider stochastic dynamics where the probability
density distribution on the phase space at time $t$, denoted
by $P(t, p, r)$, evolves following the Fokker-Planck equation
\begin{equation*}
\partial_t P = N^2 {\mathcal L}^* P
\end{equation*}
Here ${\mathcal L} = {\mathcal A} + \gamma {\mathcal S}+{\mathcal B}_{1, T_\ell} +{\mathcal B}_{N, T_r}$ is the generator of the process and ${\mathcal L}^*$ the
adjoint operator. The factor $N^2 $ in front of ${\mathcal L}^*$ is here because we have speeded up the time by $N^2 $, this corresponds to a diffusive scaling. ${\mathcal A}$ is the usual Hamiltonian vector field
\begin{eqnarray*}
{\mathcal A} = \sum_{x=1 }^{N-1 } (p_{x+1 } - p_x) \partial_{r_x}
+ \sum_{x=2 }^{N-1 } (r_x - r_{x-1 }) \partial_{p_x}\\
+ (r_1 -\ell) \partial_{p_1 } - (r_{N-1 }-\ell) \partial_{p_{N}}
\end{eqnarray*}
The constant $\ell$ fix the deformation at the boundaries. $\mathcal S$ is the generator of the stochastic perturbation and $\gamma>0 $ is a positive parameter that regulates its strength. The operator $S$ acts only on momenta $\{p_x\}$ and generates a diffusion on the surface of constant kinetic energy. This is defined as follows. For every nearest neigbor atoms $x$ and $x+1 $, consider the following one dimensional surface of constant kinetic energy $e$
\begin{equation*}
{\mathbb S}_e^1 =\{ (p_x, p_{x+1 }) \in {\mathbb R}^2 ; p_x^2 +p_{x+1 }^2 =e\}
\end{equation*}
The following vector field $X_{x, x+1 }$ is tangent to ${\mathbb S}_{e}^1 $
\begin{equation}
\label{eq:4 }
X_{x, x +1 } = p_{x+1 } \partial_{p_x} - p_x \partial_{p_{x+1 }}
\end{equation}
so $X_{x, x+1 }^2 $ generates a diffusion on ${\mathbb S}_e^1 $ (Brownian motion on the circle). We define
\begin{equation*}
{\mathcal S}= \frac {1 }{2 } \sum_{x=1 }^{N-1 } X_{x, x +1 }^2
\end{equation*}
${\mathcal B}_{1, T_\ell}$ and ${\mathcal B}_{N, T_r}$ are two boundary generators of Langevin baths at temperature $T_\ell$ and $T_r$
\begin{equation*}
{\mathcal B}_{x, T}= \frac 12 \left(T \partial_{p_x}^2 - p_x \partial_{p_x} \right)
\end{equation*}
The bulk dynamics conserve two quantities: the total energy ${\mathcal H}^N=\sum_{x=1 }^{N} {e}_x$ and the total deformation ${\mathcal R}^N=\sum_{x=1 }^{N-1 } r_x$. The energy conservation law can be read locally as (\cite{Ber1 }, \cite{BO})
\begin{equation*}
e_x (t) - e_x (0 ) = J^e_{x} (t) -J_{x+1 }^e (t)
\end{equation*}
where $J^e_{x} (t)$ is the total energy current between $x-1 $ and $x$ up to time $t$. This can be written as
\begin{equation*}
J^e_{x} (t)=N^2 \int_0 ^t j^e_{x} (s)ds + M_{x} (t)
\end{equation*}
In the above, $M_{x} (t)$ is a martingale, i. e. a stochastic noise with mean $0 $. The instantaneous energy current $j^{e}_{x}$ can be written as
\begin{equation*}
j_{x}^e= -r_{x-1 }p_{x} -\cfrac{\gamma}{2 }\nabla(p_{x}^2 )
\end{equation*}
The first term $-r_{x-1 } p_x$ is the Hamiltonian contribution to the energy current while the noise contribution is given by the discrete gradient $-(\gamma/2 ) \nabla(p_{x}^2 )=(\gamma/2 )(p_{x}^2 -p_{x+1 }^2 )$. Similarly, the deformation instantaneous current $j^{r}_{x}$ between $x-1 $ and $x$ is given by
\begin{equation*}
j^{r}_{x}=-p_{x}
\end{equation*}
We denote by $\mu^{ss}=<\cdot>_{ss}$ the invariant probability measure for the process. In the case $T_\ell = T_r=T$, the system is in thermal equilibrium. There is no heat flux and the Gibbs invariant measure (or canonical measure) is a product Gaussian measure $\mu_{ss}=\mu^{T, \ell}$ depending on the temperature $T$ and the mean deformation $\ell$:
\begin{equation}
\label{eq:mu}
\mu^{T, \ell} = Z_{T}^{-1 } \exp \left\{ -\cfrac{1 }{2 T}\sum_{x=1 }^{N} p_x^2 -\cfrac{1 }{2 T}\sum_{x=1 }^{N-1 } (r_x-\ell)^2 \right\}
\end{equation}
\section{{\label{sec:3 }} Fluctuation-dissipation equation and Hydrodynamic limit}
