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ITP–UU–02/43\
SPIN–02/25\
hep-th/0208145
[**Instantons in the Double-Tensor Multiplet**]{}\
Ulrich Theis and Stefan Vandoren\
[ *Spinoza Instituut, Universiteit Utrecht\
Postbus 80.195, 3508 TD Utrecht, The Netherlands\
U.Theis, S.Vandoren@phys.uu.nl*]{}
------------------------------------------------------------------------
**Abstract**
The double-tensor multiplet naturally appears in type IIB superstring compactifications on Calabi-Yau threefolds, and is dual to the universal hypermultiplet. We revisit the calculation of instanton corrections to the low-energy effective action, in the supergravity approximation. We derive a Bogomol’nyi bound for the double-tensor multiplet and find new instanton solutions saturating the bound. They are characterized by the topological charges and the asymptotic values of the scalar fields in the double-tensor multiplet.
------------------------------------------------------------------------
Introduction
============
Instanton effects in string and M-theory are still relatively poorly understood. This is due to the lack of a conventional instanton calculus as we know it from (supersymmetric) field theory. A well-known open problem is to determine the instanton corrections to the hypermultiplet moduli space of type II superstrings or M-theory compactified on a Calabi-Yau (CY) threefold down to four or five dimensions. Supersymmetry requires the hypermultiplet moduli space ${\cal M}_H$ to be quaternion-Kähler [@BW]. The four- (or five-) dimensional dilaton lives in a multiplet which can be dualized into the universal hypermultiplet. Hence, ${\cal M}_H$ | 0 | non_member_984 |
receives quantum corrections, and the instantons correspond to Euclidean $p$-branes wrapping $p+1$ cycles of the CY [@BBS].
The simplest setup for studying this problem, is to consider CY-compactifications of M-theory/type IIA superstrings with Hodge number $h_{2,1}=0$, or, for type IIB, $h_{1,1}=0$ [^1], since this yields a low-energy effective action of $N=2$ supergravity coupled to a single hypermultiplet, such that the moduli space ${\cal M}_H$ has dimension four. From a type IIB perspective, this hypermultiplet arises from dualizing the *double-tensor* multiplet, whose bosonic components descend from the $NS$-$NS$ and $R$-$R$ two-forms and scalars in ten dimensions. This suggests that instanton calculations should be done on the double-tensor multiplet side. In the next section, we shall make another argument, which also applies to type IIA and M-theory, why the double-tensor multiplet is more appropriate for our purposes.
Yet, even in the case of a single hypermultiplet, it is difficult to compute instanton effects directly in string theory, without explicit knowledge of the instanton measure and the details of the wrapped branes along the CY cycles. Therefore, we will study this problem in a pure supergravity context, in which semi-classical instanton calculations can be done in the more conventional and “field-theoretic” way, following | 0 | non_member_984 |
a similar strategy as in [@BB; @GS1], or as in [@R] for matter coupled to $N=1$ supergravity. Although being an approximation of the exact result, the hope is that the leading supergravity corrections, combined with the constraints from quaternion-Kähler geometry, and together with some knowledge from string theory on the isometries and singularity structure of ${\cal M}_H$, should fix the answer uniquely. Such a program has worked succesfully in the context of supersymmetric field theories in three dimensions with eight supercharges, where the hypermultiplet moduli space is hyperkähler [@SW; @DKMTV]. See [@OV; @K] for related issues.
In this paper, we carry out the first steps of the supergravity instanton calculation. In section 2, we explain how the Euclidean theory is best understood in terms of the double-tensor multiplet, since then the action is manifestly positive definite, a requirement needed for a semiclassical approximation. In section 3, we derive a Bogomol’nyi bound and show that the instanton action is purely topological and given by a surface term. We then solve the BPS equation explicitly and compute the instanton action for the solutions. A similar approach was followed in [@BB] and [@GS1]. Compared to these papers, we propose a different Euclidean version | 0 | non_member_984 |
of the universal hypermultiplet Lagrangian, so our results, where comparable, are somehow different. Moreover, we have found new instanton solutions, which will play an important role in understanding the quantum corrected hypermultiplet moduli space, as explained in the discussion at the end of the paper.
The double-tensor multiplet
===========================
As mentioned in the introduction, we are interested in the case of a single hypermultiplet coupled to $N=2$ supergravity. Classically, the four scalars of the universal hypermultiplet parametrize the homogeneous quaternion-Kähler manifold [@CFG; @FS] $$\label{UHM-QK}
{\mathcal{M}}_H = {\frac{\raisebox{-2pt}{$\mathrm{SU}(1,2)$}}{\mathrm{U}(2)}}\ .$$ In a basis of real fields $\{\phi,\chi,\varphi,\sigma\}$, the bosonic Lagrangian takes the form[^2] $$\label{UHM-action}
{\mathcal{L}}_\mathrm{UH} = - {\mathrm{d}}^D x\, \sqrt{g\,} R + {\tfrac{1}{2}}|{\mathrm{d}}\phi|^2 + {\tfrac{1}{2}}\,
{\mathrm{e}}^{-\phi} \big( |{\mathrm{d}}\chi|^2 + |{\mathrm{d}}\varphi|^2 \big) + {\tfrac{1}{2}}\,
{\mathrm{e}}^{-2\phi}\, |{\mathrm{d}}\sigma + \chi {\mathrm{d}}\varphi|^2\ ,$$ with $D=4$ or $5$, depending on whether one is interested in type II or M-theory compactifications. The Lagrangian has a global SU(1,2) isometry group.
For our purposes, it will be convenient to discuss the dual version of ${\mathcal{L}}_\mathrm{UH}$ in terms of a double-tensor multiplet. Consider the first-order Lagrangian $$\label{DTM-action}
{\mathcal{L}}_\mathrm{DT} = - {\mathrm{d}}^Dx\, \sqrt{g\,} R + {\tfrac{1}{2}}|{\mathrm{d}}\phi|^2 + {\tfrac{1}{2}}\,
{\mathrm{e}}^{-\phi} |{\mathrm{d}}\chi|^2 + {\tfrac{1}{2}}M_{ab} *\! H^a {\wedge}H^b - \lambda_a\,
{\mathrm{d}}H^a\ ,$$ where the $H^a$ | 0 | non_member_984 |
are a doublet of $(D-1)$ forms, the $\lambda_a$ are two scalars, and $$M = {\mathrm{e}}^{\phi} \begin{pmatrix} 1 & - \chi \\[2pt] - \chi & {\mathrm{e}}^{\phi}
+ \chi^2 \end{pmatrix}\ .$$ The two scalars $\phi$ and $\chi$ parametrize the coset SL$(2,\fieldR)/
\mathrm{O}(2)$; in terms of the complex combination $$\label{tau}
\tau \equiv \chi + 2{\mathrm{i}}\, {\mathrm{e}}^{\phi/2}$$ the scalar part of ${\mathcal{L}}_\mathrm{DT}$ can be written as $2|d\tau/\operatorname{Im}\tau|^2$. The tensor terms, however, break the global SL$(2,\fieldR)$ symmetry, leaving only shift symmetries of $\phi$ and $\chi$. The shift in $\chi$ acts as $$\label{shift-chi}
\tau \rightarrow \tau + b\ ,\quad \begin{pmatrix} H^1 \\[2pt] H^2
\end{pmatrix} \rightarrow \begin{pmatrix} 1 & b \\[2pt] 0 & 1
\end{pmatrix} \begin{pmatrix} H^1 \\[2pt] H^2 \end{pmatrix}\ ,
\quad \begin{pmatrix} \lambda^1 \\[2pt] \lambda^2 \end{pmatrix}
\rightarrow \begin{pmatrix} 1 & 0 \\[2pt] -b & 1 \end{pmatrix}
\begin{pmatrix} \lambda^1 \\[2pt] \lambda^2 \end{pmatrix}\ ,$$ whereas the shift in $\phi$ acts as $$\label{shift-phi}
\tau \rightarrow {\mathrm{e}}^\kappa \tau\ ,\quad \begin{pmatrix} H^1 \\[2pt]
H^2 \end{pmatrix} \rightarrow \begin{pmatrix} {\mathrm{e}}^{-\kappa} H^1
\\[2pt] {\mathrm{e}}^{-2\kappa} H^2 \end{pmatrix}\ ,\quad \begin{pmatrix}
\lambda^1 \\[2pt] \lambda^2 \end{pmatrix} \rightarrow \begin{pmatrix}
{\mathrm{e}}^\kappa \lambda^1 \\[2pt] {\mathrm{e}}^{2\kappa} \lambda^2 \end{pmatrix}\ .$$ Note that the latter acts like an SL$(2,\fieldR)$ transformation on $\tau$, but not on the $H^a$. The full type IIB theory compactified to | 0 | non_member_984 |
four dimensions (classically) has SL$(2,\fieldR)$ symmetry, due to the presence of additional tensor multiplets which transform nontrivially [@BGHL]. Setting the scalars in these multiplets to nonvanishing constants results in a breakdown of the symmetry and leaves only the above transformations as residual invariances.
The equations of motion for the Lagrange multipliers $\lambda_a$ imply that the $H^a$ are closed. Writing $H^a={\mathrm{d}}B^a$, one obtains the double-tensor multiplet. Integrating out the tensors instead gives the duality relation $$\label{dual-rel}
{\mathrm{d}}\lambda_a = - M_{ab} *\! H^b\ .$$ Substituting this back yields the action for the universal hypermultiplet , upon identifying $$\label{mult1}
\lambda_1 = \varphi\ ,\quad \lambda_2 = \sigma\ .$$ The dual formulation in terms of the double-tensor multiplet is not unique[^3]. We can start with , but write everywhere $\varphi$ instead of $\chi$. Dualizing the tensors and identifying $$\label{mult2}
\lambda_1 = - \chi\ ,\quad \lambda_2 = \sigma + \varphi\, \chi$$ yields the same hypermultiplet action, as one can easily check.
In addition, the dualization procedure yields a boundary term which has to be added to the hypermultiplet action, $$\label{surf-term}
{\mathcal{L}}_\mathrm{bnd} = (-)^D\, {\mathrm{d}}\big[ \lambda_a\, (M^{-1})^{ab} *{\mathrm{d}}\lambda_b\big]\ ,$$ where we used that, when acting on a $p$-form in Minkowski space, $**=
-(-)^{(D-1)p}$. The different choices corresponding | 0 | non_member_984 |
to and would now give different boundary terms. However, due to the isometries of the scalar manifold, they are related to each other by a field redefinition of the multipliers, $\tilde{\sigma}=\sigma+\varphi
\chi$, $\tilde{\varphi}=-\chi$, $\tilde{\chi}=\varphi$. Substituting into , we get $${\mathcal{L}}_\mathrm{bnd} = (-)^D\, {\mathrm{d}}\big[ {\mathrm{e}}^{-\phi} \chi *\! {\mathrm{d}}\chi +
{\mathrm{e}}^{-2\phi} \sigma *\! ({\mathrm{d}}\sigma + \chi {\mathrm{d}}\varphi) \big]\ .$$ The total action for the universal hypermultiplet is then $${\mathcal{L}}= {\mathcal{L}}_\mathrm{UH} + {\mathcal{L}}_\mathrm{bnd}\ .$$
The fermions have been suppressed here. For hypermultiplets, the supersymmetry transformation rules and the fermion-terms in the Lagrangian are known in general. For the double-tensor multiplet Lagrangian , the fermion-terms and susy rules can be determined by dualization. However, the most general self-interacting supersymmetric double-tensor multiplet Lagrangian has not been worked out. For a discussion on this in the context of rigid $N=2$ supersymmetry, we refer to [@B].
Euclidean formulation {#euclidean-formulation .unnumbered}
---------------------
To find instanton solutions, we need the Euclidean formulation of the universal hypermultiplet, or, equivalently, the Euclidean double-tensor multiplet Lagrangian. For the latter, apart from the usual complications with the Euclidean Einstein-Hilbert term, the Wick rotation acts in the standard way on the scalars and tensors. While the double-tensor multiplet Lagrangian formally stays the same, $$\label{E-DTM}
| 0 | non_member_984 |
{\mathcal{L}}_\mathrm{DT}^E = {\mathrm{d}}^Dx\, \sqrt{g\,} R + {\tfrac{1}{2}}|{\mathrm{d}}\phi|^2 + {\tfrac{1}{2}}\,
{\mathrm{e}}^{-\phi} |{\mathrm{d}}\chi|^2 + {\tfrac{1}{2}}M_{ab} *\! H^a {\wedge}H^b\ ,$$ the dual Euclidean universal hypermultiplet Lagrangian has two sign flips in the kinetic terms, due to the fact that we now have $**=(-)^{(D-1)p}$ when acting on a $p$-form in Euclidean space. The dualization procedure yields $$\label{EUHM-action}
{\mathcal{L}}_\mathrm{UH}^E = {\mathrm{d}}^D x\, \sqrt{g\,} R + {\tfrac{1}{2}}|{\mathrm{d}}\phi|^2 + {\tfrac{1}{2}}\,
{\mathrm{e}}^{-\phi} \big( |{\mathrm{d}}\chi|^2 - |{\mathrm{d}}\varphi|^2 \big) - {\tfrac{1}{2}}\, {\mathrm{e}}^{-2
\phi}\, |{\mathrm{d}}\sigma + \chi {\mathrm{d}}\varphi|^2\ ,$$ together with the boundary term $$\label{E-BT}
{\mathcal{L}}_\mathrm{bnd}^E = - (-)^D\, {\mathrm{d}}\big[ {\mathrm{e}}^{-\phi} \chi *\! {\mathrm{d}}\chi +
{\mathrm{e}}^{-2\phi} \sigma *\! ({\mathrm{d}}\sigma + \chi {\mathrm{d}}\varphi) \big]\ .$$ By setting $\varphi=\chi=0$, this boundary term is the same as for the $D$-instanton of type IIB in ten dimensions, obtained by dualizing the nine-form field strength into the $R$-$R$ scalar $\sigma$ [@GGP; @GG]. In four dimensions, we generate more terms due to the fact that we dualize two tensors. The sign flips of the kinetic terms of the two dual fields $\lambda_a$ are compatible with the prescription of Wick rotating pseudoscalars $\lambda_a\rightarrow{\mathrm{i}}\lambda_a$ [@vNW]. This is consistent with the duality relation .
A Euclidean version of the universal hypermultiplet action was also proposed in [@GS1]. Both their bulk | 0 | non_member_984 |
Lagrangian and boundary term differ from ours. This has important consequences since the instanton action defines the weight in the path integral, and hence correlation functions and eventually the quantum-corrected hypermultiplet moduli space will be different.
Due to the sign changes in , the geometry of the scalar manifold is no longer SU(1,2)/U(2). Instead, it is given by the coset space $$\label{SL3}
{\mathcal{M}}_H^E = {\frac{\raisebox{-2pt}{$\mathrm{SL}(3,\fieldR)$}}{\mathrm{SL}(2,\fieldR) \times
\mathrm{SO}(1,1)}}\ ,$$ which is *not* a quaternion-Kähler manifold. This is not in contradiction with supersymmetry, since only *Minkowskian* supersymmtry requires the target space to be quaternionic [@BW]. A brief discussion on the geometry of the space is given in appendix \[SL3R\].
In four dimensions, the same target space can be obtained by applying the **c**-map [@CFG] to pure $N=2$ Euclidean supergravity [@TvN]. This turns the four bosonic degrees of freedom contained in the metric and graviphoton into the four scalars of the universal hypermultiplet and gives rise to the two sign flips. Moreover, the **c**-map maps Reissner-Nordstrom black hole solutions to D-instantons in the universal hypermultiplet, as was shown in [@BGLMM].
We remark that it is the inverted signs in the Euclidean hypermultiplet action that make instanton solutions in flat space possible. Indeed, the | 0 | non_member_984 |
trace of the Einstein equation sets the bulk Lagrangian to zero, hence nontrivial field configurations would require a nonvanishing curvature scalar if the sigma model part of the Lagrangian were positive definite. The negative signs in allow for cancellations that are compatible with $R=0$. Note also that since on the hypermultiplet side the bulk action vanishes for any solution, the instanton action comes entirely from the boundary term discussed above. As already stated, the boundary term is different from the one proposed in [@GS1]. For this reason, we get different results for the instanton action, and eventually for the instanton corrected hypermultiplet moduli space.
What is more important from the point of view of instanton calculations, is that the Euclidean Lagrangian is no longer positive definite. In a path integral formulation, this makes the finite action configurations irrelevant, since the action is not bounded from below. Moreover, perturbative fluctuations around the instanton yield diverging non-Gaussian integrals, and the semiclassical approximation would break down. Similar considerations apply to the $N=2$ tensor multiplet, whose Euclidean action is not positive definite.
On the other hand, the Euclidean double-tensor multiplet Lagrangian is bounded from below, since the matrix $M_{ab}$ is positive definite. This leads | 0 | non_member_984 |
to a well-defined semiclassical treatment, in which the instantons dominate the Euclidean path integral. For this reason, it is important to perform all calculations on the double-tensor multiplet side, and after having computed the instanton corrections there, we can dualize to the hypermultiplet formulation.
Instanton solutions
===================
Asymptotics
-----------
Before finding the explicit instanton solutions, it will be useful to discuss the asymptotic behaviour of the fields that can lead to a finite action. Since the Euclidean action consists of three positive definite terms, each term individually should integrate to a finite quantity. For simplicity we consider for the moment flat four-dimensional space. This determines the following behaviour at infinity: $$\label{large-r}
\phi \rightarrow \phi_\infty + \mathcal{O} \Big( {\frac{\raisebox{-2pt}{$1$}}{r^2}} \Big)\
,\quad \chi \rightarrow \chi_\infty + \mathcal{O} \Big( {\frac{\raisebox{-2pt}{$1$}}{r^2}}
\Big)\ ,\quad H_{\mu\nu\rho} \propto {\frac{\raisebox{-2pt}{$1$}}{r^3}}\ .$$ The asymptotic value of $\phi$ is identified with the four- (or five-) dimensional string coupling constant, $$g_s \equiv {\mathrm{e}}^{-\phi_\infty/2}\ .$$ The field strengths determine topological charges, defined by integrating the tensors $H^a$ over spheres at infinity, $$\label{HQ}
\int_{S^{D-1}_\infty}\! H^a = Q^{(a)} \ ,\quad a = 1,2\ .$$ In the dual (hypermultiplet) formulation, topological charges become Noether charges, corresponding to the Peccei-Quinn symmetries which act as constant shifts | 0 | non_member_984 |
in the Lagrange multipliers $\lambda_a$. These charges descend from the brane charges in ten or eleven dimensions, and, in the appropriate units, are expected to be quantized.
The Euclidean space we shall concentrate on is actually flat space with a countable number of points, the locations of the instantons, excised[^4], $$\label{space}
{\mathcal{M}}= \fieldR^D - \cup_i\, \{\vec{x}_i\}\ ,$$ such that non-trivial cycles with corresponding charges exist. Stated differently, in the supergravity approximation it will typically not be possible to find regular solutions at the locations of the instantons, as we will explicitly see below. The only singularity which can still lead to a finite action is a logarithmic singularity in $\phi$ at the origin, $$\label{small-r}
\phi \rightarrow c\, \ln r\ ,$$ for some constant $c$. In our examples below, $\chi$ will tend to a constant $\chi_0$, and the tensors have the same $1/r^3$ behaviour such that the charges stay the same when the $H^a$ are integrated around an infinitesimal sphere around the origin.
The Bogomol’nyi bound
---------------------
The Euclidean double-tensor multiplet action is positive semi-definite (apart from the Einstein-Hilbert term). In fact, we can derive a lower bound by writing it as $${\mathcal{L}}_\mathrm{DT}^E = {\mathrm{d}}^Dx\, \sqrt{g\,} R + {\tfrac{1}{2}}*\! \big( N\! | 0 | non_member_984 |
*\! H +
O E \big)^t {\wedge}\big( N\! *\! H + O E \big) + (-)^D H^t {\wedge}N^t O E\ .$$ Here we have defined $$H = \begin{pmatrix} H^1 \\[2pt] H^2 \end{pmatrix}\ ,\quad E =
\begin{pmatrix} {\mathrm{d}}\phi \\[2pt] {\mathrm{e}}^{-\phi/2}\, {\mathrm{d}}\chi
\end{pmatrix}\ ,\quad N = {\mathrm{e}}^{\phi/2} \begin{pmatrix} 0 & {\mathrm{e}}^{\phi/2}\,
\\[2pt] 1 & -\chi \end{pmatrix}\ ,$$ such that $N^t N=M$, and $O$ is some orthogonal (scalar) field-dependent matrix, whose appearance is due to the fact that $N$ and the zweibein $E$ are determined only modulo local O(2) transformations.
