# Fluctuation Relation for the Dissipative Flux: The Role of Dynamics, Correlations and Heat Baths
## 2 Steady-State Fluctuation Relation
Then, the dynamics are called time-reversal invariants if there exists one transformation \(i:\mathcal{M}\to\mathcal{M}\) whose double application is the identity, such that  
\[iS^{t}\Gamma=S^{-t}i\Gamma\quad\text{for all}\quad\Gamma\in\mathcal{M}. \tag{6}\]  
Such a general definition is motivated by the fact that \(S^{-t}\Gamma=iS^{t}i\Gamma\), i.e., that the reversal operation generates the backward evolution, and it was implicit in Ref. [1], because the SLLOD equations are not invariant under the time-reversal operation defined by Equation (5), but they are under, e.g., the following mapping:  
\[i(x,y,z,p_{x},p_{y},p_{z})=(x,-y,z,-p_{x},p_{y},-p_{z}). \tag{7}\]  
Let \(f^{(0)}\) be an initial probability distribution on \(\mathcal{M}\), and define the _dissipation function_ as:  
\[\Omega^{(0)}=-G\cdot\partial_{\Gamma}\ln f^{(0)}-\Lambda. \tag{8}\]  
Its name is justified by the fact that \(\Omega^{(0)}\) represents indeed the dissipated power, if \(f^{(0)}\) is the equilibrium (non-driven) distribution that obeys the same constraints of the driven  
if \(f^{(0)}\) is even under the application of \(i\), i.e., \(f^{(0)}(i\Gamma)=f^{(0)}(\Gamma)\), as appropriate for an equilibrium distribution. The transient fluctuation relation immediately follows:  
\[\frac{\mu^{(0)}(\overline{\Omega}^{(0)}_{0,\tau}\in\mathcal{B}_{A,\delta})}{\mu^ {(0)}(\overline{\Omega}^{(0)}_{0,\tau}\in\mathcal{B}_{-A,\delta})}=\exp\{\tau[A +\epsilon(A,\delta,\tau)]\}\,,\quad\text{where}\quad\epsilon(A,\delta,\tau) \leq\delta \tag{19}\]  
with an error \(\epsilon\) that is not larger than \(\delta\), although it depends on all parameters of the theory. Therefore, \(\epsilon\) can be made as small as one wants, by taking \(\delta\) as correspondingly small. This also means:  
\[\frac{\mu^{(0)}\left(\overline{\Omega}^{(0)}_{0,\tau}\in\mathcal{B}_{A,\delta }\right)}{\mu^{(0)}\left(\overline{\Omega}^{(0)}_{0,\tau}\in\mathcal{B}_{-A, \delta}\right)}=\left\langle\exp\Bigl{(}-\tau\,\overline{\Omega}^{(0)}\Bigr{)} \right\rangle^{-1}_{\overline{\Omega}^{(0)}\in\mathcal{B}_{A,\delta}} \tag{20}\]  
where the right-hand side means the inverse of the average of the exponential of \(\Omega^{(0)}\) integrated from time \(0\) to time \(\tau\), over the trajectories that yield a value close to \(A\), with a difference not larger than \(\delta\). Indeed,  
\[\dot{q}=p\,,\quad\dot{p}=-q \tag{2}\]0  
is the conditional average of \(\exp\bigl{(}-\Omega^{(0)}_{0,\tau}\bigr{)}\) over the trajectories whose initial conditions lie in \(\mathcal{B}_{-A,\delta}\), which is to say the average with respect to the distribution  
\[f^{(0)}_{\mathcal{B}_{-A,\delta}}(\Gamma)=\left\{ \begin{array}{ll}\frac{1}{ \int_{\mathcal{B}_{A,\delta}}f^{(0)}(\Gamma)\mathrm{d}\Gamma}\,f^{(0)}( \Gamma)&\text{if}\quad\Gamma\in\mathcal{B}_{A,\delta}\\ \\ 0&\text{if}\quad\Gamma\
otin\mathcal{B}_{A,\delta},\end{array} \right. \tag{22}\]  
which assigns zero probability to the sets outside \(\mathcal{B}_{-A,\delta}\). The transient fluctuation relation is very robust because it is based on minimal assumptions: time reversibility and conservation of probability in phase space. It is an identity that holds exact for all \(\tau>0\), for an ensemble of experiments all starting in the initial state characterized by \(f^{(0)}\), analogously to other fluctuations such as the Jarzynski equality. To derive the steady-state fluctuation relation, we can now start from Equation (19). First, we move from the probability \(\mu^{(0)}\) to the statistic resulting at time \(t\), under the assumption that probability is conserved and moves around the phase space together with the phases \(S^{\dagger}\Gamma\). This means that the probability of a set \(E\subset\mathcal{M}\) at time \(t\), \(\mu^{(t)}(E)\) say, equals the probability at time \(0\) of the set of points that reach \(E\) at time \(t\), \(S^{-\dagger}E\), which can be written as  
\[\mu^{(t)}(E)=\mu^{(0)}(S^{-\dagger}E). \tag{23}\]  
We will then try to extrapolate the result to the steady state, by letting \(t\) grow without bounds.