Diffusive interacting particle systems can be classified in two categories: gradient systems and non-gradient systems (\cite{KL}). For the first, we can write the curent of the conserved quantities as a spatial discrete gradient. For example SSEP and KMP process are gradient systems. A powerful approach introduced by Varadhan (\cite{V}) to study non-gradient systems is to obtain a fluctuation-dissipation equation, meaning a decomposition of the current $j$ of conserved quantities as the sum of a microscopic gradient $\nabla h$ and of a fluctuating term of the form ${\mathcal L} u$:
\begin{equation}
\label{eq:fde0 }
j=\nabla h + {\mathcal L}u
\end{equation}
where ${\mathcal L}$ is the generator of the interacting particle system. In fact, the equality (\ref{eq:fde0 }) is only an approximation in a suitable Hilbert space (\cite{KL}). Fortunately, for our system, we can write an equality like (\ref{eq:fde0 }) without approximations. The fluctuation-dissipation equation for the deformation current $j^r$ and the energy current $j^e$ is given by (\cite{Ber1 })
\begin{equation}
\label{eq:fde}
\begin{cases}
j^{r}_{x}= -\gamma^{-1 }\nabla(r_x) + {\mathcal L} h_x\\
j^e_{x}=\nabla\left[\phi_x \right]+{\mathcal L} g_x
\end{cases}
\end{equation}
where
$$\phi_x = \cfrac{1 }{2 \gamma}r_x^2 + \cfrac{\gamma}{2 } p_x^2 +\cfrac{1 }{2 \gamma} p_x p_{x+1 } +\cfrac{\gamma}{4 }\nabla(p_{x+1 }^2 )$$
and
$$h_x= \gamma^{-1 } p_x, \quad g_x =\cfrac{p_x^2 }{4 }+\cfrac{p_x}{2 \gamma} (r_x + r_{x-1 })$$
Assume that initially the system starts from a local equilibrium $<\cdot>$ with macroscopic deformation profile $u_0 (q)$ and energy profile ${\varepsilon}_0 (q)$, $q \in [0,1 ]$. This means that if the macroscopic point $q \in [0,1 ]$ is related to the microscopic point $x$ by $q=x/N$ then at time $t=0 $
\begin{equation*}
<r_{[Nq]} (0 )> \to u_0 (q), \quad <e_{[Nq]} (0 )> \to {\varepsilon}_0 (q)
\end{equation*}
as $N$ goes to infinity. The currents are related to conserved quantities by the conservation law
\begin{eqnarray*}
\partial_t <r_{[Nq]}(t)> \approx -N \partial_q <j_{[Nq]}^r (t)>, \\
\partial_{t} <e_{[Nq]}(t)> \approx - N \partial_q <j_{[Nq]}^e (t)>. \end{eqnarray*}
By (\ref{eq:fde}) and the fact that the terms $N<{\mathcal L} h_x>$ and $N<{\mathcal L} g_x>$ are of order ${\mathcal O} (N^{-1 })$ and do not contribute to the limit (\cite{Ber1 }) we get
\begin{equation*}
\begin{cases}
\partial_t <r_{[Nq]} (t)> \approx \gamma^{-1 } \Delta <r_{[Nq]}(t)>\\
\partial_t <e_{[Nq]} (t)> \approx \Delta <\phi_{[Nq]} (t)>\\
\end{cases}
\end{equation*}
To close the hydrodynamic equations, one has to replace the term $<\phi_{[Nq]} (t)>$ by a function of the conserved quantities $<r_{[Nq]} (t)>$ and $<e_{[Nq]}(t)>$. The replacement is obtained through a "thermal local equilibrium" statement (see \cite{Ber1 }, \cite{BO}, \cite{ELS1 }, \cite{ELS2 }, \cite{KL}, \cite{S1 }) for a rigorous justification in the context of conservative interacting particle systems). We repeat here the arguments of \cite{BL3 } for convenience of the reader. Thermal local equilibrium assumption corresponds to assume that each given macroscopic region of the system is in equilibrium, but different regions may be in different equilibrium states, corresponding to different values of the parameters. Let us consider an atom with position $q=x/N$ which is far from the boundary and introduce a very large number $2 L+1 $ of atoms in microscopic units ($L\gg 1 $), but still an infinitesimal number at the macroscopic level ($(2 L+1 )/N \ll 1 $). We choose hence $L=\epsilon N$ where $\epsilon \ll 1 $ in order to have these two conditions. We consider the system in the box $\Lambda_L (x)$ composed of the atoms labeled by $x-L, \ldots, x+L$.