Clearly, the action is bounded from below by $$\label{bound}
S^E \geq \int_{{\mathcal{M}}} \big( {\mathrm{d}}^Dx\, \sqrt{g\,} R + (-)^D H^t {\wedge}N^t
O E \big)\ ,$$ where the second term is topological, as it is independent of the spacetime metric. The bound is saturated by field configurations satisfying the BPS condition $$\label{BPS}
* H = - N^{-1} O E\ .$$ A similar Bogomol’nyi equation was derived for an $N=1$, $D=4$ tensor multiplet (containing one tensor and one scalar) in [@R]. Notice that, if the matrix $O$ is invariant, this equation transforms covariantly under and .
Equation is a proper BPS condition only if it implies the equations of motion, and this will fix the O(2) | 0 | non_member_984 |
degeneracy. It is easily verified that field configurations satisfying have vanishing energy-momentum tensors, hence they can exist only in Ricci-flat spaces. We therefore have to amend our BPS condition by the equation $R_{\mu\nu}(g)=0$.
For the field equations of the tensors, ${\mathrm{d}}(M\!*\!H)=0$, to hold we must have $$\label{closed}
{\mathrm{d}}(N^t O E) = 0\ .$$ This condition also guarantees that the topological term in is closed and hence can locally be written as a total derivative. As a consequence, it does not contribute to the equations of motion such that also the field equations for the scalars are guaranteed to be satisfied. The latter follow from requiring that the solution of correspond to closed forms for $H^a$, $${\mathrm{d}}(N^{-1} O *\! E)=0\ .$$
To determine the O(2) matrices that are compatible with , we parametrize $O$ by $$O = \begin{pmatrix} 1 & 0 \\[2pt] 0 & {\epsilon}\end{pmatrix}
\begin{pmatrix} c & -s \\[2pt] s & c \end{pmatrix}\ ,$$ where the functions $c(\phi,\chi)$ and $s(\phi,\chi)$ are constrained by $c^2+s^2=1$, and ${\epsilon}=\pm 1$ for the two components of O(2) with $\det O={\epsilon}$. Equation then gives rise to the differential equations $$\begin{aligned}
0 & = {\partial}_\phi c - {\mathrm{e}}^{\phi/2} {\partial}_\chi s \notag \\*
0 & = {\partial}_\phi | 0 | non_member_984 |
s + {\mathrm{e}}^{\phi/2} {\partial}_\chi c - {\tfrac{1}{2}}(2{\epsilon}- 1) s\ .
\end{aligned}$$ We derive the general solution in appendix \[appO2\]. The result is that there are *three* distinct BPS conditions corresponding to the O(2) matrices $$O_{1,2} = \pm \begin{pmatrix} 1 & 0 \\[2pt] 0 & {\epsilon}\end{pmatrix}\
,\quad O_3 = \pm {\frac{\raisebox{-2pt}{$1$}}{|\tau'|}} \begin{pmatrix} \operatorname{Re}\tau' & -\operatorname{Im}\tau'\, \\[4pt] \operatorname{Im}\tau' & \operatorname{Re}\tau' \end{pmatrix}\ ,$$ invariant under both and . Here $\tau'=\tau-\chi_0$ with $\tau$ as in , $\chi_0$ is a real integration constant, and the plus and minus signs refer to the instanton and anti-instanton, respectively.
For these three O(2) matrices the 1-form $N^t OE$ is exact, $$\label{dY}
N^t O E = \pm\, {\mathrm{d}}Y\ ,$$ where modulo an additive constant $$Y_{1,2} = \begin{pmatrix} {\epsilon}\chi \\[2pt] {\mathrm{e}}^\phi - {\tfrac{1}{2}}{\epsilon}\chi^2
\end{pmatrix}\ ,\quad Y_3 = {\tfrac{1}{2}}\sqrt{4 {\mathrm{e}}^{\phi} + (\chi -
\chi_0)^2\,} \begin{pmatrix} 2 \\ - \chi - \chi_0 \end{pmatrix}\ .$$ It follows that the action for BPS configurations is given by a topological boundary term $$\label{top-BT}
S^E\,|_\mathrm{BPS} = (-)^D \int_{\mathcal{M}}H^t {\wedge}N^t O E = \mp \int_{{\partial}{\mathcal{M}}}
Y^t H\ .$$ The instanton action is therefore determined by the charges $Q^{(a)}$ and the values of the fields $\chi$ and ${\mathrm{e}}^{\phi}$ at the boundaries.
It is easy to find the corresponding | 0 | non_member_984 |
BPS equation in the dual hypermultiplet formulation. Using , , and the fact that $M=N^t N$, we find for the Lagrange multipliers $${\mathrm{d}}\lambda = \pm\, {\mathrm{d}}Y\ ,$$ such that, up to a constant, the solutions for the two extra scalars are completely determined in terms of $\phi$ and $\chi$.
Solutions and instanton action
------------------------------
We can solve the BPS condition for the three possible matrices $O$. For $O_1=\pm 1$, the condition reads $$\label{O1_BPS}
* H = \pm \begin{pmatrix} \chi {\overset{\leftrightarrow}{{\mathrm{d}}}} {\mathrm{e}}^{-\phi} \\[2pt] {\mathrm{d}}{\mathrm{e}}^{-\phi} \end{pmatrix}\ .$$ Applying ${\mathrm{d}}*$ to the equation and using the Bianchi identities of $H$, we find that ${\mathrm{e}}^{-\phi}$ must be harmonic, and from the first component it then follows that also $\chi$ satisfies the Laplace equation, $$O_1:\quad {\mathrm{d}}*\! {\mathrm{d}}{\mathrm{e}}^{-\phi} = 0\ ,\quad {\mathrm{d}}*\! {\mathrm{d}}\chi = 0\ .$$ As mentioned above, scalars satisfying these conditions will also solve their field equations.
In the following, we consider for simplicity spherically symmetric configurations (single instantons) in flat space only. The dilaton equation of motion is then solved by $$\label{O1_dil}
{\mathrm{e}}^{-\phi} = {\mathrm{e}}^{-\phi_\infty} + {\frac{\raisebox{-2pt}{$|Q^{(2)}|$}}{\Omega_D\,
r^{D-2}}}\ .$$ Here $\Omega_D=(D-2)\mathrm{Vol}(S^{D-1})$, and we have chosen the location of the instanton ($\vec{x}_1$ in ) as the origin. The integration constant $Q^{(2)}$ appearing in the | 0 | non_member_984 |
solution is equal to the topological charge associated with $H^2$, as follows from the second equation of . The ‘selfdual’ instanton (upper sign in ) is taken for negative $Q^{(2)}$, the ‘anti-selfdual’ instanton for positive $Q^{(2)}$.
Since, up to proportionality factors, there is only a unique spherically symmetric harmonic function, $\chi$ must be of the form $\chi=\chi_1
{\mathrm{e}}^{-\phi}+\chi_0$ with $\chi_0$, $\chi_1$ constant. It then follows from that $\chi_0$ is determined by $$\chi_0 = {\frac{\raisebox{-2pt}{$Q^{(1)}$}}{Q^{(2)}}}\ ,$$ and this relation is consistent with the shift symmetries and , since the charges transform non-trivially. The instanton action for $O_1$ is given by $$S_1^E = \mp \int_{{\partial}{\mathcal{M}}} \big[ \chi H^1 + ({\mathrm{e}}^{\phi} - {\tfrac{1}{2}}\chi^2)
H^2 \big]\ ,$$ where the boundary consists of the disjoint union of two spheres, ${\partial}{\mathcal{M}}=S^{D-1}_\infty\,\cup\,S^{D-1}_0$, with radii as indicated. The terms involving $\chi$ will diverge on $S^{D-1}_0$ since $\chi$ is harmonic, so in order to obtain a finite action we have to take $\chi=
\chi_0$ constant. This was already anticipated from the asymptotic behaviour of the fields, discussed in the beginning of this section. The action then reads $$S_1^E = \frac{\big| Q^{(2)} \big|}{g_s^2}\ .$$ This solution was also found in [@BB; @GS1], and should correspond to the fivebrane wrapping the | 0 | non_member_984 |
entire Calabi-Yau [@BBS]. The instanton action is positive and hence does not distinguish instantons from anti-instantons. Imaginary theta-angle-like terms will have to be added to make this distinction.
Turning to $O_2$, we have the BPS condition $$\label{H-O2}
* H = \pm\, {\mathrm{d}}\begin{pmatrix} {\mathrm{e}}^{-\phi} \chi \\[2pt] {\mathrm{e}}^{-\phi}
\end{pmatrix}\ .$$ Again, ${\mathrm{e}}^{-\phi}$ is harmonic, and the same now applies to ${\mathrm{e}}^{-\phi}
\chi$. If one imposes rotational symmetry then $$\label{O2-sol}
O_2:\quad {\mathrm{d}}*\! {\mathrm{d}}{\mathrm{e}}^{-\phi} = 0\ ,\quad \chi = \chi_1 {\mathrm{e}}^{\phi} +
\chi_0\ ,$$ and from , it follows again that $Q^{(1)}=\chi_0 Q^{(2)}$. Notice that the field $\chi$ is now completely regular everywhere, and interpolates between the boundaries according to $$\Delta \chi \equiv \chi_\infty - \chi_0 = {\frac{\raisebox{-2pt}{$\chi_1$}}{g_s^2}}\ .$$ The complete solution agrees with the asymptotics derived in and .
For this solution, with the dilaton again given by , the instanton action then becomes $$\label{S2-inst}
S_2^E = \big| Q^{(2)} \big|\, \Big( {\frac{\raisebox{-2pt}{$1$}}{g_s^2}} + {\tfrac{1}{2}}\,
(\Delta \chi)^2 \Big)\ .$$ For the particular case of $\Delta \chi=0$, the solution and instanton action are the same as for the $O_1$ solution. Notice also that both terms are positive and invariant under the shift symmetries and , as guaranteed by the properties of the original action. For | 0 | non_member_984 |
$\Delta\chi\neq 0$, our instanton solution is new, and this term in the instanton action does not depend on the string coupling constant $g_s$. The appearance of $\Delta\chi$ in the instanton action is one of the new results in this paper. Its presence was somehow anticipated in [@BBS], and here we have computed it explicitly.
We now turn to $O_3$. The BPS equation for this case reads $$* H = \pm {\frac{\raisebox{-2pt}{$1$}}{|\tau'|}} \begin{pmatrix} -2\, {\mathrm{d}}\phi + {\mathrm{e}}^{-\phi}
(\chi + \chi_0)\, {\mathrm{d}}\chi + \chi (\chi - \chi_0)\, {\mathrm{d}}{\mathrm{e}}^{-\phi}\,
\\[4pt] (\chi - \chi_0)\, {\mathrm{d}}{\mathrm{e}}^{-\phi} + 2 {\mathrm{e}}^{-\phi} {\mathrm{d}}\chi
\end{pmatrix}\ .$$ We have been unable to find the general solution[^5]. Instead, let us consider two Ansätze for which we can explicitly solve the equations. First, we set $\chi=2\chi_1{\mathrm{e}}^{\phi/2}
+\chi_0$. Then the equations simplify to $$* H = \pm 2\, {\mathrm{d}}{\mathrm{e}}^{-\phi/2} \begin{pmatrix} \sqrt{1 + \chi_1^2\,}\,
\\[2pt] 0 \end{pmatrix}\ .$$ It follows that now ${\mathrm{e}}^{-\phi/2}$ is harmonic, with solution $${\mathrm{e}}^{-\phi/2} = {\mathrm{e}}^{-\phi_{\infty}/2} + \frac{\big|Q^{(1)}\big|}
{2 \sqrt{1 + \chi_1^2\,}\, \Omega_D\, r^{D-2}}\ .$$ The scalar $\chi$ is then completely regular and interpolates between the boundaries as $$\Delta \chi = \chi_{\infty} - \chi_0 = \frac{2\chi_1}{g_s}\ .$$ Since $H^2=0$ we have $Q^{(2)}=0$, and for the instanton action we find $$\label{Sinst_O3_1}
| 0 | non_member_984 |
S_3^E = \big| Q^{(1)} \big|\, \sqrt{{\frac{\raisebox{-2pt}{$4$}}{g_s^2}} + (\Delta \chi)^2}
= \big| Q^{(1)} \big|\, \big| \tau'_{\infty} \big|\ ,$$ where $\tau'_{\infty}=(\chi_\infty-\chi_0)+2{\mathrm{i}}\,{\mathrm{e}}^{\phi_{\infty}/2}$ is the value of $\tau'$ at infinity. For $\Delta\chi=0$, a similar solution was also found in [@GS1]. Following the discussion in [@BBS], it should correspond, from a IIA point of view, to the D2-brane wrapping a three-cycle in the Calabi-Yau, or to the D1+D3+D5-branes wrapping even cycles in type IIB. Notice again consistency with the symmetries and . Observe also that for $\Delta
\chi=0$, the solution is inversely proportional to $g_s$, and is for small $g_s$ dominating over the fivebrane instanton .
As a second Ansatz, consider $\chi=2\chi_1{\mathrm{e}}^\phi+\chi_0$. This differs from the first Ansatz in the power of ${\mathrm{e}}^\phi$. The BPS condition turns into $$* H = \pm 2\, {\mathrm{d}}\sqrt{{\mathrm{e}}^{-\phi} + \chi_1^2\,} \begin{pmatrix} 1 -
\chi_1 \chi_0 \\[2pt] -\chi_1 \end{pmatrix}\ .$$ Accordingly, the square root must be harmonic, and we find for $\chi_1
\neq 0$ (the case $\chi_1=0$ is included in the previous Ansatz), $${\mathrm{e}}^{-\phi} = (h - \chi_1) (h + \chi_1)\ ,\quad h = \sqrt{
{\mathrm{e}}^{-\phi_\infty} + \chi_1^2\,} + \Big| {\frac{\raisebox{-2pt}{$Q^{(2)}$}}{2\chi_1}} \Big|\,
{\frac{\raisebox{-2pt}{$1$}}{\Omega_D\, r^{D-2}}}\ .$$ The scalar field $\chi$ is regular everywhere and interpolates between zero and infinity as $$\Delta | 0 | non_member_984 |
\chi = \frac{2\chi_1}{g_s^2}\ .$$ The BPS equation further fixes the constant $\chi_1$ to be $$\chi_1 = - {\frac{\raisebox{-2pt}{$Q^{(2)}$}}{Q^{(1)} - \chi_0 Q^{(2)}}}\ ,$$ and the instanton action is easily computed from , $$S_3^E = \big| \tau'_\infty \big|\, \Big( \big| \hat{Q}^{(1)} \big|
+ {\tfrac{1}{2}}\big| \Delta \chi\, Q^{(2)} \big| \Big)\ .$$ We have redefined the $Q^{(1)}$ charge according to $$\hat{Q}^{(1)} \equiv Q^{(1)} - \chi_0 Q^{(2)}\ ,$$ such that it is invariant under . For $Q^{(2)}=0$, the instanton action then clearly reduces to .
The obtained results for the instanton action carry over to the hypermultiplet side, because the dualization procedure does not affect the real part of the instanton action.
Discussion
==========
In this paper, we have carried out the first steps of calculating instanton corrections to the hypermultiplet moduli space. An important ingredient was to derive a Bogomol’nyi bound for the double-tensor multiplet Lagrangian, and to solve the corresponding BPS equation. In a supersymmetric formulation, adapted to Euclidean space, we expect our instanton solutions to preserve one half of the supersymmetries. A more general formulation for the double-tensor multiplet Lagrangian, including the fermions and supersymmetry transformation rules, is presently under study. This will be important for finding the fermionic zero modes and | 0 | non_member_984 |
eventually for computing instanton corrections to the relevant correlation functions that determine the hypermultiplet quantum-geometry. The exact moduli space must be consistent with the results derived in our paper. In particular, our supergravity instanton solutions should match with the results obtained from wrapping branes in the full ten-dimensional string theory. Stated differently, the universal hypermultiplet metric must contain exponential corrections which, at leading order in the string coupling constant and $\alpha'$, agree with the form of our instanton action. Using some results about quaternionic geometry [@CP; @DWRV], it should be possible to find quaternionic metrics which asymptotically reproduce our results. We intend to report further on these issues in the near future.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank Michael Gutperle and Thomas Mohaupt for discussions and reading an earlier draft of this paper. U.T. thanks the Deutsche Forschungsgemeinschaft for financial support.
SL(3,R) / SL(2,R) $\times$ SO(1,1) {#SL3R}
==================================
In this appendix, we discuss some geometrical aspects related to the sigma model corresponding to , with target space . The easiest way to study this space is by using the fact that the Minkowskian version of the universal hypermultiplet moduli space is both Kähler and quaternion-Kähler. Since Kähler geometry is simpler | 0 | non_member_984 |
to analyze, we will study the coset from the point of view of Kähler geometry. Because of the sign flips compared to the Minkowskian version, the target space will no longer be Kähler. It is therefore not possible to define complex coordinates together with a Kähler potential that determines the metric. As we show below, it is still possible to define coordinates and a potential from which the metric can be computed. To see this, we first define the fields $$a = \sigma + {\tfrac{1}{2}}\chi \varphi\ ,\quad C_\pm = {\tfrac{1}{2}}(\varphi \pm
\chi)\ ,$$ in terms of which the sigma model part of the Euclidean Lagrangian reads $${\mathcal{L}}_\mathrm{UH}^E = {\tfrac{1}{2}}|{\mathrm{d}}\phi|^2 - 2 {\mathrm{e}}^{-\phi} *\! {\mathrm{d}}C_+ {\wedge}{\mathrm{d}}C_- - {\tfrac{1}{2}}{\mathrm{e}}^{-2\phi}\, |{\mathrm{d}}a + C_+ {\overset{\leftrightarrow}{{\mathrm{d}}}} C_-|^2\ .$$ If we further pass to coordinates $u^1_\pm,u^2_\pm\in\fieldR$ via the relations $$S_\pm = {\mathrm{e}}^{\phi} \mp a - C_+ C_- = {\frac{\raisebox{-2pt}{$1 \mp u^1_\pm$}}{1 \pm
u^1_\pm}}\ ,\quad C_\pm = {\frac{\raisebox{-2pt}{$u^2_\pm$}}{1 \pm u^1_\pm}}\ ,$$ then the Lagrangian can be written as $${\mathcal{L}}^E_\mathrm{UH} = 2 g_{ij} *\! {\mathrm{d}}u^i_+ {\wedge}{\mathrm{d}}u^j_-$$ with a metric $$\label{K+-}
g_{ij} = - {\frac{\raisebox{-2pt}{${\partial}^2$}}{{\partial}u^i_+ {\partial}u^j_-}}\, \ln\! \big( 1 + u^1_+
u^1_- + u^2_+ u^2_- \big)\ .$$ The $u^i_\pm$ are inhomogeneous coordinates of the coset space , transforming under | 0 | non_member_984 |
$M\in\mathrm{SL}(2,\fieldR)$ as $u_+
\rightarrow M u_+$, $u_-\rightarrow (M^{-1})^tu_-$. We have therefore identified a potential in terms of real coordinates which determines the metric components. Such spaces are called para-Kähler[^6]. The metric for the Minkowskian universal hypermultiplet moduli space is of the same form as in , but with $u^i_\pm$ treated as complex coordinates, where $u^i_-=-
\bar{u}^i_+$ under complex conjugation.