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and will not have any effect on the hydrodynamical scales and the slow observables which are locally conserved by the dynamics and need much longer times to relax. We can then replace the term $<\phi_{[Nq]} (t)>$ by $\lambda_{{\bar r}_q (t), {\bar e}_q (t)} (\phi_0 )$. By equivalence of ensembles, in the thermodynamic limit $N \to \infty$ and then $\varepsilon \to 0 $, this last quantity is equivalent to
\begin{equation*}
\cfrac{\gamma+\gamma^{-1 }}{2 } <e_{[Nq]} (t)> + \cfrac{\gamma^{-1 }-\gamma}{4 } (<r_{[Nq]} (t)>)^2
\end{equation*}
We have obtained the time evolution of the deformation/energy profiles $u(t, q)=\lim <r_{[Nq]} (t)>$, $\varepsilon (t, q)= \lim <e_{[Nq]} (t)>$ in the bulk. At the boundaries, Langevin baths fix temperature at $T_\ell$ and $T_r$. Hence it is more natural to introduce the couple of deformation/temperature profiles rather than deformation/energy profiles. The temperature profile $T(t, q)$ is related to $u(t, q)$ and $\varepsilon (t, q)$ by $\varepsilon (t, q) = T(t, q) +u(t, q)^2 /2 $. Deformation and temperature profiles evolve according to the following equations
\begin{equation}
\label{eq:hl}
\begin{cases}
\partial_t T = \cfrac{1 }{2 }(\gamma+\gamma^{-1 }) \Delta T + \gamma^{-1 }(\nabla u)^2, \\
\partial_t u =\gamma^{-1 } \Delta u, \\
T(t,0 )=T_\ell, \;\; T(t,1 )=T_r, \\
u(t,0 )=u(t,1 )=\ell, \\
T(0, q)=T_0 (q), \; u(0, q)=u_{0 }(q). \end{cases}
\end{equation}
As $t$ goes to infinity, the system reaches its steady state characterized in the thermodynamic limit by a linear temperature profile ${\bar T}(q)=T_\ell + (T_r -T_\ell) q$ and a constant deformation profile ${\bar r}(q)= \ell$. The system satisfies Fourier's law and the conductivity is given by $(\gamma+\gamma^{-1 })/2 $ (\cite{BO}). \section{{\label{sec:4 }} Interpretation of the fluctuation-dissipation equation}
We have seen that functions $h_x$ and $g_x$ had no influence on the form of the hydrodynamic equations. This is well understood by the fact that they are related to \textit{first order} corrections to local equilibrium as we explain below. Assume $T_{\ell (r)}=T \pm \delta T /2 $ with $\delta T$ small. For $\delta T = 0 $, the stationary state $<\cdot>_{ss}$ equals the Gibbs measure $\mu_{T}^{\ell}$ (see \ref{eq:mu}). If $\delta T$ is small, it is suggestive to try an ansatz for $<\cdot>_{ss}$ in the form:
$$\tilde{\mu} = Z^{-1 } \, \prod_x dp_x dr_x \exp\left(- \cfrac{1 }{2 T(x/N)} (p_x^2 + (r_x -\ell)^2 ) \right)$$
where $T(\cdot)$ is the linear interpolation on $[0,1 ]$ between $T_\ell$ and $T_r$. ${\tilde \mu}$ is the "local equilibrium" approximation of $<\cdot>_{ss}$. Let $f_{ss}$ be the density of the stationary state $<\cdot>_{ss}$ with respect to ${\tilde \mu}$, i. e. the solution of ${\mathcal L}^{*, T(\cdot)} f_{ss} =0 $. Here ${\mathcal L}^{*, T(\cdot)}$ is the adjoint operator of ${\mathcal L}$ in ${\mathbb L}^2 ({\tilde \mu})$. It turns out that
\begin{eqnarray*}
{\mathcal L}^{*, T(\cdot)}&=& -{\mathcal A} +\gamma {\mathcal S} + B_{1, T_\ell} +B_{N, T_r} \\
&+&\cfrac{\delta T}{T^2 } \left( \cfrac{1 }{N} \sum_{x=1 }^{N-2 } {\tilde j}^{e}_{x, x+1 } -\ell \, \cfrac{1 }{N} \sum_{x=1 }^{N-1 } {\tilde j}^r_{x, x+1 }\right)\\
&+&\cfrac{\delta T}{T^2 } \left( \cfrac{1 }{N} \sum_{x=1 }^{N-1 } p_x p_{x+1 } X_{x, x+1 }\right) +\cfrac{\delta T}{4 } (\partial^2 _{p_1 } -\partial^{2 }_{p_N})\\
&+& {\mathcal O} ((\delta T)^2 ) +{\mathcal O}(N^{-1 })
\end{eqnarray*}
where ${\hat j}^{e}$ and ${\hat j}^r$ are the energy and deformation currents for the reversed dynamics at equilibrium. They are obtained from $j^e$ and $j^r$ by reversing momenta $p \to -p$. Expanding $f_{ss}$ at first order $f_{ss}= 1 +\delta T\, v + o(\delta T)$, we get that for large $N$ and small gradient temperature $\delta T$, $v$ has to satisfy the following Poisson equation:
\begin{equation*}
(-{\mathcal A} + \gamma {\mathcal S}) v = T^{-2 } \left(\cfrac{1 }{N} \sum_{x=1 }^{N-2 } {\tilde j}^{e}_{x} -\ell \, \cfrac{1 }{N} \sum_{x=1 }^{N-1 } {\tilde j}^r_{x}\right)
\end{equation*}
Let ${\hat v}$ the function obtained from $v$ by reversing momenta. By the fluctuation-dissipation equation (\ref{eq:fde}) we get
\begin{equation*}
{\hat v}=\cfrac{1 }{NT^2 }\sum_{x=1 }^{N-1 } (g_x- \ell h_x ) + {\mathcal O} (N^{-1 })
\end{equation*}
Therefore the functions $g_x$ and $h_x$ are directly related to first order corrections to local equilibrium. \section{{\label{sec:5 }} Non-equilibrium Fluctuations and steady State Correlations}
Assume that initially the system starts from a local equilibrium $<\cdot>$ with macroscopic deformation profile $u_0 (q)$ and temperature profile $T_0 (q)$, $q \in [0,1 ]$. The time-dependant deformation fluctuation field $R_t^N$ and energy fluctuation field $Y_t^N$ are defined by
$$R_t^N (H)=\cfrac{1 }{\sqrt N}\sum_{x=1 }^{N} H\left (x/N \right)\left(r_x (t) - u\left(t, x/N\right) \right)$$
$$Y_t^N (G)=\cfrac{1 }{\sqrt N}\sum_{x=1 }^{N} G\left(x/N\right)\left(e_x (t) -\varepsilon(t, x/N) \right)$$
where $H, G$ are smooth test functions, $(T(t, \cdot), u(t, \cdot))$ are solutions of the hydrodynamic equations (\ref{eq:hl}) with $\varepsilon=T+u^2 /2 $. The fluctuation-dissipation equations (section \ref{SEC:3 }) give (\cite{FNO}, \cite{S1 }):
\begin{equation*}
\begin{cases}
R_t^N (H)=R_0 ^N (H)+ \cfrac{1 }{\gamma}\int_{0 }^t R_s^N(\Delta H)ds+{\ensuremath{\mathcal M}}_t^{1, N}\\
Y_t^N (G)=Y_0 ^N (G)+\gamma \int_{0 }^{t} Y_s^N (\Delta G)ds\\
\phantom{Y_t^N (G)} + \int_{0 }^{t}ds\left\{\cfrac{1 }{\sqrt{N}}\sum_{x\in {\mathbb T}_N}(\Delta G)(x/N)f_x(\omega_s)\right\}\\
\phantom{Y_t^N (G)} +{\ensuremath{\mathcal M}}_t^{2, N}
\end{cases}
\end{equation*}
where ${\ensuremath{\mathcal M}}^{1, N}$ and ${\ensuremath{\mathcal M}}^{2, N}$ are martingales and $f_x$ is the function defined by
$$f_x (\omega)= \cfrac{\left(\gamma^{-1 }-\gamma\right)}{2 } r_x^2 -\left(\cfrac{1 }{2 \gamma}p_{x+1 }p_x -\cfrac{\gamma}{4 }\nabla^{*} p_x^2 \right)$$
Covariance of the limit martingales are computed using standard stochastic calculus and thermal equilibrium property (\cite{S1 }, \cite{FNO}):
\begin{equation*}
\left<\left({\ensuremath{\mathcal M}}^{1, N}_t\right)^2 \right> \rightarrow \cfrac{2 }{\gamma}\int_0 ^t ds \int_{[0,1 ]}dq T(q, s)(\nabla H)^{2 }(q)
\end{equation*}
\begin{equation*}
\begin{split}
\left<\left({\ensuremath{\mathcal M}}^{2, N}_t\right)^2 \right> \rightarrow \cfrac{2 }{\gamma}\int_{[0,1 ]}dq \int_0 ^t ds u^{2 } (q, s) T(q, s)(\nabla G)^{2 }(q)\\
+(\gamma+\gamma^{-1 })\int_0 ^t ds \int_{[0,1 ]}dq T^{2 }(q, s)(\nabla G)^{2 }(q)
\end{split}
\end{equation*}
\begin{eqnarray*}
\left<{\ensuremath{\mathcal M}}^{1, N}_t {\ensuremath{\mathcal M}}^{2, N}_t\right> \rightarrow \\
\cfrac{2 }{\gamma}\int_0 ^t ds\int_{[0,1 ]}dq u(s, q) T(s, q)(\nabla G)(q)(\nabla H)(q)
\end{eqnarray*}
Hence $R_t^N$ converges as $N$ goes to infinity to the solution of the linear stochastic differential equation:
\begin{equation}
\label{eq:R}
\partial_t R =\cfrac{1 }{\gamma}\Delta R -\nabla\left[\sqrt{\cfrac{2 }{\gamma}T(t, q)}W_{1 }(t, q)\right]
\end{equation}
where $W_1 (t, q)$ is a standard space time white noise. \\
The description of the limit for the energy fluctuation field is more demanding. We have first to close the equation. In order to do it, we use a "dynamical Boltzmann-Gibbs lemma" (\cite{KL}, \cite{S1 }). Observables are divided into two classes: non-hydrodynamical and hydrodynamical. The first one are non conserved quantities and fluctuate in a much faster scale than the others (in the time scale where these last change). Hence, they should average out and only their projection on the hydrodynamical variables should persist in the limit. One expects there exist constants $C, D$ such that
\begin{eqnarray*}
\cfrac{1 }{\sqrt N} \int_0 ^t ds \sum_{x=1 }^{N} (\Delta G)(x/N) \left\{ f_x (\omega_s)\right. \\
\left. -C(r_x -u(s, x/N)) -D\left(\ensuremath{\mathcal E}_x-\varepsilon (s, x/N)\right)\right\}
\end{eqnarray*}
vanishes as $N$ goes to infinity. Constants $C$ and $D$ depend on the macroscopic point $q=x/N$ and on the time $t$. In order to compute these constants, we assume thermal local equilibrium. Around the macroscopic point $q$, the system is considered in equilibrium with a fixed value of the deformation $u(t, q)$ and of the temperature $T(t, q)$. The constant $C, D$ are then computed by projecting the function $f_x$ on the deformation and energy fields (\cite{KL}, \cite{S1 }). If $\mu^{T, \ell}$ is the Gibbs equilibrium measure with temperature $T$ and mean deformation $\ell$ (the mean energy is then ${\varepsilon}={\ell}^{2 }/2 +T$), we have $\Phi(\ell, {\varepsilon})=\mu^{T, \ell}(f_{x})=\varepsilon +{\ell}^{2 }/2 $ and then
\begin{equation*}
C=\partial_\ell \Phi (u(s, q), {\varepsilon} (s, q)), \quad D= \partial_{\varepsilon} \Phi (u(s, q), {\varepsilon} (s, q))
\end{equation*}