Determination of O(2) matrices {#appO2}
==============================
We need to solve the differential equations $$\begin{aligned}
0 & = {\partial}_\phi c - {\mathrm{e}}^{\phi/2} {\partial}_\chi s \label{O2_1} \\[2pt]
0 & = {\partial}_\phi s + {\mathrm{e}}^{\phi/2} {\partial}_\chi c - {\tfrac{1}{2}}(2{\epsilon}- 1) s\ ,
\label{O2_2}
\end{aligned}$$ where $c$ and $s$ are subject to the constraint $c^2+s^2=1$. We first multiply by $s$ and by $c$, respectively, and use $-s{\partial}s=c{\partial}c$ to write the equations as $$\begin{aligned}
0 & = s\, {\partial}_\phi c + {\mathrm{e}}^{\phi/2} c\, {\partial}_\chi c \label{O2_3} \\[2pt]
0 & = c\, {\partial}_\phi s + {\mathrm{e}}^{\phi/2} c\, {\partial}_\chi c - {\tfrac{1}{2}}(2{\epsilon}- 1)
c s\ . \label{O2_4}
\end{aligned}$$ Multiplying the difference of these equations by $c$ gives $$\label{O2_5}
0 = c \big[ c\, {\overset{\leftrightarrow}{{\partial}_\phi}} s - {\tfrac{1}{2}}(2{\epsilon}- 1) c s \big] =
{\partial}_\phi s - {\tfrac{1}{2}}(2{\epsilon}- 1)\, (1 - s^2) s\ ,$$ which involves only $s$ and can | 0 | non_member_984 |
easily be integrated: $${\frac{\raisebox{-2pt}{$s^2$}}{1 - s^2}} = {\frac{\raisebox{-2pt}{$1 - c^2$}}{c^2}} = f^2(\chi)\, {\mathrm{e}}^{(2{\epsilon}- 1)
\phi}\ .$$ The positive integration constant $f^2$ may depend on $\chi$. These expressions we plug into the sum of and , $$\begin{aligned}
0 & = {\mathrm{e}}^{\phi/2} {\partial}_\chi c^2 + {\partial}_{\phi} (cs) - {\tfrac{1}{2}}(2{\epsilon}- 1) cs
\notag \\[2pt]
& = - {\frac{\raisebox{-2pt}{$2 f\, {\mathrm{e}}^{(3 - 4{\epsilon})\phi/2}$}}{(f^2 + {\mathrm{e}}^{(1 - 2{\epsilon})\phi}
)^2}}\ \big[ {\partial}_\chi f \pm {\tfrac{1}{2}}(2{\epsilon}- 1)\, {\mathrm{e}}^{({\epsilon}- 1)\phi}
f^2 \big]\ ,
\end{aligned}$$ where the sign ambiguity originates from taking the square root of $(cs)^2$. The equation is satisfied if the expression in square brackets vanishes. For ${\epsilon}=-1$, this is only possible if $f=0$ since $f$ is independent of $\phi$. For ${\epsilon}=+1$, which corresponds to $O\in
\mathrm{SO}(2)$, we find $$f = 0 \quad\text{or}\quad f = \pm {\frac{\raisebox{-2pt}{$2$}}{\chi - \chi_0}}\ ,$$ with $\chi_0$ an integration constant. $f=0$ implies $c=\pm 1$ and $s=
0$. For nontrivial $f$ we obtain (with the relative sign fixed by the original equations and $\eqref{O2_2}$) $$c = \pm \frac{\chi - \chi_0}{\sqrt{4 {\mathrm{e}}^\phi + (\chi - \chi_0)^2\,}
\,}\ ,\quad s = \pm {\frac{\raisebox{-2pt}{$2\, {\mathrm{e}}^{\phi/2}$}}{\sqrt{4 {\mathrm{e}}^\phi + (\chi -
\chi_0)^2\,}\,}}\ ,$$ or in terms of $\tau'=(\chi-\chi_0)+2{\mathrm{i}}\,{\mathrm{e}}^{\phi/2}$, $$c + {\mathrm{i}}s = \pm {\frac{\raisebox{-2pt}{$\tau'$}}{|\tau'|}}\ .$$
[99]{} J. Bagger | 0 | non_member_984 |
---
abstract: |
We show that, consistently, for some regular cardinals $\theta<\lambda$, there exist a Boolean algebra ${{\mathbb B}}$ such that $|{{\mathbb B}}|=\lambda^+$ and for every subalgebra ${{\mathbb B}}'\subseteq{{\mathbb B}}$ of size $\lambda^+$ we have ${{\rm Depth}}({{\mathbb B}}')=
\theta$.
address:
- |
Department of Mathematics\
University of Nebraska at Omaha\
Omaha, NE 68182-0243, USA\
and Mathematical Institute of Wroclaw University\
50384 Wroclaw, Poland
- |
Institute of Mathematics\
The Hebrew University of Jerusalem\
91904 Jerusalem, Israel\
and Department of Mathematics\
Rutgers University\
New Brunswick, NJ 08854, USA
author:
- 'Andrzej Ros[ł]{}anowski'
- Saharon Shelah
title: 'Historic forcing for ${{\rm Depth}}$'
---
Introduction
============
The present paper is concerned with forcing a Boolean algebra which has some prescribed properties of ${{\rm Depth}}$. Let us recall that, for a Boolean algebra ${{\mathbb B}}$, its depth is defined as follows: $$\begin{array}{lcl}
{{\rm Depth}}({{\mathbb B}})&=&\sup\{|X|: X\subseteq{{\mathbb B}}\mbox{ is well-ordered by the Boolean
ordering}\;\},\\
{{\rm Depth}}^+({{\mathbb B}})&=&\sup\{|X|^+: X\subseteq{{\mathbb B}}\mbox{ is well-ordered by the
Boolean ordering}\;\}.
\end{array}$$ (${{\rm Depth}}^+({{\mathbb B}})$ is used to deal with attainment properties in the definition of ${{\rm Depth}}({{\mathbb B}})$, see e.g. [@RoSh:534 §1].) The depth (of Boolean algebras) is among cardinal functions that have more algebraic origins, and their relations to | 0 | non_member_985 |
“topological fellows” is often indirect, though sometimes very surprising. For example, if we define $${{\rm Depth}}_{{\rm H}+}({{\mathbb B}})=\sup\{{{\rm Depth}}({{\mathbb B}}/I): I\mbox{ is an ideal in }{{\mathbb B}}\;\},$$ then for any (infinite) Boolean algebra ${{\mathbb B}}$ we will have that ${{\rm Depth}}_{{\rm
H}+}({{\mathbb B}})$ is the tightness $t({{\mathbb B}})$ of the algebra ${{\mathbb B}}$ (or the tightness of the topological space ${\rm Ult}({{\mathbb B}})$ of ultrafilters on ${{\mathbb B}}$), see [@M2 Theorem 4.21]. A somewhat similar function to ${{\rm Depth}}_{{\rm H}+}$ is obtained by taking $\sup\{{{\rm Depth}}({{\mathbb B}}'): {{\mathbb B}}'$ is a subalgebra of ${{\mathbb B}}\;\}$, but clearly this brings nothing new: it is the old Depth. But if one wants to understand the behaviour of the depth for subalgebras of the considered Boolean algebra, then looking at the following [*subalgebra ${{\rm Depth}}$ relation*]{} may be very appropriate: $$\begin{array}{lr}
{{\rm Depth}}_{\rm Sr}({{\mathbb B}})=\{(\kappa,\mu):&\mbox{there is an infinite subalgebra
${{\mathbb B}}'$ of ${{\mathbb B}}$ such that }\ \\
&|{{\mathbb B}}'|=\mu\mbox{ and }{{\rm Depth}}({{\mathbb B}}')=\kappa\;\}.
\end{array}$$ A number of results related to this relation is presented by Monk in [@M2 Chapter 4]. There he asks if there are a Boolean algebra ${{\mathbb B}}$ and an infinite cardinal $\theta$ such that $(\theta,(2^\theta)^+)\in {{\rm Depth}}_{\rm
Sr}({{\mathbb | 0 | non_member_985 |
B}})$, while $(\omega, (2^\theta)^+)\notin{{\rm Depth}}_{\rm Sr}({{\mathbb B}})$ (see Monk [@M2 Problem 14]; we refer the reader to Chapter 4 of Monk’s book [@M2] for the motivation and background of this problem). Here we will partially answer this question, showing that it is consistent that there is such ${{\mathbb B}}$ and $\theta$. The question if that can be done in ZFC remains open.
Our consistency result is obtained by forcing, and the construction of the required forcing notion is interesting [*per se*]{}. We use the method of [*historic forcing*]{} which was first applied in Shelah and Stanley [@ShSt:258]. The reader familiar with [@ShSt:258] will notice several correspondences between the construction here and the method used there. However, we do not relay on that paper and our presentation here is self-contained.
Let us describe how our historic forcing notion is built. So, we fix two (regular) cardinals $\theta,\lambda$ and our aim is to force a Boolean algebra ${\dot{{{\mathbb B}}}^\theta_\lambda}$ such that $|{\dot{{{\mathbb B}}}^\theta_\lambda}|=\lambda^+$ and for every subalgebra ${{\mathbb B}}\subseteq{\dot{{{\mathbb B}}}^\theta_\lambda}$ of size $\lambda^+$ we have ${{\rm Depth}}({{\mathbb B}})=\theta$. The algebra ${\dot{{{\mathbb B}}}^\theta_\lambda}$ will be generated by $\langle x_i:i\in\dot{U}\rangle$ for some set $\dot{U}\subseteq\lambda^+$. A condition $p$ will be an approximation to the algebra | 0 | non_member_985 |
${\dot{{{\mathbb B}}}^\theta_\lambda}$, it will carry the information on what is the subalgebra ${{\mathbb B}}_p=\langle x_i: i\in u^p\rangle_{{\dot{{{\mathbb B}}}^\theta_\lambda}}$ for some $u^p\subseteq\lambda^+$. A natural way to describe algebras in this context is by listing ultrafilters (or: homomorphisms into $\{0,1\}$):
\[0.C\] For a set $w$ and a family $F\subseteq 2^{\textstyle w}$ we define
${{\rm cl}}(F)=\{g\in 2^{\textstyle w}: (\forall u\in [w]^{\textstyle
<\omega})(\exists f\in F)(f{\restriction}u=g{\restriction}u)\}$,
${{\mathbb B}}_{(w,F)}$ is the Boolean algebra generated freely by $\{x_\alpha:\alpha\in w\}$ except that
if $u_0,u_1\in [w]^{\textstyle <\omega}$ and there is no $f\in F$ such that $f{\restriction}u_0\equiv 0$, $f{\restriction}u_1\equiv 1$
then $\bigwedge\limits_{\alpha\in u_1} x_\alpha\wedge
\bigwedge\limits_{\alpha\in u_0} (-x_\alpha)=0$.
This description of algebras is easy to handle, for example:
\[see [[@Sh:479 2.6]]{}\] \[0.D\] Let $F\subseteq 2^{\textstyle w}$. Then:
1. Each $f\in F$ extends (uniquely) to a homomorphism from ${{\mathbb B}}_{(w,F)}$ to $\{0,1\}$ (i.e. it preserves the equalities from the definition of ${{\mathbb B}}_{(w,F)}$). If $F$ is closed, then every homomorphism from ${{\mathbb B}}_{(w,F)}$ to $\{0,1\}$ extends exactly one element of $F$.
2. If $\tau(y_0,\ldots,y_\ell)$ is a Boolean term and $\alpha_0,\ldots,
\alpha_\ell\in w$ are distinct then $$\begin{array}{l}
{{\mathbb B}}_{(w,F)}\models\tau(x_{\alpha_0},\ldots,x_{\alpha_\ell})\neq 0\qquad
\qquad\mbox{ if and only if}\\
(\exists f\in F)(\{0,1\}\models\tau(f(\alpha_0),\ldots,f(\alpha_k))=1).
\end{array}$$
3. If $w\subseteq w^*$, $F^*\subseteq 2^{\textstyle w^*}$ and $$(\forall f\in F)(\exists g\in F^*)(f\subseteq g)\quad\mbox{ | 0 | non_member_985 |
and }\quad
(\forall g\in F^*)(g{\restriction}w\in{{\rm cl}}(F))$$ then ${{\mathbb B}}_{(w,F)}$ is a subalgebra of ${{\mathbb B}}_{(w^*,F^*)}$.
So each condition $p$ in our forcing notion ${{{\mathbb P}^\theta_\lambda}}$ will have a set $u^p\in
[\lambda^+]^{<\lambda}$ and a closed set $F^p\subseteq 2^{\textstyle u^p}$ (and the respective algebra will be ${{\mathbb B}}_p={{\mathbb B}}_{(u^p,F^p)}$). But to make the forcing notion work, we will have to put more restrictions on our conditions, and we will be taking only those conditions that have to be taken to make the arguments work. For example, we want that cardinals are not collapsed by our forcing, and demanding that ${{{\mathbb P}^\theta_\lambda}}$ is $\lambda^+$-cc (and somewhat $({<}\lambda)$–closed) is natural in this context. How do we argue that a forcing notion is $\lambda^+$–cc? Typically we start with a sequence of $\lambda^+$ distinct conditions, we carry out some “cleaning procedure” (usually involving the $\Delta$–lemma etc), and we end up with (at least two) conditions that “can be put together”. Putting together two (or more) conditions that are approximations to a Boolean algebra means amalgamating them. There are various ways to amalgamate conditions - we will pick one that will work for several purposes. Then, once we declare that some conditions forming a “clean” $\Delta$–sequence of | 0 | non_member_985 |
length $\theta$ are in ${{{\mathbb P}^\theta_\lambda}}$, we will be bound to declare that the amalgamation is in our forcing notion. The amalgamation (and natural limits) will be the only way to build new conditions from the old ones, but the description above still misses an important factor. So far, a condition does not have to know what are the reasons for it to be called to ${{{\mathbb P}^\theta_\lambda}}$. This information is [*the history of the condition*]{} and it will be encoded by two functions $h^p,g^p$. (Actually, these functions will give histories of all elements of $u^p$ describing why and how those points were incorporated to $u^p$. Thus both functions will be defined on $u^p\times{{\rm ht}}(p)$, were ${{\rm ht}}(p)$ is the height of the condition $p$, that is the step in our construction at which the condition $p$ is created.) We will also want that our forcing is suitably closed, and getting “$({<}\lambda)$–strategically closed” would be fine. To make that happen we will have to deal with two relations on on ${{{\mathbb P}^\theta_\lambda}}$: $\leq_{\rm pr}$ and $\leq$. The first (“pure”) is $({<}\lambda)$–closed and it will help in getting the strategic closure of the second (main) one. In some sense, the relation | 0 | non_member_985 |
$\leq_{\rm pr}$ represents “the official line in history”, and sometimes we will have to rewrite that official history, see Definition \[defptran\] and Lemma \[3y1\] (on changing history see also Orwell [@Or49]).
The forcing notion ${{{\mathbb P}^\theta_\lambda}}$ has some other interesting features. (For example, conditions are very much like fractals, they contain many self-similar pieces (see Definition \[defcompo\] and Lemma \[3.4x\]).) The method of historic forcing notions could be applicable to more problems, and this is why in our presentation we separated several observations of general character (presented in the first section) from the problem specific arguments (section 2)
[**Notation:**]{}Our notation is standard and compatible with that of classical textbooks on set theory (like Jech [@J]) and Boolean algebras (like Monk [@M1], [@M2]). However in forcing considerations we keep the older tradition that
[*the stronger condition is the greater one.* ]{}
Let us list some of our notation and conventions.
1. Throughout the paper, $\theta,\lambda$ are fixed regular infinite cardinals, $\theta<\lambda$.
2. A name for an object in a forcing extension is denoted with a dot above (like $\dot{X}$) with one exception: the canonical name for a generic filter in a forcing notion ${{\mathbb P}}$ will be called $\Gamma_{{\mathbb P}}$. For | 0 | non_member_985 |
a ${{\mathbb P}}$–name $\dot{X}$ and a ${{\mathbb P}}$–generic filter $G$ over ${{\bf V}}$, the interpretation of the name $\dot{X}$ by $G$ is denoted by $\dot{X}^G$.
3. $i,j,\alpha,\beta,\gamma,\delta,\ldots$ will denote ordinals.
4. For a set $X$ and a cardinal $\lambda$, $[X]^{\textstyle<\lambda}$ stands for the family of all subsets of $X$ of size less than $\lambda$. The family of all functions from $Y$ to $X$ is called $X^{\textstyle Y}$. If $X$ is a set of ordinals then its order type is denoted by ${{\rm otp}}(X)$.
5. In Boolean algebras we use $\vee$ (and $\bigvee$), $\wedge$ (and $\bigwedge$) and $-$ for the Boolean operations. If ${{\mathbb B}}$ is a Boolean algebra, $x\in{{\mathbb B}}$ then $x^0=x$, $x^1=-x$.
6. For a subset $Y$ of an algebra ${{\mathbb B}}$, the subalgebra of ${{\mathbb B}}$ generated by $Y$ is denoted by $\langle Y\rangle_{{{\mathbb B}}}$.
[**Acknowledgements:**]{}We would like to thank the referee for valuable comments and suggestions.
The forcing and its basic properties
====================================
Let us start with the definition of the forcing notion ${{{\mathbb P}^\theta_\lambda}}$. By induction on $\alpha<\lambda$ we will define sets of conditions $P^{\theta,
\lambda}_\alpha$, and for each $p\in P^{\theta,\lambda}_\alpha$ we will define $u^p,F^p,{{\rm ht}}(p),h^p$ and $g^p$. Also we will define relations $\leq^\alpha$ and $\leq^\alpha_{\rm | 0 | non_member_985 |
pr}$ on $P^{\theta,\lambda}_\alpha$. Our inductive requirements are:
1. for each $p\in P^{\theta,\lambda}_\alpha$:\
$u^p\in [\lambda^+]^{\textstyle<\lambda}$, ${{\rm ht}}(p)\leq\alpha$, $F^p
\subseteq 2^{\textstyle u^p}$ is a non-empty closed set, $g^p$ is a function with domain ${{\rm dom}}(g^p)=u^p\times{{\rm ht}}(p)$ and values of the form $(\ell,
\tau)$, where $\ell<2$ and $\tau$ is a Boolean term, and $h^p:u^p\times
{{\rm ht}}(p)\longrightarrow\theta+2$ is a function,
2. $\leq^\alpha,\leq^\alpha_{\rm pr}$ are transitive and reflexive relations on $P^{\theta,\lambda}_\alpha$, and $\leq^\alpha$ extends $\leq^\alpha_{\rm pr}$,
3. if $p,q\in P^{\theta,\lambda}_\alpha$, $p\leq^\alpha
q$, then $u^p\subseteq u^q$, ${{\rm ht}}(p)\leq{{\rm ht}}(q)$, and $F^p=\{f{\restriction}u^p:
f\in F^q\}$, and if $p\leq^\alpha_{\rm pr} q$, then for every $i\in u^p$ and $\xi<{{\rm ht}}(p)$ we have $h^p(i,\xi)=h^q(i,\xi)$ and $g^p(i,\xi)= g^q(i,\xi)$,
4. if $\beta<\alpha$ then $P^{\theta,\lambda}_\beta
\subseteq P^{\theta,\lambda}_\alpha$, and $\leq^\alpha_{\rm pr}$ extends $\leq^\beta_{\rm pr}$, and $\leq^\alpha$ extends $\leq^\beta$.
For a condition $p\in P^{\theta,\lambda}_\alpha$, we will also declare that ${{\mathbb B}}^p={{\mathbb B}}_{(u^p,F^p)}$ (the Boolean algebra defined in Definition \[0.C\]).