Therefore the time-dependant energy fluctuation field $Y_t^N$ converges as $N$ goes to infinity to the solution of the linear stochastic differential equation:
\begin{widetext}
\begin{equation}
\label{eq:Y}
\partial_t Y =\cfrac{1 }{2 }\left(\gamma+\cfrac{1 }{\gamma}\right)\Delta Y +\cfrac{1 }{2 }\left(\cfrac{1 }{\gamma}-\gamma\right)\Delta (u(t, q) R)
-\nabla\left[\sqrt{\gamma +\gamma^{-1 }}T(t, q)W_{2 }(t, q)+u(t, q)\sqrt{\cfrac{2 T(t, q)}{\gamma}}W_1 (t, q)\right]
\end{equation}
\end{widetext}
where $W_2 (t, q)$ is a standard space-time white noise independent of $W_1 (t, q)$. Remark that the deterministic terms in (\ref{eq:R}) and (\ref{eq:Y}) result from linearizing the nonlinear equation as (\ref{eq:hl}). We now compute the fluctuations fields for the NESS $<\cdot>_{ss}$ which is obtained as the stationary solution of the Langevin equations (\ref{eq:R}-\ref{eq:Y}). The field $L_t$ defined by $L_t= -\ell R_t +Y_t$ is solution of the Langevin equation
\begin{equation*}
\partial_t L = b\Delta L -\nabla \left[\sqrt{2 b}{T} (t, q) W_2 (q, t) \right]
\end{equation*}
with $b=\cfrac{1 }{2 } (\gamma +\gamma^{-1 })$. The fields $R_t$ and $L_t$ are solutions of independent decoupled linear Langevin equations and converge as $t$ goes to infinity to independent Gaussian fields. It follows that $R_t$ and $L_t$ converge to stationary fluctuation fields ${R}_{ss}$ and ${Y}_{ss}$ such that
\begin{eqnarray*}
\text{Cov}({R}_{ss}(G), {R}_{ss}(H))=\int_{0 }^{1 }dq G(q)H(q){\bar T}(q)\\
\text{Cov}({Y}_{ss}(G), {Y}_{ss} (H))=\int_{0 }^{1 }dq G(q) H(q)\left\{{\bar T}^2 (q)+\ell^2 {\bar T}(q)\right\}\\
+2 (T_\ell -T_r)^{2 } \int_0 ^1 G(q) (\Delta^{-1 } H) (q) dq\\
\text{Cov}({Y}_{ss} (G), {R}_{ss}(H))=\ell\int_{0 }^{1 } H(q) G(q){\bar T}(q)dq
\end{eqnarray*}
Observe that the covariance of the fluctuations of energy is composed of two terms. The first one corresponds to Gaussian fluctuations for the energy under local equilibrium state while the second term represents the contribution to the covariance due to the long range correlations in the NESS. As in the case of SSEP and KMP process, the correction is given by the Green function of the Dirichlet Laplacian (\cite{BL3 }, \cite{S2 }).
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KL}): we perturb the dynamics in such a way that the prescribed paths become typical and we compute the cost of such perturbation. \\
Fix a path ${\mathcal Y} (t, \cdot)=(u(t, \cdot), {\varepsilon} (t, \cdot))$. The empirical deformation profile ${\mathcal R}^N_t$ and empirical energy profile ${\mathcal E}_t^N$ are defined by
\begin{eqnarray}
\label{eq:RE0 }
{\mathcal R}_t^N (q)= N^{-1 } \sum_{x=1 }^N r_x (t) {\bf 1 }_{\left[x/N, (x+1 )/N \right)} (q), \\
{\mathcal E}_t^N (q)= N^{-1 } \sum_{x=1 }^N e_x (t) {\bf 1 }_{\left[x/N, (x+1 )/N \right)} (q). \nonumber
\end{eqnarray}
In appendix, we explain how to define a Markovian dynamics associated to a couple of functions $H(t, q), G(t, q)$, $q \in [0,1 ]$, such that the perturbed system has hydrodynamic limits given by $u$ and $\varepsilon$. This is possible if the function $F=(H, G)$ solves the Poisson equation
\begin{equation}
\label{eq:poisson}
\begin{cases}
\partial_t {\mathcal Y}= \Delta {\mathcal Y} -\nabla(\sigma \nabla F))\\
F(t,0 )=F(t,1 )=(0,0 )
\end{cases}
\end{equation}
where the mobility $\sigma:=\sigma (u, \varepsilon)$ is given by
\begin{equation}
\label{eq:mobility}
\sigma(u, \varepsilon)=
2 \left(
\begin{array}{cc}
T & uT\\
uT & u^2 T +T^2
\end{array}
\right), \quad T={\varepsilon}-u^2 /2
\end{equation}
The perturbed process defined a probability measure $\tilde{\mathbb P}$ on the deformation/energy paths space by mean of the empirical deformation and energy profiles (see (\ref{eq:RE0 })). Our goal is to estimate the probability
\begin{eqnarray*}
\phantom{a}&\phantom{=}&{\mathbb P} \left[({\mathcal R}_s^N, {\mathcal E}_{s}^N)\sim(u(s, \cdot), \varepsilon (s, \cdot)), \; s\in [0, t]\right]\\
&=&{\tilde {\mathbb E}} \left[ \cfrac{d{\mathbb P}}{d{\tilde{{\mathbb P}}}} {\bf 1 }_{\left\{({\mathcal R}_s^N, {\mathcal E}_{s}^N )\sim(r(s, \cdot), \varepsilon (s, \cdot)), \; s\in[0, t]\right\}}\right]
\end{eqnarray*}
To avoid irrelevant complications due to the fluctuations of the initial state which have no incidence on the derivation of the quasi-potential, we assume that the initial profiles $u_0 $ and $T_0 $ are the stationary profiles ${\bar r} (q)=\ell$ and ${\bar T} (q) =T_\ell +(T_r -T_\ell)q$. The function $F$ is such that