We define $P^{\theta,\lambda}_0=\{\langle\xi\rangle:\xi<\lambda^+\}$ and for $p=\langle\xi\rangle$ we let $F^p=2^{\textstyle\{\xi\}}$, ${{\rm ht}}(p)=0$ and $h^p=\emptyset=g^p$. The relations $\leq^0_{\rm pr}$ and $\leq^0$ both are the equality. \[Clearly these objects are as declared, i.e, clauses (i)$_0$–(iv)$_0$ hold true.\]
If $\gamma<\lambda$ is a limit ordinal, then we put $$\begin{array}{l}
P^*_\gamma=\big\{\langle p_\xi:\xi<\gamma\rangle: (\forall\xi<\zeta<
\gamma)(p_\xi\in P^{\theta,\lambda}_\xi\ \&\ {{\rm ht}}(p_\xi)=\xi \ \&\ p_\xi
\leq^\zeta_{\rm pr} | 0 | non_member_985 |
p_\zeta)\big\},\\
P^{\theta,\lambda}_\gamma=\bigcup\limits_{\alpha<\gamma}P^{\theta,
\lambda}_\alpha\cup P^*_\gamma,
\end{array}$$ and for $p=\langle p_\xi:\xi<\gamma\rangle\in P^*_\gamma$ we let $$u^p=\bigcup\limits_{\xi<\gamma} u^{p_\xi},\quad F^p=\{f\in 2^{\textstyle
u^p}:(\forall\xi<\gamma)(f{\restriction}u^{p_\xi}\in F^{p_\xi})\},\quad {{\rm ht}}(p)=
\gamma$$ and $h^p=\bigcup\limits_{\xi<\gamma} h^{p_\xi}$ and $g^p=\bigcup\limits_{\xi<
\gamma} g^{p_\xi}$. We define $\leq^\gamma$ and $\leq^\gamma_{\rm pr}$ by:
$p\leq^\gamma_{\rm pr}q$if and only if
[*either*]{} $p,q\in P^{\theta,\lambda}_\alpha$, $\alpha<\gamma$ and $p\leq^\alpha_{\rm pr}q$,
[*or*]{} $q=\langle q_\xi:\xi<\gamma\rangle\in P^*_\gamma$, $p\in
P^{\theta,\lambda}_\alpha$ and $p\leq^\alpha_{\rm pr}q_\alpha$ for some $\alpha<\gamma$,
[*or*]{} $p=q$;
$p\leq^\gamma q$if and only if
[*either*]{} $p,q\in P^{\theta,\lambda}_\alpha$, $\alpha<\gamma$ and $p\leq^\alpha q$,
[*or*]{} $q=\langle q_\xi:\xi<\gamma\rangle\in P^*_\gamma$, $p\in
P^{\theta,\lambda}_\alpha$ and $p\leq^\alpha q_\alpha$ for some $\alpha<
\gamma$,
[*or*]{} $p=\langle p_\xi:\xi<\gamma\rangle\in P^*_\gamma$, $q=\langle
q_\xi:\xi<\gamma\rangle\in P^*_\gamma$ and $$(\exists\delta<\gamma)(\forall\xi<\gamma)(\delta\leq\xi\ \Rightarrow\
p_\xi\leq^\xi q_\xi).$$
\[It is straightforward to show that clauses (i)$_\gamma$–(iv)$_\gamma$ hold true.\]
Suppose now that $\alpha<\lambda$. Let $P^*_{\alpha+1}$ consist of all tuples $$\langle\zeta^*,\tau^*,n^*,u^*,\langle p_\xi,v_\xi:\xi<\theta\rangle\rangle$$ such that for each $\xi_0<\xi_1<\theta$:
1. $\zeta^*<\theta$, $n^*<\omega$, $\tau^*=\tau^*(y_1,\ldots,y_{
n^*})$ is a Boolean term, $u^*\in [\lambda^+]^{\textstyle<\lambda}$,
2. $p_{\xi_0}\in P^{\theta,\lambda}_\alpha$, ${{\rm ht}}(p)=\alpha$, $v_{\xi_0}\in [u^{p_{\xi_0}}]^{\textstyle n^*}$,
3. the family $\{u^{p_\xi}:\xi<\theta\}$ forms a $\Delta$–system with heart $u^*$ and $u^{p_{\xi_0}}\setminus u^*\neq
\emptyset$ and $$\sup(u^*)<\min(u^{p_{\xi_0}}\setminus u^*)\leq\sup (u^{p_{\xi_0}}\setminus
u^*)<\min(u^{p_{\xi_1}}\setminus u^*),$$
4. ${{\rm otp}}(u^{p_{\xi_0}})={{\rm otp}}(u^{p_{\xi_1}})$ and if $H:u^{p_{
\xi_0}}\longrightarrow u^{p_{\xi_1}}$ is the order isomorphism then $H{\restriction}u^*$ is the identity on $u^*$, $F^{p_{\xi_0}}=\{f{\circ}H:f\in F^{p_{\xi_1}}
\}$, $H[v_{\xi_0}]=v_{\xi_1}$ and $$(\forall j\in u^{p_{\xi_0}})(\forall\beta<\alpha)(h^{p_{\xi_0}}(j,\beta)=
h^{p_{\xi_1}}(H(j),\beta)\ \&\ g^{p_{\xi_0}}(j,\beta)=g^{p_{\xi_1}}(H(j),
\beta)).$$
We put $P^{\theta,\lambda}_{\alpha+1}=P^{\theta,\lambda}_\alpha\cup P^*_{
| 0 | non_member_985 |
\alpha+1}$ and for $p=\langle\zeta^*,\tau^*,n^*,u^*,\langle p_\xi,v_\xi:\xi<
\theta\rangle\rangle\in P^*_{\alpha+1}$ we let $u^p=\bigcup\limits_{\xi<
\theta} u^{p_\xi}$ and $$\begin{array}{ll}
F^p=\{f\in 2^{\textstyle u^p}:& (\forall\xi<\theta)(f{\restriction}u^{p_\xi}\in
F^{p_{\xi}})\mbox{ and for all }\xi<\zeta<\theta\\
\ &f(\sigma_{\rm maj}(\tau_{3\cdot\xi},\tau_{3\cdot\xi+1},\tau_{3\cdot\xi+
2}))\leq f(\sigma_{\rm maj}(\tau_{3\cdot\zeta},\tau_{3\cdot\zeta+1},\tau_{3
\cdot\zeta+2}))\},
\end{array}$$ where $\tau_\xi=\tau^*(x_i:i\in v_\xi)$ for $\xi<\theta$ (so $\tau_\xi$ is an element of the algebra ${{\mathbb B}}^{p_\xi}={{\mathbb B}}_{(u^{p_\xi},F^{p_\xi})}$), and $\sigma_{\rm maj}(y_0,y_1,y_2)=(y_0\wedge y_1)\vee (y_0\wedge y_2)\vee
(y_1\wedge y_2)$. Next we let ${{\rm ht}}(p)=\alpha+1$ and we define functions $h^p,g^p$ on $u^p\times (\alpha+1)$ by $$h^p(j,\beta)=\left\{\begin{array}{lll}
h^{p_{\xi}}(j,\beta)&\mbox{if}&j\in u^{p_\xi},\ \xi<\theta,\ \beta<\alpha,\\
\theta &\mbox{if}&j\in u^*,\ \beta=\alpha,\\
\theta+1 &\mbox{if}&j\in u^{p_{\zeta^*}}\setminus u^*,\
\beta=\alpha,\\
\xi &\mbox{if}&j\in u^{p_\xi}\setminus u^*,\ \xi<\theta,\
\xi\neq\zeta^*,\ \beta=\alpha,
\end{array}\right.$$ $$g^p(j,\beta)=\left\{\begin{array}{lll}
g^{p_{\xi}}(j,\beta)&\mbox{if}&j\in u^{p_\xi},\ \xi<\theta,\ \beta<\alpha,\\
(1,\tau^*) &\mbox{if}&j\in v_\xi,\ \xi<\theta,\ \beta=\alpha,\\
(0,\tau^*) &\mbox{if}&j\in u^{p_\xi}\setminus v_\xi,\ \xi<\theta,\
\beta=\alpha.
\end{array}\right.$$ Next we define the relations $\leq^{\alpha+1}_{\rm pr}$ and $\leq^{\alpha+1}$ by:
$p\leq^{\alpha+1}_{\rm pr}q$if and only if
[*either*]{} $p,q\in P^{\theta,\lambda}_\alpha$ and $p\leq^\alpha_{\rm
pr}q$,
[*or*]{} $q=\langle \zeta^*,\tau^*,n^*,u^*,\langle q_\xi,v_\xi:\xi<\theta
\rangle\rangle\in P^*_{\alpha+1}$, $p\in P^{\theta,\lambda}_\alpha$, and $p\leq^\alpha_{\rm pr} q_{\zeta^*}$,
[*or*]{} $p=q$;
$p\leq^{\alpha+1} q$if and only if
[*either*]{} $p,q\in P^{\theta,\lambda}_\alpha$ and $p\leq^\alpha q$,
[*or*]{} $q=\langle\zeta^*,\tau^*,n^*,u^*,\langle q_\xi,v_\xi:\xi<\theta
\rangle\rangle\in P^*_{\alpha+1}$, $p\in P^{\theta,\lambda}_\alpha$, and $p\leq^\alpha q_\xi$ for some $\xi<\theta$,
[*or*]{} $p=\langle\zeta^{**},\tau^*,n^*,u^*,\langle p_\xi,v_\xi:\xi<
\theta\rangle\rangle$, $q=\langle\zeta^*,\tau^*,n^*,u^*,\langle q_\xi,v_\xi:
\xi<\theta\rangle\rangle$ are from $P^*_{\alpha+1}$ and $$(\forall\xi<\theta)(p_\xi\leq^\alpha q_\xi\ \&\ u^{p_\xi}=u^{q_\xi}).$$
\[Again, it is easy to show that clauses (i)$_{\alpha+1}$–(iv)$_{
\alpha+1}$ are satisfied.\]
After the construction is carried out | 0 | non_member_985 |
we let $${{{\mathbb P}^\theta_\lambda}}=\bigcup\limits_{\alpha<\lambda} P^{\theta,\lambda}_\alpha\quad\mbox{
and }\quad{}\leq_{\rm pr}{}={}\bigcup\limits_{\alpha<\lambda}{\leq^\alpha_{
\rm pr}}\quad\mbox{ and }\quad{}\leq{}={}\bigcup\limits_{\alpha<\lambda}
{\leq^\alpha}.$$ One easily checks that $\leq_{\rm pr}$ is a partial order on ${{{\mathbb P}^\theta_\lambda}}$ and that the relation $\leq$ is transitive and reflexive, and that ${\leq_{\rm pr}}
\subseteq {\leq}$.
\[3.1\] Let $p,q\in{{{\mathbb P}^\theta_\lambda}}$.
1. If $p\leq q$ then ${{\rm ht}}(p)\leq{{\rm ht}}(q)$, $u^p\subseteq u^q$ and $F^p=\{
f{\restriction}u^p: f\in F^q\}$ (so ${{\mathbb B}}^p$ is a subalgebra of ${{\mathbb B}}^q$). If $p\leq q$ and ${{\rm ht}}(p)={{\rm ht}}(q)$, then $q\leq p$.
2. For each $j\in u^p$, the set $\{\beta<{{\rm ht}}(p): h^p(j,\beta)<\theta\}$ is finite.
3. If $p\leq_{\rm pr} q$ and $i\in u^p$, then $h^q(i,\beta)\geq\theta$ for all $\beta$ such that ${{\rm ht}}(p)\leq\beta<{{\rm ht}}(q)$.
4. If $i,j\in u^p$ are distinct, then there is $\beta<{{\rm ht}}(p)$ such that $\theta\neq h^p(i,\beta)\neq h^p(j,\beta)\neq\theta$.
5. For each finite set $X\subseteq{{\rm ht}}(p)$ there is $i\in u^p$ such that $$\{\beta<{{\rm ht}}(p):h^p(i,\beta)<\theta\}=X.$$
6. If $p\leq_{\rm pr} q$ then there is a $\leq_{\rm pr}$–increasing sequence $\langle p_\xi:\xi\leq{{\rm ht}}(p)\rangle\subseteq{{{\mathbb P}^\theta_\lambda}}$ such that $p_{{{\rm ht}}(p)}=p$, $p_{{{\rm ht}}(q)}=q$ and ${{\rm ht}}(p_\xi)=\xi$ (for $\xi\leq{{\rm ht}}(p)$). (In particular, if $p\leq_{\rm pr} q$ and ${{\rm ht}}(p)={{\rm ht}}(q)$ then $p=q$.)
7. If ${{\rm ht}}(p)=\gamma$ is a limit ordinal, $p=\langle p_\xi:\xi<\gamma
\rangle$, then for each $i\in u^p$ and $\xi<\gamma$: $$i\in u^{p_\xi}\quad\mbox{ | 0 | non_member_985 |
if and only if }\quad (\forall\zeta<\gamma)(\xi
\leq\zeta\ \Rightarrow\ h^p(i,\zeta)\geq\theta).$$
1)Should be clear (an easy induction).
2)Suppose that $p\in{{{\mathbb P}^\theta_\lambda}}$ and $j\in u^p$ are a counterexample with the minimal possible value of ${{\rm ht}}(p)$. Necessarily ${{\rm ht}}(p)$ is a limit ordinal, $p=\langle p_\xi:\xi<{{\rm ht}}(p)\rangle$, ${{\rm ht}}(p_\xi)=\xi$ and $\zeta<
\xi<{{\rm ht}}(p)\ \Rightarrow\ p_\zeta\leq_{\rm pr} p_\xi$. Let $\xi<{{\rm ht}}(p)$ be the first ordinal such that $j\in u^{p_\xi}$. By the choice of $p$, the set $\{\beta\leq\xi: h^p(j,\beta)<\theta\}$ is finite, but clearly $h^p(j,\beta)
\geq\theta$ for all $\beta\in (\xi,{{\rm ht}}(p))$.
3)An easy induction on ${{\rm ht}}(q)$ (with fixed $p$).
4)We show this by induction on ${{\rm ht}}(p)$. Suppose that ${{\rm ht}}(
p)=\alpha+1$, so $p=\langle\zeta^*,\tau^*,n^*,u^*,\langle p_\xi,v_\xi:\xi<
\theta\rangle\rangle$, and $i,j\in u^p$ are distinct. If $i,j\in u^{p_\xi}$ for some $\xi<\theta$, then by the inductive hypothesis we find $\beta<
\alpha$ such that $$\theta\neq h^p(i,\beta)=h^{p_\xi}(i,\beta)\neq h^{p_\xi}(j,\beta)=
h^p(j,\beta)\neq\theta.$$ If $i\in u^{p_\xi}\setminus u^*$, $j\in u^{p_\zeta}\setminus u^*$ and $\xi,
\zeta<\theta$ are distinct, then look at the definition of $h^p(i,\alpha)$, $h^p(j,\alpha)$ – these two values cannot be equal (and both are distinct from $\theta$). Finally suppose that ${{\rm ht}}(p)$ is limit, so $p=\langle
p_\xi:\xi<{{\rm ht}}(p)\rangle$. Take $\xi<{{\rm ht}}(p)$ such that $i,j\in u^{p_\xi}$ and apply the inductive hypothesis to $p_\xi$ getting $\beta<\xi$ such that $h^p(i,\beta)\neq h^p(j,\beta)$ | 0 | non_member_985 |
(and both are not $\theta$).
5)Again, it goes by induction on ${{\rm ht}}(p)$. First consider a limit stage, and suppose that ${{\rm ht}}(p)=\gamma$ is a limit ordinal, $X\in[\gamma]^{\textstyle{<}\omega}$ and $p=\langle p_\xi:\xi<\gamma
\rangle$. Let $\xi<\gamma$ be such that $X\subseteq\xi$. By the inductive hypothesis we find $i\in u^{p_\xi}$ such that $\{\beta<\xi:h^p(i,\beta)<
\theta\}=X$. Applying clause (3) we may conclude that this $i$ is as required. Now consider a successor case ${{\rm ht}}(p)=\alpha+1$. Let $p=\langle
\zeta^*,\tau^*,n^*,u^*,\langle p_\xi,v_\xi:\xi<\theta\rangle\rangle$, and let $\xi<\theta$ be $\zeta^*$ if $\alpha\in X$, and be $\zeta^*+1$ otherwise. Apply the inductive hypothesis to $p_\xi$ and $X\cap \alpha$ to get suitable $i\in u^{p_\xi}$, and note that this $i$ works for $p$ and $X$ too.
6), 7)Straightforward.
\[defiso\] We say that conditions $p,q\in{{{\mathbb P}^\theta_\lambda}}$ are [*isomorphic*]{} if ${{\rm ht}}(p)=
{{\rm ht}}(q)$, ${{\rm otp}}(u^p)={{\rm otp}}(u^q)$, and if $H:u^p\longrightarrow u^q$ is the order isomorphism, then for every $\beta<{{\rm ht}}(p)$ $$(\forall j\in u^p)(h^p(j,\beta)=h^q(H(j),\beta)\ \&\ g^p(j,\beta)=g^p(H(j),
\beta)).$$ \[In this situation we may say that $H$ is the isomorphism from $p$ to $q$.\]
\[3x1\] Suppose that $q_0,q_1\in{{{\mathbb P}^\theta_\lambda}}$ are isomorphic conditions and $H$ is the isomorphism from $q_0$ to $q_1$.
1. If ${{\rm ht}}(q_0)={{\rm ht}}(q_1)=\gamma$ is a limit ordinal, $q_\ell=\langle
q^\ell_\xi: \xi<\gamma\rangle$ (for $\ell<2$), then $H{\restriction}u^{q_\xi^0}$ is an isomorphism from | 0 | non_member_985 |
$q^0_\xi$ to $q^1_\xi$.
2. If ${{\rm ht}}(q_0)={{\rm ht}}(q_1)=\alpha+1$, $\alpha<\lambda$, and $q_\ell=
\langle\zeta^*_\ell,\tau^*_\ell,n^*_\ell,u^*_\ell,\langle q^\ell_\xi,
v^\ell_\xi:\xi<\theta\rangle\rangle$ (for $\ell<2$), then $\zeta^*_0=
\zeta^*_1$, $\tau^*_0=\tau^*_1$, $n^*_0=n^*_1$, $H{\restriction}u^{q_\xi^0}$ is an isomorphism from $q^0_\xi$ to $q^1_\xi$ and $H[v^0_\xi]=v^1_\xi$ (for $\xi<
\theta$).
3. $F^{q_0}=\{f{\circ}H:f\in F^{q_1}\}$.
4. Assume $p_0\leq q_0$. Then there is a unique condition $p_1\leq q_1$ such that $H{\restriction}u^{p_0}$ is the isomorphism from $p_0$ to $p_1$.\
1), 2)Straightforward (for (1) use Lemma \[3.1\](7)).\
3), 4)Easy inductions on ${{\rm ht}}(q_0)$ using (1), (2) above.
\[defptran\] By induction on $\alpha<\lambda$, for conditions $p,q\in P^{\theta,
\lambda}_\alpha$ such that $p\leq^\alpha q$, we define [*the $p$–transformation $T_p(q)$ of $q$*]{}.
- If $\alpha=0$ (so necessarily $p=q$) then $T_p(q)=p$.
- Assume that ${{\rm ht}}(q)=\alpha+1$, $q=\langle\zeta^*,\tau^*,n^*,u^*,
\langle q_\xi,v_\xi:\xi<\theta\rangle\rangle$.
If $p\leq q_\xi$ for some $\xi<\theta$, then let $\xi^*$ be such that $p\leq
q_{\xi^*}$. Next for $\xi<\theta$ let $q_\xi'=T_{H_{\xi^*, \xi}(p)}(q_\xi)$, where $H_{\xi^*,\xi}$ is the isomorphism from $q_{\xi^*}$ to $q_\xi$. Define $T_p(q)=\langle\xi^*,\tau^*,n^*,u^*,\langle q_\xi',v_\xi: \xi<\theta\rangle
\rangle$.
Suppose now that $p=\langle\zeta^{**},\tau^*,n^*,u^*,\langle p_\xi,v_\xi:\xi
<\theta\rangle\rangle$ and $u^{p_\xi}=u^{q_\xi}$, $p_\xi\leq q_\xi$ (for $\xi<\theta$). Let $q_\xi'=T_{p_\xi}(q_\xi)$ and put $T_p(q)=\langle
\zeta^{**},\tau^*,n^*,u^*,\langle q_\xi',v_\xi:\xi<\theta\rangle\rangle$.
- Assume now that ${{\rm ht}}(q)$ is a limit ordinal and $q=\langle q_\xi:\xi<
{{\rm ht}}(q)\rangle$.
If ${{\rm ht}}(p)<{{\rm ht}}(q)$ then $p\leq q_{\varepsilon}$ for some ${\varepsilon}<{{\rm ht}}(q)$, and we may choose $q_\xi'$ (for $\xi<{{\rm ht}}(q)$) | 0 | non_member_985 |
such that ${{\rm ht}}(q_\xi')=\xi$, $\xi<
\xi'<{{\rm ht}}(q)\ \Rightarrow\ q_\xi'\leq_{\rm pr} q_{\xi'}'$, and $q_\zeta'
=T_p(q_\zeta)$ for $\zeta\in [{\varepsilon},{{\rm ht}}(q))$. Next we let $T_p(q)=\langle
q_\zeta':\zeta<\theta\rangle$.
If ${{\rm ht}}(p)={{\rm ht}}(q)$, $p=\langle p_\xi:\xi<{{\rm ht}}(p)\rangle$ and $p_\xi\leq
q_\xi$ for $\xi>\delta$ (for some $\delta<{{\rm ht}}(p)$) then we define $T_p(q)=p$.