$${\tilde {\mathbb P}} \left[({\mathcal R}_s^N, {\mathcal E}_{s}^N)\sim(u(s, \cdot), {\varepsilon} (s, \cdot)), \; s \in [0, t]\right]\approx 1 $$
In the appendix we show that in the large $N$ limit, under ${\tilde {\mathbb P}}$, the Radon-Nikodym derivative is given by
\begin{equation*}
\cfrac{d{\mathbb P}}{d{\tilde{{\mathbb P}}}} \approx \exp\left\{ -N J_{[0, t]}(u, {\varepsilon})\right\}
\end{equation*}
where
\begin{equation}
\label{eq:29 }
J_{[0, t]} (u, {\varepsilon})=\cfrac{1 }{2 }\int_0 ^{t} ds<\nabla F (s, \cdot), \sigma \nabla F (s, \cdot)>_q
\end{equation}
where $\sigma$ is here for $\sigma (u(s, \cdot), \varepsilon(s, \cdot))$ and $<\cdot, \cdot>_q$ for the usual scalar product in ${\mathbb L}^2 ([0,1 ], dq)$. Hence we have obtained
\begin{eqnarray*}
{\mathbb P} \left[({\mathcal R}_s^N, {\mathcal E}_{s}^N)\sim(u(s, \cdot), {\varepsilon}(s, \cdot)), \; s\in [0, t]\right] \\
\approx \exp\left\{ -N J_{[0, t]}(u, {\varepsilon})\right\}
\end{eqnarray*}
\subsection{The quasi-potential}
To understand what is the quasi-potential, consider the following situation. Assume the system is macroscopically in the stationary profile $(u(-\infty, \cdot), {\varepsilon} (-\infty, \cdot))= (\ell, {\bar T}(\cdot)+\ell^2 /2 )$ at $t=-\infty$ but at $t=0 $ we find it in the state $(u(q), \varepsilon(q))$. We want to determine the most probable trajectory followed in the spontaneous creation of this fluctuation. According to the precedent subsection this trajectory is the one that minimizes $J_{[-\infty,0 ]}$ among all trajectories $({\hat u}, {\hat \varepsilon})$ connecting the stationary profiles to $(u, \varepsilon)$. The quasi-potential is then defined by
\begin{equation*}
W (u, \varepsilon) = \inf_{({\hat u}, {\hat \varepsilon})} J_{[0, t]} ({\hat u}, {\hat \varepsilon})
\end{equation*}
MFT postulates the quasi-potential $W$ is the appropriate generalization of the free energy for non-equilibrium systems and this has been proven rigorously for SSEP (\cite{BL2 }). $W$ is solution of an infinite-dimensional Hamilton-Jacobi equation which is in general very difficult to solve. It has been solved for specific models (SSEP and KMP) having a single conservation law (\cite{BL2 }, \cite{BL3 }). For the system we consider, two quantities are conserved and we are not able to solve this Hamilton-Jacobi equation. Nevertheless we can compute the quasi-potential for the temperature profile (\ref{eq:30 }) in the case $\gamma=1 $. The latter is obtained by projecting the quasi-potential $W$ on the deformation/energy profiles with a prescribed temperature profile. Consider the system in its steady state $<\cdot>_{ss}$. Our aim is here to estimate the probability that the empirical kinetic energy defined by
\begin{equation}
\label{eq:30 }
\Theta^N (q) = N^{-1 } \sum_{x=1 }^N p_x^2 {\bf 1 }_{\left[x/N, (x+1 )/N \right)} (q)
\end{equation}
is close to some prescribed temperature profile $\pi (q)$ different form the linear profile ${\bar T} (q)= T_\ell +(T_r -T_\ell)q$. This probability will be exponentially small in $N$
\begin{equation*}
\left< \left[ \Theta^N (q) \sim \pi (q) \right] \right>_{ss} \approx \exp( -N V(\pi))
\end{equation*}
By MFT, the rate function $V(\pi)$ coincides with the following \textit{projected quasi-potential}
\begin{equation*}
V(\pi)=\inf_{t >0 } \inf_{(u, \varepsilon)\in \ensuremath{\mathcal A}_{t, \pi}} J_{[0, t]} (u, \varepsilon)
\end{equation*}
where the paths set ${\ensuremath{\mathcal A}}_{t, \pi}$ is defined by
\begin{eqnarray*}
{\ensuremath{\mathcal A}}_{t, \pi} = \left\{ (u, \varepsilon); \quad {\varepsilon}(t, \cdot)-\cfrac{u^2 (t, \cdot)}{2 }=\pi(\cdot); \right. \\
\left. {\phantom{\cfrac{u^2 (t, \cdot)}{2 }}}u (0, \cdot)=\ell, \, T(0, \cdot)= {\bar T}(\cdot)\right\}
\end{eqnarray*}
Paths ${\mathcal Y}=(u, \varepsilon) \in {\mathcal A}_{t, \pi}$ must also satisfy the boundary conditions
\begin{equation}
\label{eq:bc}
u(t,0 )=u(t,1 )=\ell, \quad \varepsilon (t,0 )=T_\ell+\ell^2 /2, \, \varepsilon (t,1 )=T_r +\ell^2 /2
\end{equation}
In fact, it can be shown that $J_{[0, t]} (u, \varepsilon) =+\infty$ if the path ${\mathcal Y}$ does not satisfy these boundary conditions. \\
Our main result is the computation of the projected quasi-potential:
\begin{equation}
\label{eq:32 }
V (\pi)= \inf_{\tau \in \ensuremath{\mathcal T}} [{\ensuremath{\mathcal F}} (\pi, \tau) ]
\end{equation}
where $\ensuremath{\mathcal T}=\{ \tau \in C^{1 }([0,1 ]); \; \tau' (q)>0, \; \tau(0 )=T_\ell, \; \tau (1 )=T_r\}$ and
\begin{equation*}
{\cal F} (\pi, \tau)= \int_0 ^1 dq \left[\cfrac{\pi (q)}{\tau(q)} -1 - \log\cfrac{\pi(q)}{\tau (q)} -\log \cfrac{\tau' (q)}{(T_r - T_{\ell})}\right]
\end{equation*}
Before proving (\ref{eq:32 }) let us make some remarks. First, $V(\pi)$ is equal to the rate function for the KMP process (\cite{BL3 }). Nevertheless, it is not easy to understand the deep reason. The symmetric part ${\mathcal S}$ of the generator ${\mathcal L}$ is more or less a time-continuous version of the KMP process for the kinetic energy but the Hamiltonian part has a non-trivial effect on the latter since it mixes momenta with positions. Hence, the derivation of the quasi-potential for the kinetic energy can not be derived from the the computations for the KMP process. Secondly, we are able to compute $V$ only for $\gamma=1 $. When $\gamma$ is equal to $1 $ hydrodynamic equations for the deformation and for the energy are decoupled but since temperature is a non-linear function of deformation and energy, it is not clear why it helps-- but it does. Formula (\ref{eq:32 }) shows that the large deviation functional $V$ is nonlocal and consequently not additive: the probability of temperature profile in disjoint macroscopic regions is not given by the product of the separate probabilities. Nonlocality is a generic feature of NESS and is related to the ${\mathcal O} (N^{-1 })$ corrections to local thermal equilibrium. \vspace{0,5 cm}
Let us call $S(\pi)$ the right hand side of equality (\ref{eq:32 }). \vspace{0,5 cm}
For every time independent deformation/energy profiles $(r(q), e(q))$ and $\tau (q) \in \ensuremath{\mathcal T}$ we define the functional
\begin{equation}
\label{eq:U}
U(r, e, \tau)=\int_0 ^1 dq \left\{ \cfrac{T}{\tau} -1 - \log \cfrac{T}{\tau} -\log \cfrac{\tau'}{T_\ell - T_r} +\cfrac{(r-\ell)^2 }{2 \tau}\right\}
\end{equation}
where $T(q)=e(q)-r(q)^2 /2 $ the temperature profile corresponding to $(r(q), e(q))$. Define the function $\tau:=\tau(r, e)$ of ${\ensuremath{\mathcal T}}$ as the unique increasing solution of:
\begin{equation}
\label{eq:tau}
\begin{cases}
\tau^2 \cfrac{\Delta \tau}{(\nabla \tau)^2 } = \tau -T -\cfrac{1 }{2 } (r-\ell)^2 \\
\tau(0 )=T_\ell, \; \tau (1 )= T_r
\end{cases}
\end{equation}
Fix deformation/energy paths satisfying boundary conditions (\ref{eq:bc}) and define ${\mathcal Z}$ by
\begin{equation}
\label{eq:YZ}
{\mathcal Y}=\left(\begin{array}{c}u\\\varepsilon \end{array}\right), \qquad {\mathcal Z}=\left[\partial_t {\mathcal Y}-\Delta {\mathcal Y}+\nabla(\sigma \nabla(\delta U))\right], \\
\end{equation}
In the appendix we show the following formula
\begin{widetext}
\begin{eqnarray}
\label{eq:43 }
J_{[0, t]}(u, \varepsilon)=U(u(t, \cdot), {\varepsilon} (t, \cdot), \tau({\varepsilon} (t, \cdot), u(t, \cdot)))-U(u(0, \cdot), {\varepsilon}(0, \cdot), \tau(u(0, \cdot), {\varepsilon} (0, \cdot)))\\
\nonumber \\
+\cfrac{1 }{2 }\int_0 ^t ds \left<\nabla^{-1 }{\mathcal Z}, \sigma^{-1 }\nabla^{-1 }{\mathcal Z} \right>_q+\cfrac{1 }{4 }\int_0 ^t ds \int_0 ^1 dq (u(s, q)-\ell)^4 \cfrac{(\nabla \tau)^2 (s, q)}{\tau^4 (s, q)}\nonumber
\end{eqnarray}
\end{widetext}
where
\begin{equation*}
\delta U=\left(\begin{array}{c}\cfrac{\delta U}{\delta r}\\ \phantom{a}\\ \cfrac{\delta U}{\delta {e}} \end{array}\right) (u, \varepsilon, \tau(u, \varepsilon))
\end{equation*}
If $(u, \varepsilon)$ belongs to ${\mathcal A}_{t, \pi}$,
\begin{eqnarray*}
U(\, u(0, \cdot), {\varepsilon}(0, \cdot), \tau(u(0, \cdot), {\varepsilon} (0, \cdot))\, )\\
=U(\, \ell, \bar{T}+\ell^2 /2, \tau(\ell, {\bar T} +\ell^2 /2 )\, )=0
\end{eqnarray*}
and
$$U(u(t, \cdot), {\varepsilon} (t, \cdot), \tau(u(t, \cdot), \varepsilon(t, \cdot))) \ge {\mathcal F} (\pi, \tau (u(t, \cdot), {\varepsilon} (t, \cdot)). $$
The two last terms on the right hand side of (\ref{eq:43 }) are positive so that for every paths in ${\mathcal A}_{t, \pi}$, we have
\begin{eqnarray*}
J_{[0, t]}(u, \varepsilon) \geq S(\pi)
\end{eqnarray*}
and we obtain hence
\begin{equation}
\label{eq:firstinequality}
V(\pi)\geq S(\pi)
\end{equation}
To obtain the other sense of the inequality, we have to construct an optimal path $(u^*, {\varepsilon}^*) \in \ensuremath{\mathcal A}_{t, \pi}$ such that the two last terms in the right hand side of $(\ref{eq:43 })$ are equal to $0 $, i. e. \begin{equation}
\label{eq:44 }
\begin{cases}
\partial_t {\mathcal Y}=\Delta {\mathcal Y}-\nabla(\sigma \nabla(\delta U))\\
u(t, q)=\ell
\end{cases}
\end{equation}
We note $T^{*}={\varepsilon}^* -{u^*}^2 /2 $ the corresponding temperature.