To show that the definition of $T_p(q)$ is correct one proves inductively (parallely to the definition of the $p$–transformation of $q$) the following facts.
\[3y1\] Assume $p,q\in{{{\mathbb P}^\theta_\lambda}}$, $p\le q$. Then:
1. $T_p(q)\in{{{\mathbb P}^\theta_\lambda}}$, $u^{T_p(q)}=u^q$, ${{\rm ht}}(T_p(q))={{\rm ht}}(q)$,
2. $p\leq_{\rm pr} T_p(q)\leq q\leq T_p(q)$,
3. ${{\rm ht}}(p)={{\rm ht}}(q)\ \Rightarrow\ T_p(q)=p$,
4. if $q'\in{{{\mathbb P}^\theta_\lambda}}$ is isomorphic to $q$ and $H:u^q\longrightarrow u^{q'}$ is the isomorphism from $q$ to $q'$, then $H$ is the isomorphism from $T_p(q)$ to $T_{H(p)}(q')$,
5. if $q\leq_{\rm pr}q'$ then $T_p(q)\leq_{\rm pr} T_p(q')$.
\[3y2\] Every $\leq_{\rm pr}$–increasing chain in ${{{\mathbb P}^\theta_\lambda}}$ of length $<\lambda$ has a $\leq_{\rm pr}$–upper bound, that is the partial order $({{{\mathbb P}^\theta_\lambda}},\leq_{\rm
pr})$ is $(<\lambda)$–closed.
Let us recall that a forcing notion $({{\mathbb Q}},\leq)$ is [*$({<}\lambda)$–strategically closed*]{} if the second player has a winning strategy in the following game $\Game_\lambda({{\mathbb Q}})$.
The game $\Game_\lambda({{\mathbb Q}})$ lasts $\lambda$ moves. The first player starts with choosing a condition $p^*\in{{\mathbb Q}}$. Later, in her | 0 | non_member_985 |
$i^{\rm th}$ move, the first player chooses an open dense subset $D_i$ of ${{\mathbb Q}}$. The second player (in his $i^{\rm th}$ move) picks a condition $p_i\in{{\mathbb Q}}$ so that $p_0\geq
p^*$, $p_i\in D_i$ and $p_i\geq p_j$ for all $j<i$. The second player looses the play if for some $i<\lambda$ he has no legal move.
It should be clear that $({<}\lambda)$–strategically closed forcing notions do not add sequences of ordinals of length less than $\lambda$. The reader interested in this kind of properties of forcing notions and iterating them is referred to [@Sh:587], [@Sh:667].
\[3.3\] Assume that $\theta<\lambda$ are regular cardinals, $\lambda^{<\lambda}=
\lambda$. Then $({{{\mathbb P}^\theta_\lambda}},\leq)$ is a $(<\lambda)$–strategically closed $\lambda^+$–cc forcing notion.
It follows from Lemma \[3y1\](2) that if $D\subseteq{{{\mathbb P}^\theta_\lambda}}$ is an open dense set, $p\in{{{\mathbb P}^\theta_\lambda}}$, then there is a condition $q\in D$ such that $p\leq_{\rm
pr} q$. Therefore, to win the game $\Game_\lambda({{{\mathbb P}^\theta_\lambda}})$, the second player can play so that the conditions $p_i$ that he chooses are $\leq_{\rm
pr}$–increasing, and thus there are no problems with finding $\leq_{\rm
pr}$–bounds (remember Proposition \[3y2\]).
Now, to show that ${{{\mathbb P}^\theta_\lambda}}$ is $\lambda^+$–cc, suppose that $\langle p_\delta:
\delta<\lambda^+\rangle$ is a sequence of distinct conditions from ${{{\mathbb P}^\theta_\lambda}}$. We | 0 | non_member_985 |
may find a set $A\in [\lambda^+]^{\textstyle \lambda^+}$ such that
- conditions $\{p_\delta:\delta\in A\}$ are pairwise isomorphic,
- the family $\{u^{p_\delta}:\delta\in A\}$ forms a $\Delta$–system with heart $u^*$,
- if $\delta_0<\delta_1$ are from $A$ then $$\sup(u^*)<\min(u^{p_{\delta_0}}\setminus u^*)\leq \sup(u^{p_{\delta_0}}
\setminus u^*)<\min(u^{p_{\delta_0}}\setminus u^*).$$
Take an increasing sequence $\langle\delta_\xi:\xi<\theta\rangle$ of elements of $A$, let $\tau^*={\bf 1}$, $v_\xi=\emptyset$ (for $\xi<\theta$), and look at $p=\langle 0,\tau^*,0,u^*,\langle p_{\delta_\xi},v_\xi:\xi<
\theta\rangle\rangle$. It is a condition in ${{{\mathbb P}^\theta_\lambda}}$ stronger than all $p_{\delta_\xi}$’s.
\[defcompo\] By induction on ${{\rm ht}}(p)$ we define [*$\alpha$–components of $p$*]{} (for $p\in{{{\mathbb P}^\theta_\lambda}}$, $\alpha\leq{{\rm ht}}(p)$).
- First we declare that the only ${{\rm ht}}(p)$–component of $p$ is the $p$ itself.
- If ${{\rm ht}}(p)=\beta+1$, $p=\langle\zeta^*,\tau^*,n^*,u^*,\langle p_\xi,
v_\xi:\xi<\theta\rangle\rangle$ and $\alpha=\beta$, then $\alpha$–components of $p$ are $p_\xi$ (for $\xi<\theta$); if $\alpha<\beta$, then $\alpha$–components of $p$ are those $q$ which are $\alpha$–components of $p_\xi$ for some $\xi<\theta$.
- If ${{\rm ht}}(p)$ is a limit ordinal, $p=\langle p_\xi:\xi<{{\rm ht}}(p)\rangle$ and $\alpha<{{\rm ht}}(p)$, then $\alpha$–components of $p$ are $\alpha$–components of $p_\xi$ for $\xi\in [\alpha,{{\rm ht}}(p))$.
\[3.4x\] Assume $p\in {{{\mathbb P}^\theta_\lambda}}$ and $\alpha<{{\rm ht}}(p)$.
1. If $q$ is an $\alpha$–component of $p$ then $q\leq p$, ${{\rm ht}}(q)=
\alpha$, and for all $j_0,j_1\in u^q$ and every $\beta\in [\alpha,{{\rm ht}}(p))$: $$h^p(j_0,\beta)\neq\theta\ \&\ h^p(j_1,\beta)\neq \theta\quad\Rightarrow
| 0 | non_member_985 |
\quad h^p(j_0,\beta)=h^p(j_1,\beta).$$ Moreover, for each $i\in u^p$ there is a unique $\alpha$–component $q$ of $p$ such that $i\in u^q$ and $$(\forall j\in u^q)(\forall\beta\in [\alpha,{{\rm ht}}(p)))(h^p(i,\beta)\geq
\theta\ \Rightarrow\ h^p(j,\beta)\geq\theta).$$
2. If $H$ is an isomorphism from $p$ onto $p'\in{{{\mathbb P}^\theta_\lambda}}$, and $q$ is an $\alpha$–component of $p$, then $H(q)$ is an $\alpha$–component of $p'$. If $q_0,q_1$ are $\alpha$–components of $p$ then $q_0,q_1$ are isomorphic.
3. There is a unique $\alpha$–component $q$ of $p$ such that $q\leq_{\rm
pr} p$.
Easy inductions on ${{\rm ht}}(p)$.
\[closed\] By induction on ${{\rm ht}}(p)$ we define when a set $Z\subseteq \lambda$ is $p$–closed for a condition $p\in{{{\mathbb P}^\theta_\lambda}}$.
- If ${{\rm ht}}(p)=0$ then every $Z\subseteq\lambda$ is $p$–closed;
- if ${{\rm ht}}(p)$ is limit, $p=\langle p_\xi:\xi<{{\rm ht}}(p)\rangle$, then $Z$ is $p$–closed provided it is $p_\xi$–closed for each $\xi<{{\rm ht}}(p)$;
- if ${{\rm ht}}(p)=\alpha+1$, $p=\langle\zeta^*,\tau^*,n^*,u^*,\langle
p_\xi,v_\xi:\xi<\theta\rangle\rangle$ and $\alpha\notin Z$, then $Z$ is $p$–closed whenever it is $p_{\zeta^*}$–closed;
- if ${{\rm ht}}(p)=\alpha+1$, $p=\langle\zeta^*,\tau^*,n^*,u^*,\langle
p_\xi,v_\xi:\xi<\theta\rangle\rangle$ and $\alpha\in Z$, then $Z$ is $p$–closed provided it is $p_{\zeta^*}$–closed and $$\{\beta<\alpha:(\exists j\in v_{\zeta^*}\cup\{\min(u^{p_{\zeta^*}}
\setminus u^*)\})(h^{p_{\zeta^*}}(j,\beta)<\theta)\}\subseteq Z.$$
\[3.5.1\]
1. If $p\in{{{\mathbb P}^\theta_\lambda}}$ and $w\in [{{\rm ht}}(p)]^{\textstyle<\omega}$, then there is a finite $p$–closed set $Z\subseteq{{\rm ht}}(p)$ such that $w\subseteq Z$.
2. If $p,q\in{{{\mathbb P}^\theta_\lambda}}$ are | 0 | non_member_985 |
isomorphic and $Z$ is $p$–closed, then $Z$ is $q$–closed. If $Z$ is $p$–closed, $\alpha<{{\rm ht}}(p)$ and $p^*$ is an $\alpha$–component of $p$, then $Z\cap\alpha$ is $p^*$–closed.
Easy inductions on ${{\rm ht}}(p)$ (remember Lemma \[3.1\](2)).
\[UUpsilon\] Suppose that $p\in{{{\mathbb P}^\theta_\lambda}}$ and $Z\subseteq {{\rm ht}}(p)$ is a finite $p$–closed set. Let $Z=\{\alpha_0,\ldots,\alpha_{k-1}\}$ be the increasing enumeration.
1. We define $$U[p,Z]\stackrel{\rm def}{=}\{j\in u^p: (\forall\beta<{{\rm ht}}(p))(h^p(j,\beta)
<\theta\ \Rightarrow\ \beta\in Z)\}.$$
2. We let $$\Upsilon_p(Z)=\langle\zeta_\ell,\tau_\ell,n_\ell,\langle g_\ell,h^\ell_0,
\ldots,h^\ell_{n_\ell-1}\rangle:\ell<k\rangle,$$ where, for $\ell<k$, $\zeta_\ell$ is an ordinal below $\theta$, $\tau_\ell$ is a Boolean term, $n_\ell<\omega$ and $g_\ell,h^\ell_0,\ldots,h^\ell_{
n_\ell-1}:\ell\longrightarrow 2$, and they all are such that for every (equivalently: some) $\alpha_\ell+1$–component $q=\langle\zeta^*,\tau^*,
n^*,u^*,\langle q_\xi,v_\xi:\xi<\theta\rangle\rangle$ of $p$ we have:$\zeta_\ell=\zeta^*$, $\tau_\ell=\tau^*$, $n_\ell=n^*$ and if $v_\xi=\{j_0,
\ldots,j_{n_\ell-1}\}$ (the increasing enumeration) then $$(\forall m<n_\ell)(\forall\ell'<\ell)(h^\ell_m(\ell')=h^q(j_m,
\alpha_{\ell'})),$$ and if $i_0=\min(u^{q_{\zeta^*}}\setminus u^*)$ then $(\forall\ell'<\ell)(
g_\ell(\ell')=h^q(i_0,\alpha_{\ell'}))$. (Note that $\zeta_\ell,\tau_\ell,
n_\ell$, $g_\ell,h^\ell_0,\ldots,h^\ell_{n_\ell-1}$ are well-defined by Lemma \[3.4x\]. Necessarily, for all $m<n_\ell$ and $\beta\in
\alpha_\ell\setminus Z$ we have $h^q(i_0,\beta),h^q(j_m,\beta)\geq\theta$; remember that $Z$ is $p$–closed.)
Note that if $Z\subseteq{{\rm ht}}(p)$ is a finite $p$–closed set, $\alpha=\max(Z)$ and $p^*$ is the $\alpha+1$–component of $p$ satisfying $p^*\leq_{\rm pr} p$ (see \[3.4x\](3)), then $U[p,Z]\subseteq u^{p^*}$.
\[3.6.x\] Suppose that $p\in{{{\mathbb P}^\theta_\lambda}}$ and $Z_0,Z_1\subseteq{{\rm ht}}(p)$ are finite $p$–closed sets such that $\Upsilon_p(Z_0)=\Upsilon_p(Z_1)$. Then ${{\rm otp}}(U[p,Z_0])=
| 0 | non_member_985 |
{{\rm otp}}(U[p,Z_1])$, and the order preserving isomorphism $\pi:U[p,Z_0]\longrightarrow U[p,Z_1]$ satisfies
1. $(\forall\ell<k)(h^p(i,\alpha^0_\ell)=h^p(\pi(i),
\alpha^1_\ell))$,
where $\{\alpha^x_0,\ldots,\alpha^x_{k-1}\}$ is the increasing enumeration of $Z_x$ (for $x=0,1$).
We prove this by induction on $|Z_0|=|Z_1|$ (for all $p,Z_0,Z_1$ satisfying the assumptions).
[Step]{} $|Z_0|=|Z_1|=1$; $Z_0=\{\alpha^0_0\}$, $Z_1=\{\alpha^1_0\}$.\
Take the $\alpha^x_0+1$–component $q_x$ of $p$ such that $q_x\leq_{\rm pr}
p$. Then, for $x=0,1$, $q_x=\langle\zeta,\tau,n, u^x,\langle q^x_\xi,
v^x_\xi:\xi<\theta\rangle\rangle$, and for each $i\in v^x_\xi$, $\beta<
\alpha^x_0$ we have $h^{q^x_\xi}(i,\beta)\geq\theta$. Also, if $i^x_0=
\min(u^{q^x_\zeta}\setminus u^x)$ and $\beta<\alpha^x_0$, then $h^{q^x_\zeta}(i^x_0,\beta)\geq\theta$. Consequently, $n=|v^x_\xi|\leq 1$, and if $n=1$ then $\{i^x_0\}=v^x_\zeta$ (remember Lemma \[3.1\](4)). Moreover, $$U[p,Z_x]=U[q_x,Z_x]=\{H^x_{\xi,\zeta}(i^x_0):\xi<\theta\},$$ where $H^x_{\xi,\zeta}$ is the isomorphism from $q^x_\zeta$ to $q^x_\xi$. Now it should be clear that the mapping $\pi:H^0_{\xi,\zeta}(
i^0_0)\mapsto H^1_{\xi,\zeta}(i^1_0):U[p,Z_0]\longrightarrow U[p,Z_1]$ is the order preserving isomorphism (remember clause $(\gamma)$ of the definition of $P^*_{\alpha+1}$), and it has the property described in $(\otimes)$.
[Step]{} $|Z_0|=|Z_1|=k+1$; $Z_0=\{\alpha^0_0,\ldots,\alpha^0_k
\}$, $Z_1=\{\alpha^1_0,\ldots,\alpha^1_k\}$.\
Let $$\Upsilon_p(Z_0)=\Upsilon_p(Z_1)=\langle\zeta_\ell,\tau_\ell,n_\ell,\langle
g_\ell,h^\ell_0,\ldots,h^\ell_{n_\ell-1}\rangle:\ell\leq k\rangle.$$ For $x=0,1$, let $q_x=\langle\zeta,\tau,n, u^x,\langle q^x_\xi,v^x_\xi:\xi<
\theta\rangle\rangle$ be the $\alpha^x_k+1$–component of $p$ such that $q_x
\leq_{\rm pr} p$. The sets $Z_x\cap\alpha^x_k$ (for $x=0,1$) are $q^x_\xi$–closed for every $\xi<\theta$, and clearly $\Upsilon_p(Z_0\cap
\alpha^0_k)= \Upsilon_p(Z_1\cap\alpha^1_k)$. Hence, by the inductive hypothesis, ${{\rm otp}}(U[q^0_\xi,Z_0\setminus\{\alpha^0_k\}])={{\rm otp}}(U[q^1_\xi,Z_1
\setminus\{\alpha^1_k\}])$ (for each $\xi<\theta$), and the order preserving mappings $\pi_\xi:U[q^0_\xi,Z_0\setminus\{ \alpha^0_k\}]\longrightarrow
U[q^1_\xi, Z_1\setminus\{\alpha^1_k\}]$ satisfy the demand in $(\otimes)$. | 0 | non_member_985 |
Let $i^x_\xi=\min(u^{q^x_\xi}\setminus u^x)$. Then, as $q^x_\xi$ and $q^x_\zeta$ are isomorphic and the isomorphism is the identity on $u^x$, we have $(\forall\ell<k)(h^p(i^x_\xi,\alpha_\ell^x)=g_k(\ell))$. Hence $\pi_\xi
(i^0_\xi)=i^1_\xi$, and therefore $\pi_\xi[u^0\cap U[q^0_\xi,Z_0\setminus
\{\alpha^0_k\}]]=u^1\cap U[q^1_\xi,Z_1\setminus\{\alpha^1_k\}]$. But since the mappings $\pi_\xi$ are order preserving, the last equality implies that $\pi_\xi\restriction (u^0\cap U[q^0_\xi,Z_0\setminus\{\alpha^0_k\}])=
\pi_\zeta\restriction (u^0\cap U[q^0_\zeta,Z_0\setminus\{\alpha^0_k\}])$, and hence $\pi=\bigcup\limits_{\xi<\theta}\pi_\xi$ is a function, and it is an order isomorphism from $U[q_0,Z_0]=U[p,Z_0]$ onto $U[q_1,Z_1]=U[p,Z_1]$ satisfying $(\otimes)$.
The algebra and why it is OK (in ${{\bf V}}^{{{{\mathbb P}^\theta_\lambda}}}$)
==============================================================================
Let ${\dot{{{\mathbb B}}}^\theta_\lambda}$ and $\dot{U}$ be ${{{\mathbb P}^\theta_\lambda}}$–names such that $${\Vdash}_{{{{\mathbb P}^\theta_\lambda}}}\mbox{`` }{\dot{{{\mathbb B}}}^\theta_\lambda}=\bigcup\{{{\mathbb B}}^p:p\in \Gamma_{{{{\mathbb P}^\theta_\lambda}}}\}\mbox{
''}\quad\mbox{ and }\quad{\Vdash}_{{{{\mathbb P}^\theta_\lambda}}}\mbox{`` }\dot{U}=\bigcup\{u^p:p\in
\Gamma_{{{{\mathbb P}^\theta_\lambda}}}\}\mbox{ ''.}$$ Note that $\dot{U}$ is (a name for) a subset of $\lambda^+$. Let $\dot{F}$ be a ${{{\mathbb P}^\theta_\lambda}}$–name such that $${\Vdash}_{{{{\mathbb P}^\theta_\lambda}}}\mbox{`` }\dot{F}=\{f\in 2^{\textstyle\dot{U}}: (\forall
p\in\Gamma_{{{{\mathbb P}^\theta_\lambda}}})(f\restriction u^p\in\dot{F}^p)\}\mbox{ ''.}$$
\[3.4\] Assume $\theta<\lambda$ are regular, $\lambda^{<\lambda}=\lambda$. Then in ${{\bf V}}^{{{{\mathbb P}^\theta_\lambda}}}$:
1. $\dot{F}$ is a non-empty closed subset of $2^{\textstyle\dot{U}}$, and ${\dot{{{\mathbb B}}}^\theta_\lambda}$ is the Boolean algebra generated ${{\mathbb B}}_{(\dot{U},\dot{F})}$ (see Definition \[0.C\]);
2. $|\dot{U}|=|{\dot{{{\mathbb B}}}^\theta_\lambda}|=\lambda^+$;
3. For every subalgebra ${{\mathbb B}}\subseteq{\dot{{{\mathbb B}}}^\theta_\lambda}$ of size $\lambda^+$ we have ${{\rm Depth}}^+({{\mathbb B}})>\theta$.
2)Note that if $p\in{{{\mathbb P}^\theta_\lambda}}$, $\sup(u^p)<j<\lambda^+$ then there is a condition $q\geq p$ such that $j\in u^q$. | 0 | non_member_985 |
Hence ${\Vdash}|\dot{U}|=
\lambda^+$. To show that, in ${{\bf V}}^{{{{\mathbb P}^\theta_\lambda}}}$, the algebra ${\dot{{{\mathbb B}}}^\theta_\lambda}$ is of size $\lambda^+$ it is enough to prove the following claim.
\[3.4.1\] Let $p\in{{{\mathbb P}^\theta_\lambda}}$, $j\in u^p$. Then $x_j\notin\langle x_i:i\in j\cap u^p
\rangle_{{{\mathbb B}}^p}$.