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\begin{equation*}
V(\pi)=\inf_{t >0 } \inf_{\ensuremath{\mathcal A}_{t, \pi}} J_{[0, t]}(u, \varepsilon) \leq S(\pi)
\end{equation*}
This inequality with (\ref{eq:firstinequality}) shows that $V(\pi)=S(\pi)$.
It remains to prove that such ``good'' path exists. The proof is similar to \cite{BL3 } and we shall merely outline it. Equation (\ref{eq:44 }) is equivalent to the following one
\begin{equation}
\label{eq:45 }
\begin{cases}
\begin{array}{l}
\partial_t T^{*}=-\Delta(T^*)+2 \nabla\left[\cfrac{(T^*)^2 }{(\tau^*)^2 }\nabla(\tau^*)\right]\\
u^*(t, q)=\ell
\end{array}
\end{cases}
\end{equation}
where $\tau^*(t, \cdot)= \tau(\ell, T^*(t, \cdot) +\ell^2 /2 )$. Let us denote by $\theta^*(s, \cdot)=T^{*}(t-s, \cdot)$ the time reversed path of $T^*$. $\theta^{*}$ can be constructed in the following procedure. We define $\theta^{*}(s, q), \; s\in[0, t], \; q\in [0,1 ]$ by
$$\theta^{*}(s, \cdot)= \rho (s, \cdot) -2 \rho(s, \cdot)^2 \cfrac{\Delta \rho (s, \cdot)}{[(\nabla \rho)(s, \cdot)]^2 }$$
where $\rho(s, q)$ is the solution of
\begin{equation*}
\label{eq47 }
\begin{cases}
\partial_s \rho =\Delta \rho\\
\rho (s,0 )=T_{\ell}, \qquad \rho (s,1 )=T_r\\
\rho(0, q)=\rho_0 (q)=\tau(\ell, \pi+\ell^2 /2 )(q)
\end{cases}
\end{equation*}
It can be checked that $T^{*}(s, q)=\theta^{*}(t-s, q)$ solves $(\ref{eq:45 })$. Moreover, we have $T^{*}(0, \cdot)=\theta^{*}(t, \cdot)$ and $T^{*}(t, \cdot)=\pi(\cdot)$. This path belongs to ${\ensuremath{\mathcal A}}_{t, \pi}$ only as $t\to \infty$ since $\theta^{*}(t, \cdot)$ goes to $\bar{T}(\cdot)$ as $t\to +\infty$. We have hence in fact to define $T^*$ by the preceding procedure in some time interval $[t_1, t]$ and to interpolate $\bar{T}(\cdot)$ to $\pi^{*}(t_1, \cdot)$ in the time interval $[0, t_1 ]$ (see \cite{BL2 }, \cite{BL3 } for details).
This optimal path is also obtained as the time reversed solution of the hydrodynamic equation corresponding to the process with generator ${\ensuremath{\mathcal L}}^{*}$. It is easy to show that this last hydrodynamic equation is in fact the same as the hydrodynamic equation corresponding to ${\ensuremath{\mathcal L}}$. This is the "generalized" Onsager-Machlup theory developed in \cite{BL1 } for NESS: "the spontaneous emergence of a macroscopic fluctuation takes place most likely following a trajectory which can be characterized in terms of the time reversed process. " Observe also the following a priori non trivial fact: the optimal path is obtained with a constant deformation profile.
\section{Conclusions}
In the present work we obtained hydrodynamic limits, Gaussian fluctuations and (partially) large fluctuations for a model of harmonic oscillators perturbed by a conservative noise. Up to now MFT has been restricted to gradient systems with a single conservation law. This work is hence the first one where MFT is applied for a non-gradient model with two conserved quantities. The quasi-potential for the temperature has been computed in the case $\gamma=1 $ and it turns out that it coincides with the one of the KMP process. Our results show this system exhibits generic features of non-equilibrium models : long range correlations and non-locality of the quasi-potential.
Nevertheless our study is not completely satisfactory. It would be interesting to extend the previous results to the case $\gamma \neq 1 $ and to compute the quasi-potential for the two conserved quantities and not only for the temperature. The difficulty is that there does not exist general strategy to solve the corresponding infinite-dimensional Hamilton-Jacobi equation.
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