Suppose not, and let $p,j$ be a counterexample with the smallest possible ${{\rm ht}}(p)$. Necessarily, ${{\rm ht}}(p)$ is a successor ordinal, say ${{\rm ht}}(p)=\alpha+1$. So let $p=\langle\zeta^*,\tau^*, n^*,u^*,\langle
p_\xi,v_\xi:\xi<\theta\rangle\rangle$ and suppose that $v\in [u^p\cap
j]^{\textstyle <\omega}$ is such that $x_j\in \langle x_i: i\in
v\rangle_{{{\mathbb B}}^p}$. If $j\in u^*$ then $v\subseteq u^*$ and we immediately get a contradiction (applying the inductive hypothesis to $p_{\zeta^*}$). So let $\xi<\theta$ be such that $j\in u^{p_\xi}\setminus u^*$. We know that $x_j\notin\langle x_i:i\in u^*\cup (v\cap u^{p_\xi})\rangle_{{{\mathbb B}}^{p_\xi}}$ (remember clause $(\gamma)$ of the definition of $P^*_{\alpha+1}$), so we may take functions $f_0,f_1\in F^{p_\xi}$ such that $f_0{\restriction}(u^*\cup (v\cap
u^{p_\xi}))=f_1{\restriction}(u^*\cup (v\cap u^{p_\xi}))$, $f_0(j)=0$, $f_1(j)=1$. Let $g_0,g_1:u^p\longrightarrow 2$ be such that $g_\ell{\restriction}u^{p_\xi}=
f_\ell$, $g_\ell{\restriction}u^{p_\zeta}=f_0{\circ}H_{\zeta,\xi}$ for $\zeta\neq\xi$ (where $H_{\zeta,\xi}$ is the order isomorphism from $u^{p_\zeta}$ to $u^{p_\xi}$). Now one easily checks that $g_0,g_1\in F^p$ (remember the definition of the term $\sigma_{\rm maj}$). By our choices, $g_0(i)=g_1(i)$ for all $i\in v$, and $g_0(j)\neq g_1(j)$, and this is a clear contradiction with the choice | 0 | non_member_985 |
of $i$ and $v$.
3)Suppose that $\langle\dot{a}_\xi:\xi<\lambda^+\rangle$ is a ${{{\mathbb P}^\theta_\lambda}}$–name for a $\lambda^+$–sequence of distinct members of ${\dot{{{\mathbb B}}}^\theta_\lambda}$ and let $p\in{{{\mathbb P}^\theta_\lambda}}$. Applying standard cleaning procedures we find a set $A\subseteq\lambda^+$ of the order type $\theta$, an ordinal $\alpha<
\lambda$ and $\tau^*,n^*,u^*$ and $\langle p_\xi,v_\xi:\xi\in A\rangle$ such that $p\leq p_\xi$, ${{\rm ht}}(p_\xi)=\alpha$, $p_\xi{\Vdash}\dot{a}_\xi=
\tau^*(x_i:i\in v_\xi)$ and $$q\stackrel{\rm def}{=}\langle 0,\tau^*,n^*,u^*,\langle p_\xi,v_\xi:\xi\in
A\rangle\rangle\in P^*_{\alpha+1},$$ where $A$ is identified with $\theta$ by the increasing enumeration (so we will think $A=\theta$). For $\xi<\theta$ let $\tau_\xi=\tau^*(x_i: i\in
v_\xi)\in{{\mathbb B}}^{p_\xi}$. Since $\dot{a}_\xi$ were (forced to be) distinct we know that ${{\mathbb B}}^q\models\tau_\xi\neq\tau_\zeta$ for distinct $\xi,\zeta$. Hence $\tau_\xi\notin\langle x_i:i\in u^*\rangle_{{{\mathbb B}}^{p_\xi}}$ (for each $\xi$) and therefore we may find functions $f^0_\xi,f^1_\xi\in F^{p_\xi}$ such that $f^0_\xi{\restriction}u^*=f^1_\xi{\restriction}u^*$, and $f^0_\xi(\tau_\xi)=0$, $f^1_\xi(\tau_\xi)=1$, and if $\xi<\zeta<\theta$, and $H_{\xi,\zeta}$ is the isomorphism from $p_\xi$ to $p_\zeta$, then $f^\ell_\xi=f^\ell_\zeta{\circ}H_{\xi,\zeta}$. Now fix $\xi<\zeta<\theta$ and let $$g\stackrel{\rm def}{=}\bigcup_{\alpha\leq 3\cdot\xi+2} f^0_\alpha\cup
\bigcup_{3\cdot\xi+2<\alpha<\theta} f^1_\alpha.$$ It should be clear that $g$ is a function from $u^q$ to $2$, and moreover $g\in F^q$. Also easily $$g(\sigma_{\rm maj}(\tau_{3\cdot\xi},\tau_{3\cdot\xi+1},\tau_{3\cdot\xi+
2}))=0\ \mbox{ and }\ g(\sigma_{\rm maj}(\tau_{3\cdot\zeta},\tau_{3\cdot
\zeta+1},\tau_{3\cdot\zeta+2}))\}=1.$$ Hence we may conclude that $${{\mathbb B}}^q\models\sigma_{\rm maj}(\tau_{3\cdot\xi},\tau_{3\cdot\xi+1},\tau_{3
\cdot\xi+2})<\sigma_{\rm maj}(\tau_{3\cdot\zeta},\tau_{3\cdot\zeta+1},\tau_{3
\cdot\zeta+2})$$ for $\xi<\zeta<\theta$ (remember the definition of $F^q$ and Proposition \[0.D\]). Consequently we get | 0 | non_member_985 |
$q{\Vdash}{{\rm Depth}}^+(\langle\dot{a}_\xi:\xi<
\lambda^+\rangle_{{\dot{{{\mathbb B}}}^\theta_\lambda}})>\theta$, finishing the proof.
\[3.5\] Assume $\theta<\lambda$ are regular, $\lambda=\lambda^{<\lambda}$. Then ${\Vdash}_{{{{\mathbb P}^\theta_\lambda}}}{{\rm Depth}}({\dot{{{\mathbb B}}}^\theta_\lambda})=\theta$.
By Proposition \[3.4\] we know that ${\Vdash}{{\rm Depth}}^+({\dot{{{\mathbb B}}}^\theta_\lambda})>\theta$, so what we have to show is that there are no increasing sequences of length $\theta^+$ of elements of ${\dot{{{\mathbb B}}}^\theta_\lambda}$. We will show this under an additional assumption that $\theta^+ <\lambda$ (after the proof is carried out, it will be clear how one modifies it to deal with the case $\lambda=\theta^+$). Due to this additional assumption, and since the forcing notion ${{{\mathbb P}^\theta_\lambda}}$ is $(<\lambda)$–strategically closed (by Proposition \[3.3\]), it is enough to show that ${{\rm Depth}}({{\mathbb B}}^p)\leq\theta$ for each $p\in{{{\mathbb P}^\theta_\lambda}}$.
So suppose that $p\in{{{\mathbb P}^\theta_\lambda}}$ is such that ${{\rm Depth}}({{\mathbb B}}^p)\geq\theta^+$. Then we find a Boolean term $\tau$, an integer $n$ and sets $w_\rho\in [u^p]^{
\textstyle n}$ (for $\rho<\theta^+$) such that $$\rho_0<\rho_1<\theta^+\quad\Rightarrow\quad{{\mathbb B}}^p\models\tau(x_i:i\in w_{
\rho_0})<\tau(x_i:i\in w_{\rho_1}).$$ For each $\rho<\theta^+$ use Lemma \[3.5.1\] to choose a finite $p$–closed set $Z_\rho\subseteq{{\rm ht}}(p)$ containing the set $$\{\beta<{{\rm ht}}(p): (\exists j\in w_\rho)(h^p(j,\beta)<\theta)\}.$$ Look at $\Upsilon_p(Z_\rho)$ (see Definition \[UUpsilon\]). There are only $\theta$ possibilities for the values of $\Upsilon_p(Z_\rho)$, so we find $\rho_0<\rho_1<\theta^+$ such that
1. $|Z_{\rho_0}|=|Z_{\rho_1}|$, $\Upsilon_p(Z_{\rho_0})=\Upsilon_p(
Z_{\rho_1})=\langle\zeta_\ell,\tau_\ell,n_\ell,\langle g_\ell,h^\ell_0,
\ldots,h^\ell_{n_\ell-1}\rangle:\ell<k\rangle$,
2. if $\pi^*:Z_{\rho_0}\longrightarrow Z_{\rho_1}$ | 0 | non_member_985 |
---
abstract: 'We consider an infinite chain of coupled harmonic oscillators with a Langevin thermostat attached at the origin and energy, momentum and volume conserving noise that models the collisions between atoms. The noise is rarefied in the limit, [that corresponds to the hypothesis]{} that in the macroscopic unit time only a finite number of collisions takes place (Boltzmann-Grad limit). We prove that, after the hyperbolic space-time rescaling, the Wigner distribution, describing the energy density of phonons in space-frequency domain, converges to a positive energy density function $W(t, y, k)$ that evolves according to a linear kinetic equation, with the interface condition at $y=0$ that corresponds to reflection, transmission and absorption of phonons. The paper extends the results of [@kors], where a thermostatted harmonic chain (with no inter-particle scattering) has been considered.'
author:
- 'Tomasz Komorowski[^1]'
- 'Stefano Olla[^2]'
title: Kinetic limit for a chain of harmonic oscillators with a point Langevin thermostat
---
Introduction {#intro}
============
The mathematical analysis of macroscopic energy transport in anharmonic chain of oscillators [constitutes a very hard mathematical problem]{}, see [@spohn2006]. One approach to it is to replace the non-linearity by a stochastic exchange of momentum between nearest neighbor particles [in such a way | 0 | non_member_986 |
that the total kinetic energy and momentum are conserved]{}. This stochastic exchange can be modeled in various ways: e.g. for each couple of nearest neighbor particles the exchange of their momenta can occur independently at an exponential time (which models their elastic collision). Otherwise, for each triple of consecutive particles, exchange of momenta can be performed in a continuous, diffusive fashion, so that its energy and momentum are preserved. In the present article we adopt the latter choice, see Section \[sec2.2.1\] for a detailed description of the dynamics, but, with no significant changes, all our results can be extended to other stochastic noises. A small parameter ${\epsilon}>0$ is introduced to rescale space and time, and the intensity of the noise is adjusted so that in a (macroscopic) finite interval of time, there is only a finite amount of momentum exchanged by the stochastic mechanism. In terms of the random exchanges, it means that, on average, each particle undergoes only a finite number of stochastic collisions in a finite time. Letting ${\epsilon}\to 0$ corresponds therefore to taking the kinetic limit for the system.
The Wigner distribution is a useful tool to localize in space the energy per frequency mode. In the | 0 | non_member_986 |
absence of the thermostat, it is proven in [@BOS], that as ${\epsilon}\to 0$, the Wigner distribution converges to the solution of the kinetic transport equation $$\label{eq:bos}
\partial_t W(t,y,k) + \bar\omega'(k) \partial_y W(t,y,k) =
2\gamma_0 \int_{{{\mathbb T}}} R(k,k') \left(W(t,y,k') - W(t,y,k)\right) dk, \qquad (t,y,k)\in [0,+\infty)\times{{\mathbb T}}\times \mathbb R,$$ with an explicitly given scattering kernel $ R(k,k') \ge 0$. It is symmetric and the total scattering kernel behaves as $$\label{Rk}
R(k):=\mathlarger{\int}_{{{\mathbb T}}}R(k,k')dk' \sim |k|^2\quad\mbox{ for }|k| \ll 1.$$ Here ${{\mathbb T}}$ is the unit torus, which is the interval $[-1/2,1/2]$, with identified endpoints. Furthermore $\gamma_0>0$ is the scattering rate for the microscopic chain (see below) and $\bar\omega(k) = \omega(k)/2\pi$, where $\omega(k)$ is the dispersion relation of the chain (see definition ).
In the present paper we are interested in the macroscopic effects of a heat bath at temperature $T$, modeled by a Langevin dynamics, applied to one particle, say the one labelled $0$, with a coupling strength $\gamma_1>0$ (see below for a detailed description). Unlike the conservative stochastic dynamics acting on the bulk, the action of the heat bath is not rescaled with ${\epsilon}$, so in the limit as ${\epsilon}\to 0$ it constitutes a singular perturbation on the dynamics. The effect in | 0 | non_member_986 |
the limit is to introduce the following interface conditions at $y=0$ on : $$\label{bc-int}
\begin{split}
W(t,0^+,k) &=p_-(k) W(t,0^+, -k) + p_+(k)W(t, 0^-,k)+{{\fgeeszett}}(k)T, \quad\hbox{ for $0< k\le 1/2$},\\
W(t,0^-, k) &=p_-(k)W(t,0^-,-k) + p_+(k) W(t,0^+, k) +{{\fgeeszett}}(k)T,\quad \hbox{ for $-1/2< k< 0$.}
\end{split}$$ Interpreting $W(t,y,k)$ as the density of the energy of the phonons of mode $k$ at time $t$ and position $y$, then $ p_+(k)$, $ p_-(k)$ and ${{\fgeeszett}}(k)$ are respectively the probabilities for transmission, reflection and absorption of a phonon of mode $k$ when it crosses $y=0$, while ${{\fgeeszett}}(k)T$ is the rate of creation of a phonon of that mode. These probabilities are functions of $k$ and depend only on the dispersion relation $\omega(\cdot)$ and the intensity $\gamma_0$ of the thermostat (cf. ). They are properly normalized, i.e. $ p_+(k) + p_-(k) + {{\fgeeszett}}(k) = 1$, so that $W(t,y,k) = T$ is a stationary solution (thermal equilibrium).
This result was recently proven in the absence of the conservative noise in the bulk (i.e. $\gamma_0= 0$ in ), see [@kors]. Then, the resulting dynamics outside the interface, given by , reduces itself to pure transport as ${\gamma}_0=0$. Obviously, the coefficients appearing in the interface conditions do not depend on the presence | 0 | non_member_986 |
of the bulk noise.
The goal of the present paper is to extend the result of [@kors] to the case when the inter-particle noise is present, i.e. ${\gamma}_0>0$, see Theorem \[main-thm\] below for the precise formulation of our main result. We emphasize that in the situation when $\gamma_0=0$, both the equations for the microscopic and macroscopic dynamics, given below by and respectively, can be solved explicitly, in terms of the initial condition, and this fact has been extensively used in the proof in [@kors]. The argument can be extended to the dynamics where only the damping terms of the noise are present, i.e. with no noise input both from the inter-particle scattering and the thermostat, see . The equation for the macroscopic limit of the respective Wigner distribution $W^{\rm un}(t,y,k)$ reads (cf ), see Theorem \[main:thm-un\] below, $$\label{eq:damp}
\partial_t W^{\rm un}(t,y,k) + \bar\omega'(k) \partial_y W^{\rm un}(t,y,k) =
-2 \gamma_0 R(k) W^{\rm un}(t,y,k), \quad y\not=0,$$ with the boundary conditions as in . In the next step we add the stochastic part corresponding to the inter-particle scattering, which corresponds to $T=0$ for the thermostat, see equation formulated for the respective wave function. Next, we use the previously described dynamics to represent the | 0 | non_member_986 |
solution of the equation with the help of the Duhamel formula, see . The corresponding representation for the Wigner distribution is given in . Having already established the macroscopic limit for the dynamics with no stochastic noise, we can use the Duhamel representation to identify the kinetic limit of the noisy microscopic dynamics when the thermostat temperature $T=0$, see Theorem \[cor020805-19\]. The extension to the case when the temperature $T>0$ is possible by another application of the Duhamel formula, see Section \[sec10\].
Concerning the organization of the paper, Section \[prelim\] is devoted to preliminaries and the formulation of the main result, see Theorem \[main-thm\]. Among things discussed is the rigorous definition of a solution of a kinetic equation with the interface condition , see Sections \[sec2.6.3\] and \[sec2.6.4\]. Section \[sec3\] deals with the basic properties of the microscopic dynamics obtained by removal of stochastic noises, both between the particles of the chain and the thermostat. This dynamics is an auxiliary tool for the mild formulation of the microscopic dynamics corresponding to the chain with inter-particle scattering and thermostat. We discuss first the case when thermostat is set at $T=0$, see Section \[sec4\]. In this section we obtain also basic estimates | 0 | non_member_986 |
for the microscopic Wigner distributions, see Proposition \[prop012105-19\], that follow from the energy balance equation established in . A similar result is also formulated for the auxiliary dynamics with no stochastic noise in Section \[sec5.1\]. In Section \[sec5.3\] we formulate the result concerning the kinetic limit for this dynamics, see Theorem \[main:thm-un\]. Its proof is quite analogous to the argument of [@kors] and is given in Appendix \[appb\]. Section \[sec5.5\] is essentially devoted to the proof of the main result (Theorem \[main-thm\]) for the case $T=0$ and the proof for $T>0$ is presented in Section \[sec10\]. Some properties of the dynamics corresponding to the macroscopic kinetic limit are proven in Appendix \[appa\]. Section \[appC\] of the appendix is devoted to the proof of some properties of the interface coefficients appearing in the limit.
Acknowledgements {#acknowledgements .unnumbered}
================
T.K. acknowledges the support of the National Science Centre: NCN grant DEC-2016/23/B/ST1/00492. S.O. acknowledges the ANR-15-CE40-0020-01 LSD grant of the French National Research Agency.
Preliminaries and statement of the main results {#prelim}
===============================================
Basic notation
--------------
We shall use the following notation: let ${{\mathbb R}}_*:={{\mathbb R}}\setminus\{0\}$, ${{\mathbb R}}_+:=(0,+\infty)$, ${{\mathbb R}}_-:=(-\infty,0)$ and likewise ${{\mathbb T}}_*:={{\mathbb T}}\setminus\{0\}$, ${{\mathbb T}}_+:=(0,1/2)$, ${{\mathbb T}}_-:=(-1/2,0)$. Throughout the paper we | 0 | non_member_986 |
use the short hand notation $$\label{011909}
{\frak s}(k):=\sin(\pi k)\quad {\frak c}(k):=\cos(\pi k),\quad k\in{{\mathbb T}}.$$ Let $
e_x(k):=\exp\{2\pi i xk\}$ for $x$ belonging to the set of integers ${{\mathbb Z}}.$ The Fourier series corresponding to a complex valued sequence $(f_x)_{x\in{{\mathbb Z}}}$ belonging to $\ell_2$ - the Hilbert space of square integrable sequences of complex numbers - is given by $$\label{fourier}
\hat f(k)=\sum_{x\in{{\mathbb Z}}}f_xe_x^\star(k), \quad k\in{{\mathbb T}}.$$ Here $z^\star$ is the complex conjugate of $z\in\mathbb C$. By the Parseval identity $\hat f\in L^2({{\mathbb T}})$ - the space of complex valued, square integrable functions - and $\|\hat f\|_{L^2({{\mathbb T}})}=\|f\|_{\ell_2}$. Given ${\epsilon}>0$ we let ${{\mathbb Z}}_{{\epsilon}}:=({\epsilon}/2){{\mathbb Z}}$ and ${{\mathbb T}}_{\epsilon}:=(2/{\epsilon}){{\mathbb T}}$. Let $\ell_{2,{\epsilon}}$ be the space made of all complex valued square integrable sequences $(f_y)_{y\in
{{\mathbb Z}}_{\epsilon}}$ equipped with the norm $$\|f\|_{\ell_{2,{\epsilon}}}:=\left\{\frac{{\epsilon}}{2}\sum_{y\in{{\mathbb Z}}_{\epsilon}}|f_y|^2\right\}^{1/2}.$$ Let $$\label{fouriereps}
\hat f(\eta)=\frac{{\epsilon}}{2}\sum_{y\in{{\mathbb Z}}_{\epsilon}}f_ye_y^\star(\eta), \quad \eta\in{{\mathbb T}}_{\epsilon}.$$ The Parseval identity takes then the form $\|\hat f\|_{L^2({{\mathbb T}}_{\epsilon})}=\|f\|_{\ell_{2,{\epsilon}}}$.
For any non-negative functions $f,g$ acting on a set $A$ the notation $f\preceq g$ means that there exists a constant $C>0$ such that $f(a)\le Cg(a)$ for $a\in A$. We shall write $f\approx g$ if $f\preceq g$ and $g\preceq f$.
Given a function $f:\bar{{\mathbb R}}_+\to\mathbb C$ satisfying $|f(t)|\le
Ce^{Mt}$, for fome $C,M>0$ we denote | 0 | non_member_986 |
by $\tilde
f({\lambda})$ its Laplace transform $$\tilde
f({\lambda})=\int_0^{+\infty}e^{-{\lambda}t}f(t)dt,\quad {\rm Re}\,{\lambda}>M.$$
Some function spaces
--------------------
For a given $ G\in {\cal S}({{\mathbb R}}\times{{\mathbb T}})$ - the class of Schwartz functions on ${{\mathbb R}}\times{{\mathbb T}}$ - we let $$\widehat G(\eta,k)=\int_{{{\mathbb R}}}e_y^\star(\eta)G(y,k)dy$$ be its Fourier transform in the first variable. Let $$\label{AC}
{\cal
A}_c:=[G:\,\hat G\in C_c^\infty({{\mathbb R}}\times{{\mathbb T}})].$$ Let ${\cal A}$ be the Banach space obtained by the completion of ${\cal A}_c$ in the norm $$\label{060805-19}
\|G\|_{\cal A}:=\int_{{{\mathbb R}}}\sup_{k\in{{\mathbb T}}}|\widehat G(\eta,k)|d\eta,\quad G\in{\cal
A}_c.$$ Space ${\cal A}'$ - the dual to ${\cal A}$ - consists of all distributions $G\in {\cal
S}'({{\mathbb R}}\times{{\mathbb T}})$ of the form $$\langle G, F\rangle=\int_{{{\mathbb R}}\times {{\mathbb T}}} \widehat G^\star(\eta,k)
\widehat F(\eta,k)d\eta dk,\quad F\in {\cal A}$$ for some measurable function $\widehat G:{{\mathbb R}}\times{{\mathbb T}}\to\mathbb C$, equipped with the norm $$\label{060805-19a}
\|G\|_{{\cal A}'}=\sup_{\eta\in{{\mathbb R}}}\int_{{{\mathbb T}}}|\widehat G(\eta,k)|d k<+\infty.$$
We shall also consider the spaces ${\cal
L}_{2,{\epsilon}}:=\ell_{2,{\epsilon}}\otimes L^2({{\mathbb T}})$. The respective norms of $G:{{\mathbb Z}}_{\epsilon}\times{{\mathbb T}}\to\mathbb C$ and $\widehat
G:{{\mathbb T}}_{\epsilon}\times{{\mathbb T}}\to\mathbb C$ are given by $$\label{011505-19}
\|G\|_{{\cal L}_{2,{\epsilon}}}:=\left\{\frac{{\epsilon}}{2}\sum_{y\in{{\mathbb Z}}_{\epsilon}}\|G_y\|_{L^2({{\mathbb T}})}^2\right\}^{1/2}
=\|\widehat
G\|_{L^2({{\mathbb T}}_{\epsilon}\times{{\mathbb T}})}.$$
Infinite system of interacting harmonic oscillators
---------------------------------------------------
### Thermostatted Hamiltonian dynamics with momentum and energy conserving noise {#sec2.2.1}
We consider a stochastically perturbed chain of harmonic oscillators thermostatted | 0 | non_member_986 |
at a fixed temperature $T\ge 0$ at $x=0$. Its dynamics is described by the system of Itô stochastic differential equations $$\begin{aligned}
&&d{\frak q}_{x}(t)={\frak p}_x(t)dt \nonumber
\\
&& d{\frak p}_x(t)=\left[-(\alpha\star{\frak
q}(t))_x-\frac{{\epsilon}{\gamma}_0}{2}(\theta\star{\frak
p}(t))_x\right]dt+\sqrt{{\epsilon}{\gamma}_0}\sum_{k=-1,0,1}(Y_{x+k}{\frak
p}_x(t))dw_{x+k}(t)\label{eq:bas1} \\
&&
+\left(-{\gamma}_1{\frak p}_0(t)dt+\sqrt{2{\gamma}_1 T}dw(t)\right)\delta_{0,x},\quad x\in{{\mathbb Z}}.\nonumber\end{aligned}$$ Here $$\label{011210}
Y_x:=({\frak p}_x-{\frak p}_{x+1})\partial_{{\frak p}_{x-1}}+({\frak p}_{x+1}-{\frak p}_{x-1})\partial_{{\frak p}_{x}}+({\frak p}_{x-1}-{\frak p}_{x})\partial_{{\frak p}_{x+1}}$$ and $\left(w_x(t)\right)_{t\ge0}$, $x\in{{\mathbb Z}}$ with $\left(w(t)\right)_{t\ge0}$, are i.i.d. one dimensional, real valued, non-anticipative standard Brownian motions, over some filtered probability space $(\Sigma,{\cal F},\left({\cal F}_t\right)_{t\ge0},{{\mathbb P}})$. In addition, $$\theta_x=\Delta\theta^{(0)}_x:=\theta^{(0)}_{x+1}+\theta^{(0)}_{x-1}-2\theta^{(0)}_x$$ with $$\theta^{(0)}_x=\left\{
\begin{array}{rl}
-4,&x=0\\
-1,&x=\pm 1\\
0, &\mbox{ if otherwise.}
\end{array}
\right.$$ A simple calculation shows that $$\label{beta}
\hat \theta(k)=8{\frak s}^2(k)\left(1+2{\frak c}^2( k)\right),\quad k\in{{\mathbb T}}.$$ Parameters ${\epsilon}\gamma_0>0$, ${\gamma}_1$ describe the strength of the inter-particle and thermostat noises, respectively. In what follows we shall assume that ${\epsilon}>0$ is small, that corresponds to the low density hypothesis that results in atoms suffering finitely many ”collisions” in a macroscopic unit of time (Boltzmann-Grad limit). Although the noise considered here is continuous we believe that the results of the present paper extend to other type of noises, such as e.g. Poisson shots.
Since the vector field $Y_x$ is orthogonal both to a sphere ${\frak
p}_{x-1}^2+{\frak p}_x^2+{\frak p}_{x+1}^2\equiv {\rm const}$ and plane ${\frak p}_{x-1}+{\frak p}_x+{\frak p}_{x+1}\equiv | 0 | non_member_986 |
{\rm const}$, the inter-particle noise conserves locally the kinetic energy and momentum.
Concerning the Hamiltonian part of the dynamics, we assume (cf [@BOS]) that the coupling constants $({\alpha}_x)_{x\in{{\mathbb Z}}}$ satisfy the following:
- they are real valued and there exists $C>0$ such that $|\alpha_x|\le Ce^{-|x|/C}$ for all $x\in {{\mathbb Z}}$,
- $\hat\alpha(k)$ is also real valued and $\hat\alpha(k)>0$ for $k\not=0$ and in case $\hat \alpha(0)=0$ we have $\hat\alpha''(0)>0$.
The above conditions imply that both functions $x\mapsto\alpha_x$ and $k\mapsto\hat\alpha(k)$ are even. In addition, $\hat\alpha\in
C^{\infty}({{\mathbb T}})$ and in case $\hat\alpha(0)=0$ we have $\hat\alpha(k)=k^2\phi(k^2)$ for some strictly positive $\phi\in
C^{\infty}({{\mathbb T}})$. The dispersion relation ${\omega}:{{\mathbb T}}\to
\bar{{\mathbb R}}_+$, given by $$\label{mar2602}
{\omega}(k):=\sqrt{\hat \alpha (k)}$$ is even. Throughout the paper it is assumed to be unimodal, i.e. increasing on $[0,1/2]$ and then, in consequence, decreasing on $[-1/2,0]$. Its unique minimum and maximum, attained at $k=0$, $k=1/2$, respectively are denoted by ${\omega}_{\rm min}\ge 0$ and ${\omega}_{\rm max}$, correspondingly. We denote the two branches of its inverse by ${\omega}_\pm:[{\omega}_{\rm min},{\omega}_{\rm max}]\to\bar {{\mathbb T}}_\pm$.
### Initial data
We assume that the initial data is random and, given ${\epsilon}>0$, distributed according to probabilistic measure $\mu_{\epsilon}$ and $$\label{mu-eps}
{\cal E}_*:=\sup_{{\epsilon}\in(0,1]}{\epsilon}\sum_{x\in{{\mathbb Z}}}\langle{\frak e}_x\rangle_{\mu_{\epsilon}}<+\infty.$$ Here $\langle\cdot\rangle_{\mu_{\epsilon}}$ is the expectation with | 0 | non_member_986 |
respect to $\mu_{\epsilon}$ and the microscopic energy density $${\frak e}_x:=\frac12\left({\frak p}_x^2+\sum_{x'\in{{\mathbb Z}}}{\alpha}_{x-x'}{\frak q}_x{\frak q}_{x'}\right).$$ The assumption ensures that the macroscopic energy density of the chain is finite.
Kinetic scaling of the wave-function
------------------------------------
To observe macroscopic effects of the inter-particle scattering consider time of the order $t/{\epsilon}$. It is also convenient to introduce the wave function that, adjusted to the macroscopic time, is given by $$\label{011307}
\psi^{({\epsilon})}(t):=\tilde{{\omega}} \star {\frak q}\left(\frac{t}{{\epsilon}}\right)+i{\frak p}\left(\frac{t}{{\epsilon}}\right),$$ where $({\frak p}(t),{\frak q}(t))$ satisfies and $\left(\tilde {\omega}_x\right)_{x\in{{\mathbb Z}}}$ are the Fourier coefficients of the dispersion relation, see . We shall consider the Fourier transform of the wave function, given by $$\label{011307a}
\hat\psi^{({\epsilon})}(t,k)={\omega}(k)\hat {\frak q}^{({\epsilon})}\left(t,k\right)+i\hat{\frak p}^{({\epsilon})}\left(t,k\right).$$ Here $\hat {\frak q}^{({\epsilon})}\left(t\right)$, $\hat{\frak
p}^{({\epsilon})}\left(t\right)$ are given by the Fourier series for ${\frak q}^{({\epsilon})}_x(t):={\frak q}_x(t/{\epsilon})$ and ${\frak
p}^{({\epsilon})}_x(t):={\frak p}_x(t/{\epsilon})$, $x\in{{\mathbb Z}}$, respectively. They satisfy $$\begin{aligned}
\label{basic:sde:2a}
&&
d\hat\psi^{({\epsilon})}(t,k)=\left\{-\frac{i}{{\epsilon}} {\omega}(k)\hat\psi^{({\epsilon})}(t,k)
-2i{\gamma}_0R(k)\hat{\frak p}^{({\epsilon})}\left(t,k\right)-\frac{i{\gamma}_1}{{\epsilon}}\int_{{{\mathbb T}}}\hat{\frak p}^{({\epsilon})}\left(t,k'\right)dk'\right\}dt
\nonumber\\
&&
-2\sqrt{{\gamma}_0}\int_{{{\mathbb T}}}r(k,k')\hat{\frak p}^{({\epsilon})}\left(t,k-k'\right)B(dt,dk')+i\sqrt{\frac{2{\gamma}_1T}{{\epsilon}}}dw(t),\\
&&
\hat\psi^{({\epsilon})}(0)= \hat\psi,\nonumber
\end{aligned}$$ where $$\begin{aligned}
&
\hat{\frak p}^{({\epsilon})}\left(t,k\right):=\frac{1}{2i}\left[\hat\psi^{({\epsilon})}(t,k)-\left(\hat\psi^{({\epsilon})}\right)^\star(t,-k)\right],\\
&
r(k,k'):=4 {\frak s}(k) {\frak s}(k-k') {\frak s}(2k-k')
\quad k,k'\in {{\mathbb T}},\\
&
R(k):=\int_{{{\mathbb T}}}r^2(k,k')dk'={\frak s}^2(2k)+2 {\frak s}^2(k)=\frac{\hat\theta(k)}{4}.
\end{aligned}$$ Here $B(t,dk)=\sum_{x\in{{\mathbb Z}}}w_x(t)e_x(k)dk$ is a cylindrical Wiener process on $L^2({{\mathbb T}})$, i.e. $$\label{cnoise}
{\mathbb E}[B(dt,dk)B^\star(ds,dk')]=\delta(k-k')\delta(t-s)dt ds dkdk'.$$
Energy density - Wigner function
--------------------------------
One can easily | 0 | non_member_986 |
check that $$\label{e1}
\|\hat\psi^{({\epsilon})}(t)\|^2_{L^2({{\mathbb T}})}=\|\psi^{({\epsilon})}(t)\|^2_{\ell^2}=\sum_{x\in{{\mathbb Z}}}\left({\frak p}^{({\epsilon})}_x(t)\right)^2+\sum_{x,x'\in{{\mathbb Z}}}{\alpha}_{x-x'}{\frak q}^{({\epsilon})}_x(t){\frak q}^{({\epsilon})}_{x'}(t).$$ After straightforward calculations one can verify that $$\label{031709}
d\|\hat\psi^{({\epsilon})}(t)\|^2_{L^2({{\mathbb T}})}
=-\frac{2{\gamma}_1}{{\epsilon}} \left[
\left({\frak p}^{({\epsilon})}_0(t)\right)^2
- T \right]dt
+\left(\frac{2{\gamma}_1T}{{\epsilon}}\right)^{1/2}{\frak p}^{({\epsilon})}_0(t) dw(t),\quad {{\mathbb P}}_{\epsilon}\,\mbox{-a.s.}$$ Here ${{\mathbb P}}_{\epsilon}:={{\mathbb P}}\otimes\mu_{\epsilon}$. By ${\mathbb E}_{\epsilon}$ we denote the expectation with respect to ${{\mathbb P}}_{\epsilon}$. From assumptions and we obtain.
\[prop012409\] Under the kinetic scaling we have $$\label{052709-18}
{\cal E}_*(t):=\sup_{{\epsilon}\in(0,1]}\frac{{\epsilon}}{2}{\mathbb E}_{\epsilon}\|\hat\psi^{({\epsilon})}(t)\|^2_{L^2({{\mathbb T}})}\le
{\cal E}_*+{\gamma}_1 T t,\quad t\ge0.$$
We can introduce the (averaged) Wigner distribution $
W_{\epsilon}(t)\in {\cal A}'$, corresponding to $\psi^{({\epsilon})}(t)$, by the formula $$\label{wigner-bas}
\langle
W_{\epsilon}(t),G\rangle:=\frac{{\epsilon}}{2}\sum_{x,x'\in{{\mathbb Z}}}{\mathbb E}_{\epsilon}\left[\left(\psi^{({\epsilon})}_{x'}(t)\right)^\star
\psi^{({\epsilon})}_{x}(t)\right]e_{x'-x}(k)G\left({\epsilon}\frac{x+x'}{2},k\right),\quad
G\in {\cal A}.$$ Thanks to it is well defined for any $t\ge0$ and ${\epsilon}\in(0,1]$. Using the Fourier transform in the first variable we can rewrite the Wigner distribution as $$\label{wigner-bas1}
\langle
W_{\epsilon}(t),G\rangle=\int_{{{\mathbb T}}_{\epsilon}\times{{\mathbb T}}}\widehat
W_{\epsilon}^\star(t,\eta,k)\widehat G\left(\eta,k\right)d\eta dk,\quad
G\in {\cal A},$$ where $$\label{W+}
\widehat
W_{\epsilon}(t,\eta,k):=\frac{{\epsilon}}{2}{\mathbb E}_{\epsilon}\left[\left(\hat\psi^{({\epsilon})}\right)^\star\left(t,k-\frac{{\epsilon}\eta}{2}\right) \hat\psi^{({\epsilon})}\left(t,k+\frac{{\epsilon}\eta}{2}\right)\right].$$ We shall refer to $\widehat
W_{\epsilon}(t)$ as the Fourier-Wigner function corresponding to the given wave function. For the sake of future reference define also $Y_{\epsilon}(t)$, by its Fourier transform $$\label{Y+}
\widehat
Y_{\epsilon}(t,\eta,k):=\frac{{\epsilon}}{2}{\mathbb E}_{\epsilon}\left[\hat\psi^{({\epsilon})}\left(t,-k+\frac{{\epsilon}\eta}{2}\right) \hat\psi^{({\epsilon})}\left(t,k+\frac{{\epsilon}\eta}{2}\right)\right].$$
Kinetic equation
----------------
An important role in our analysis will be played by the function, see Section 2 of [@kors], $$\label{eq:bessel0}
J(t) = \int_{{{\mathbb T}}}\cos\left(\omega(k) t\right) dk.$$ | 0 | non_member_986 |
Its Laplace transform $$\label{eq:2}
\tilde J({\lambda}):=\int_0^{\infty}e^{-{\lambda}t}J(t)dt= \int_{{{\mathbb T}}}
\frac{\lambda}{\lambda^2 + \omega^2(k)} dk,\quad {\lambda}\in \mathbb C_+:=[z:\,{\rm Re}\,z>0].$$ One can easily see that ${\rm Re}\,\tilde J({\lambda})>0$ for ${\lambda}\in
\mathbb C_+$, therefore we can define the function $$\label{tg}
\tilde g(\lambda) := ( 1 + \gamma_1 \tilde J(\lambda))^{-1},\quad {\lambda}\in \mathbb C_+.$$ We have $$\label{012410}
|\tilde g(\lambda)|\le 1,\quad {\lambda}\in \mathbb C_+.$$ The function $\tilde g(\cdot)$ is analytic on $ \mathbb C_+$ so, by the Fatou theorem, see e.g. p. 107 of [@koosis], we know that $$\label{nu}
\nu(k) :=\lim_{{\epsilon}\to0+}\tilde g({\epsilon}-i{\omega}(k))$$ exists a.e. in ${{\mathbb T}}$ and in any $L^p({{\mathbb T}})$ for $p\in[1,\infty)$.
Let us introduce $$\label{033110}
\wp(k):=\frac{{\gamma}_1 \nu(k)}{2|\bar{\omega}'(k)|},\quad \fgeeszett(k):=\frac{{\gamma}_1|\nu(k)|^2}{|\bar{\omega}'(k)|},\quad p_+(k):=\left|1-\wp(k)\right|^2 ,\quad
p_-(k):=|\wp(k)|^2 ,$$ where $
\bar{\omega}'(k):={\omega}'(k)/(2\pi)$. We have shown in [@kors] that $$\label{feb1402}
{\rm Re}\,\nu(k)=\left(1+\frac{{\gamma}_1}{2|\bar{\omega}'(k)|}\right)|\nu(k)|^2.$$ The functions $p_\pm(\cdot)$ and $\fgeeszett(\cdot)$ are even. Thanks to we have $$\label{012304}
p_+(k)+p_-(k)+\fgeeszett(k)=1.$$
### Linear kinetic equation with an interface
Let $L$ be the operator given by $$\label{L}
LF(k):= 2\int_{{{\mathbb T}}}R(k,k')
\left[F\left(k'\right) - F\left(k\right)\right]dk',\quad k\in{{\mathbb T}},$$ [for $F\in L^1({{\mathbb T}})$]{} and $$\begin{aligned}
\label{R}
&
R(k,k'):=\frac12\left\{r^2\left(k,k-
k'\right)+r^2\left(k,k+k'\right)\right\}\\
&
=8{\frak s}^2(k){\frak s}^2(k')\left\{{\frak s}^2(k){\frak c}^2(k')+{\frak s}^2(k'){\frak c}^2(k)\right\},\quad k,k'\in{{\mathbb T}}.\nonumber\end{aligned}$$ Note that (cf ) the total scattering kernel equals $$\label{Rk}
R(k):=\int_{{{\mathbb T}}}R(k,k')dk'=\frac{\hat\theta(k)}{4}.$$
\[df012603-19\] Given $T\in{{\mathbb R}}$, let ${\cal C}_T$ be a subclass of $C_b({{\mathbb R}}_*\times{{\mathbb | 0 | non_member_986 |
T}}_*)$ that consists of continuous functions $F$ that can be continuously extended to $\bar{{\mathbb R}}_\pm\times{{\mathbb T}}_*$ and satisfy the interface conditions $$\label{feb1408}
F(0^+,k)=p_-(k)F(0^+, -k)+p_+(k)F(0^-,k)+{{\fgeeszett}}(k)T, \quad\hbox{ for $0< k\le 1/2$},\\$$ and $$\label{feb1410}
F(0^-, k)=p_-(k)F(0^-,-k) + p_+(k)F(0^+, k) +{{\fgeeszett}}(k)T,\quad \hbox{ for $-1/2< k< 0$.}$$
Note that $F\in{\cal C}_T$ if and only if $F-T'\in{\cal C}_{T-T'}$ for any $T,T'\in{{\mathbb R}}$, because of (\[012304\]).
Let us fix $T\ge0$. We consider the kinetic interface problem given by equation $$\label{eq:8}
\begin{aligned}
&\partial_tW(t,y,k) + \bar{\omega}'(k) \partial_y W(t,y,k) = {\gamma}_0 L_k W(t,y,k),
\quad (t,y,k)\in{{\mathbb R}}_+\times {{\mathbb R}}_*\times{{\mathbb T}}_*,
\\
&
W(0,y,k)=W_0(y,k),
\end{aligned}$$ with the interface condition $$\label{feb1408aa}
W(t)\in {\cal C}_T,\quad t\ge0.$$ Here $ L_k $ denotes the operator $L$ acting on the $k$ variable. We shall omit writing the subscript if there is no danger of confusion.
### Simplified case. Explicit solution {#sec2.6.2}
We consider first the situation when equation is replaced by $$\label{eq:8p}
\begin{aligned}
&\partial_tW^{\rm un}(t,y,k) + \bar{\omega}'(k) \partial_y W^{\rm un}(t,y,k) = -2{\gamma}_0R(k) W^{\rm un}(t,y,k),
\quad (t,y,k)\in{{\mathbb R}}_+\times {{\mathbb R}}_*\times{{\mathbb T}}_*,
\\
&
W^{\rm un}(0,y,k)=W_0(y,k),
\end{aligned}$$ with the interface conditions – , with $T=0$. It can be solved explicitly, using the method of characteristics, and we obtain $$\begin{aligned}
\label{010304}
&W^{\rm un}(t,y,k)
=
e^{-2{\gamma}_0R(k)t}\left\{\vphantom{\int_0^1}W_0\left(y-\bar{{\omega}}'(k)t,k\right)
1_{[0,\bar{{\omega}}'(k)t]^c}(y)
+p_+(k)W_0\left(y-\bar{{\omega}}'(k)t,k\right)1_{[0,\bar{{\omega}}'(k)t]}(y)
\right. \nonumber
\\
| 0 | non_member_986 |
&
+\left. \vphantom{\int_0^1}p_-(k) W_0\left(-y+\bar{\omega}'(k) t,-k\right)1_{[0,\bar {\omega}'(k)t]}(y)\right\}.\end{aligned}$$ Consider a semigroup of bounded operators on $L^\infty({{\mathbb R}}\times {{\mathbb T}}_*)$ defined by $$\label{010304s}
{\frak W}^{\rm un}_t(W_0)\left(y,k\right)
:=
W^{\rm un}(t,y,k),$$ with $W_0\in L^\infty({{\mathbb R}}\times
{{\mathbb T}}_*)$, $t\ge0$ and $(y,k)\in {{\mathbb R}}_*\times {{\mathbb T}}_*$. From formula one can conclude that $\left({\frak
W}^{\rm un}_t\right)_{t\ge0}$ forms a semigroup of contractions on both $L^1({{\mathbb R}}\times{{\mathbb T}})$ and $L^\infty({{\mathbb R}}\times{{\mathbb T}})$. Thus, by interpolation, formula defines a semigorup of contractions on any $L^p({{\mathbb R}}\times{{\mathbb T}})$, $1\le p\le+\infty$.
Note that if $W_0$ is continuous in ${{\mathbb R}}_*\times {{\mathbb T}}_*$, then $ W^{\rm un}(t,y,k)$ satisfies the interface conditions and , with $T=0$ for all $t>0$. Therefore, $( {\frak W}^{\rm un}_t)_{t\ge0}$ is a semigroup on ${\cal C}_0$ with $W^{\rm un}(t,y,k)$ (cf ) satisfying the first equation of , the interface condition - and the initial condition $$\lim_{t\to0+} W^{\rm un}(t,y,k)=W_0(y,k),\quad (y,k)\in{{\mathbb R}}_*\times{{\mathbb T}}_*.$$
### Kinetic equation - classical solution {#sec2.6.3}
\[df013001-19\] We say that a function $W:\bar{{\mathbb R}}_+\times {{\mathbb R}}\times {{\mathbb T}}_*\to {{\mathbb R}}$ is a classical solution to equation with the interface conditions , at $y=0$, if it is bounded and continuous on ${{\mathbb R}}_+\times{{\mathbb R}}_*\times {{\mathbb T}}_*$, and the following conditions hold:
- the restrictions of $W$ to ${{\mathbb R}}_+\times{{\mathbb R}}_\iota\times | 0 | non_member_986 |
{{\mathbb T}}_{\iota'}$, $\iota,\iota'\in\{-,+\}$, can be extended to bounded and continuous functions on the respective closures $\bar{{\mathbb R}}_+\times\bar{{\mathbb R}}_{\iota}\times \bar{{\mathbb T}}_{\iota'}$,
- for each $(t,y,k)\in {{\mathbb R}}_+\times{{\mathbb R}}_*\times {{\mathbb T}}_*$ fixed, the function $W(t+s,y+\bar{\omega}'(k) s,k)$ is of the $C^1$ class in the $s$-variable in a neighborhood of $s=0$, and the directional derivative $$\label{Dt}
D_tW(t,y,k)=\left(\partial_t+\bar{\omega}'(k)\partial_y\right)W(t,y,k):=\frac{d}{ds}_{|s=0} W(t+s,y+\bar{\omega}'(k) s,k)$$ is bounded in ${{\mathbb R}}_+\times{{\mathbb R}}_*\times {{\mathbb T}}_*$ and satisfies $$\label{eq:8a}
\begin{array}{ll}
D_tW(t,y,k) = {\gamma}_0 L_k W(t,y,k), &
\quad (t,y,k)\in{{\mathbb R}}_+\times {{\mathbb R}}_*\times{{\mathbb T}}_*,
\end{array}$$
- $W(t)$ satisfies , and $$\label{010102-19}
\lim_{t\to0+}W(t,y,k)=W_0(y,k),\quad (y,k)\in{{\mathbb R}}_*\times{{\mathbb T}}_*.$$
The following result has been shown in [@koran], see Proposition 2.2.
\[prop013001-19\] Suppose that $W_0\in{\cal C}_T$. Then, under the above hypotheses on the scattering kernel $R(k,k')$ and the dispersion relation ${\omega}(k)$, there exists a unique classical solution to equation with the interface conditions and in the sense of Definition $\ref{df013001-19}$.
### $L^2$ solution {#sec2.6.4}
We assume that $T=0$. Define ${\frak W}_t(W_0):=W(t)$. Thanks to Proposition \[prop013001-19\] the family $\left({\frak W}_t\right)_{t\ge0}$ forms a semigroup on the linear space ${\cal C}_0$. Furthermore, we let $$\label{cR}
{\cal R}F(k):= \int_{{{\mathbb T}}}R(k,k')
F\left(k'\right) dk',\quad k\in{{\mathbb T}},\quad F\in L^1({{\mathbb T}}).$$ Let $
{\cal C}_0':= {\cal C}_0\cap L^2({{\mathbb R}}\times {{\mathbb T}}).
$ The following result holds.
\[prop011406-19\] We | 0 | non_member_986 |
have ${\frak W}_t\left( {\cal C}_0'\right)\subset {\cal C}_0'$ for all $t\ge0$. The semigroup $\left({\frak W}_t\right)_{t\ge0}$ extends by the $L^2$ closure from ${\cal C}_0'$ to a $C_0$-continuous semigroup of contractions on $L^2({{\mathbb R}}\times{{\mathbb T}})$. Moreover, it is the unique solution in $L^2({{\mathbb R}}\times{{\mathbb T}})$ of the integral equation $$\label{integral}
{\frak W}_t={\frak W}_t^{\rm un}+2{\gamma}_0\int_0^t {\frak W}_{t-s}^{\rm
un}{\cal R} {\frak W}_sds,\quad t\ge0.$$
The proof of this result is contained in Appendix \[appa\]. We shall refer to the semigroup solution described in Proposition \[prop011406-19\] as the $L^2$-solution of quation with the interface conditions and for $T=0$. To extend the definition of such a solution to the case of an arbitrary $T\ge0$ we proceed as follows. Suppose that $W_0\in L^2({{\mathbb R}}\times{{\mathbb T}})$. Let $\chi\in C^\infty_c({{\mathbb R}})$ be an arbitrary real valued, even function that satisfies $$\label{011406-19}
\chi(y)=\left\{\begin{array}{ll}
1,&\mbox{ for }|y|\le 1/2, \\
0,&\mbox{ for }|y|\ge 1,\\
\mbox{belongs to }[0,1],& \mbox{ if otherwise.}
\end{array}\right.$$
\[df011406-19\] We say that $W(t,y,k)$ is the $L^2$-solution of quation with the interface conditions and for a given $T\ge0$ and an initial condition $W_0\in L^2({{\mathbb R}}\times{{\mathbb T}})$, if it is of the form $$\label{021406-19}
W(t,y,k):={\frak W}_t(\widetilde W_0)(y,k)+\int_0^t {\frak
W}_s(F)(y,k)ds+T\chi(y),\quad (t,y,k)\in \bar{{\mathbb R}}_+\times{{\mathbb R}}\times{{\mathbb T}}.$$ Here $$\label{eqF}
F(y,k):=-T\bar{\omega}'(k)\chi'(y),\quad \widetilde W_0(y,k):= W_0(y,k)-T\chi(y).$$
\[rm011406-19\] [*Note that | 0 | non_member_986 |
the definition of the solution does not depend on the choice of function $\chi$ satisfying .* ]{}
\[rm021406-19\] [*Suppose that $W_0\in {\cal C}_T$. Then, $W(t,y,k)$ given by is the classical solution of with the interface conditions and , in the sense of Definition $\ref{df013001-19}$.*]{}
Asymptotics of the Wigner functions - the statement of the main result
----------------------------------------------------------------------
Thanks to we conclude that $$\label{030406-19}
\sup_{{\epsilon}\in(0,1]}\| W_{{\epsilon}}\|_{L^\infty([0,\tau];{\cal
A}')}<+\infty,\quad \mbox{for any }\tau>0.$$ Therefore $\left(W_{{\epsilon}}(\cdot)\right)$ is sequentially $\star$-weakly compact in $L^\infty_{\rm loc}([0,+\infty),{\cal A}')$, i.e. from any sequence ${\epsilon}_n\to0$ we can choose a subsequence, that we still denote by the same symbol, for which $\left( W_{{\epsilon}_n}(\cdot)\right)$ $\star$-weakly converges in $\left(L^1([0,t],{\cal
A})\right)'$ for any $t>0$. In our main result we identify the limit as the $L^2$ solution of the kinetic equation with the interface conditions and , in the sense of Definition $\ref{df011406-19}$.
\[main-thm\] Suppose that there exist $C,\kappa>0$ such that $$\label{011812aa}
|\widehat W_{\epsilon}(0,\eta,k)|+|\widehat Y_{\epsilon}(0,\eta,k)|\le
C\varphi(\eta),\quad (\eta,k)\in{{\mathbb T}}_{{\epsilon}}\times {{\mathbb T}}, \,{\epsilon}\in(0,1],$$ where $$\label{011812c}
\varphi(\eta):=\frac{1}{(1+\eta^2)^{3/2+\kappa}},$$ and $$ W_{{\epsilon}}(0)\mathop{\stackrel{{\rm \tiny w}^\star}{\longrightarrow}}\limits_{\tiny{{\epsilon}\to0+}} W_0\quad\mbox{ in
}{\cal A}'.$$ Then, $W_0\in L^2({{\mathbb R}}\times{{\mathbb T}})$ and for any $G\in
L^1_{\rm loc}\left([0,+\infty);{\cal A}\right)$ we have $$\label{061406-19}
\lim_{{\epsilon}\to0+}\int_0^{\tau}\langle W_{{\epsilon}}(t), G(t)\rangle dt=\int_0^{\tau}\langle W(t), G(t)\rangle dt,\quad\tau>0.$$ Here $W(t)$ is the $L^2$ solution of the kinetic equation with | 0 | non_member_986 |
the interface conditions and satisfying $W(0)=W_0$.
In the case $T=0$ the theorem is a direct consequence of Theorem \[cor020805-19\] proved below. The more general case $T\ge0$ is treated in Section \[sec10\].
Deterministic wave equation corresponding to (\[basic:sde:2a\]) {#sec3}
===============================================================
Here we consider a deterministic part of the dynamics described in (\[basic:sde:2a\]). Its corresponding energy density function will converge to the solution of . In microscopic time the evolution of the wave function is given by $$\begin{aligned}
\label{012503-19}
&&
\frac{d}{dt}\hat\phi(t,k)=-i{\omega}(k)\hat\phi(t,k)
-2i{\epsilon}{\gamma}_0R(k)\hat p\left(t,k\right)-i{\gamma}_1\int_{{{\mathbb T}}}\hat{ p}\left(t,k'\right)dk',
\\
&&
\hat\phi(0,k)= \hat\psi(k),\nonumber
\end{aligned}$$ where $$\hat p(t,k):=\frac{1}{2i}\left[\hat\phi(t,k)-\left(\hat\phi(t,-k)\right)^\star\right].$$ In fact it is convenient to deal with the vector formulation of the equation for $$\label{Ut}
\hat\Phi(t,k)=\left[
\begin{array}{c}
\hat\phi_+(t,k)\\
\\
\hat\phi_-(t,k),
\end{array}\right],\quad \hat\Psi(k)=\left[
\begin{array}{c}
\hat\psi_+(k)\\
\\
\hat\psi_-(k)
\end{array}\right].$$ Here, we use the convention $\hat\phi_+(t,k)=\hat\phi(t,k)$ and $\hat\phi_-(t,k):=\left(\hat\phi(t,-k)\right)^\star$ and similarly for $\hat\psi_\pm(k)$. The equation then takes the form $$\begin{aligned}
\label{basic:sde:2av}
&&
\frac{d}{dt}\hat\Phi(t,k)={\Omega}_{\epsilon}(k) \hat\Phi(t,k) - i\gamma_1 {\frak f}{p}_0(t)
,\nonumber\\
&&
\hat\Phi(0,k)= \hat\Psi(k).
\end{aligned}$$ Here $$\label{Omk}
{\Omega}_{\epsilon}(k):=\left[
\begin{array}{cc}
-{\gamma}_0{\epsilon}R(k)-i{\omega}(k)&{\gamma}_0{\epsilon}R(k)\\
{\gamma}_0{\epsilon}R(k)&-{\gamma}_0{\epsilon}R(k)+i{\omega}(k)
\end{array}\right]={\Omega}_0(k)-{\gamma}_0{\epsilon}R(k){\bf D}.$$ and $${\frak f}:=\left[
\begin{array}{r}
1\\
-1
\end{array}\right],\quad {\bf D}:={\frak f}^T\otimes {\frak f}=\left[
\begin{array}{rr}
1&-1\\
-1&1
\end{array}\right].$$ The momentum at $x=0$ equals $$\label{p0}
{p}_0(t)
:=\frac{1}{2i}\int_{{{\mathbb T}}}\hat\Phi(t,k)\cdot {\frak f}dk=\frac{1}{2i}\int_{{{\mathbb T}}}\left[\hat\phi (t,k)-\left(\hat\phi (t,-k)\right)^\star\right]dk.$$ The eigenvalues of the matrix ${\Omega}_{\epsilon}(k)$ equal $
{\lambda}_\pm(k)=-{\gamma}_0{\epsilon}R(k)\pm i{\omega}_{\epsilon}(k),
$ | 0 | non_member_986 |
where $$\label{030904-19}
\beta(k)=\frac{{\gamma}_0 R(k)}{{\omega}(k)},\qquad{\omega}_{\epsilon}(k):={\omega}(k) \sqrt{1-({\epsilon}\beta(k))^2}.$$ Note that $
{\lambda}_+^\star={\lambda}_-.
$
Solution of (\[basic:sde:2av\]) {#solution-of-basicsde2av .unnumbered}
-------------------------------
By the Duhamel formula, from we get $$\label{022503-19}
\hat\Phi(t,k)=e_{{\Omega}_{\epsilon}}(k,t) \hat\Psi(k) - i\gamma_1 \int_0^t e_{{\Omega}_{\epsilon}}(k,t-s){\frak f}{p}_0(s)ds .$$ Here $$\begin{aligned}
\label{011902-19ac}
&e_{{\Omega}_{\epsilon}}(k,t):=\exp\left\{{\Omega}_{\epsilon}(k)t\right\}=\left[
\begin{array}{cc}
e_{{\Omega}_{\epsilon}}^{1,1}(k,t)&e_{{\Omega}_{\epsilon}}^{1,2}(k,t)\\
e_{{\Omega}_{\epsilon}}^{1,2}(k,t)&[e_{{\Omega}_{\epsilon}}^{1,1}]^\star(k,t)
\end{array}\right]\end{aligned}$$ and $$\begin{aligned}
\label{011902-19a}
&
e_{{\Omega}_{\epsilon}}^{1,1}(k,t):=\dfrac14\left\{\left(1+\sqrt{1-({\epsilon}\beta(k))^2}\right)^2e_-(k,t)-({\epsilon}\beta(k))^2e_+(k,t)\right\}, \nonumber\\
&e_{{\Omega}_{\epsilon}}^{1,2}(k,t):=\dfrac{i{\epsilon}\beta(k)}{4} \left(1+\sqrt{1-({\epsilon}\beta(k))^2}\right)(e_-(k,t)-e_+(k,t))
,\\
&
e_\pm(k,t):=e^{{\lambda}_\pm(k)
t}.\nonumber\end{aligned}$$ Note that $e_\pm^\star(k,t)=e_\mp(k,t)$.
Multiplying scalarly both sides of by ${\frak f}$ and integrating over $k$ we conclude that $$\label{042503-19}
p_0(t)+{\gamma}_1J_{\epsilon}\star p_0(t)=p_0^0(t)$$ Here $$\label{null}
p_0^0(t):=\frac{1}{2i}\int_{{{\mathbb T}}}e_{{\Omega}_{\epsilon}}(k,t)\hat\Psi(k) \cdot {\frak f}dk$$ and $$\label{J-eps}
{J}_{\epsilon}(t):=\frac12\int_{{{\mathbb T}}} \exp\left\{{\Omega}_{\epsilon}(k)t\right\}{\frak
f}\cdot {\frak f} dk=\int_{{{\mathbb T}}}j_{\epsilon}(t,k)dk,$$ where $$\label{j}
j_{\epsilon}(t,k):=\frac12\exp\left\{{\Omega}_{\epsilon}(k)t\right\}{\frak f}\cdot
{\frak f}
=e^{-{\epsilon}{\gamma}_0 R(k) t}\left\{\sqrt{1-({\epsilon}\beta(k))^2}\cos\left({\omega}_{\epsilon}(k)t\right)-{\epsilon}\beta(k)\, \sin\left({\omega}_{\epsilon}(k)t\right)\right\}.$$ Taking the Laplace transforms of the both sides of we obtain $$\label{062503-19}
\tilde p_0({\lambda})(1+{\gamma}_1\tilde J_{\epsilon}({\lambda}))=\tilde p_0^0({\lambda}).$$ By a direct calculation one concludes that $$\label{052503-19}
{\rm Re}\, \tilde {J}_{\epsilon}({\lambda})>0,\quad\mbox{for any }{\lambda}\in \mathbb
C_+.$$ Since $J_{\epsilon}(\cdot)$ is real valued, we have $$\tilde {J}_{\epsilon}^\star({\lambda})=\tilde {J}_{\epsilon}({\lambda}^\star)\quad\mbox{for any }{\lambda}\in \mathbb
C_+.$$ From we get $$\label{072503-19}
\tilde p_0({\lambda})=\tilde g_{\epsilon}({\lambda})\tilde p_0^0({\lambda}),$$ with $\tilde g_{\epsilon}({\lambda})$ defined by $$\label{012302-19}
\tilde {g}_{\epsilon}({\lambda}):=\left(1+{\gamma}_1\tilde {J}_{\epsilon}({\lambda})\right)^{-1},\quad {\rm Re}\,{\lambda}>0.$$ Thanks to we obtain $$\label{G}
|\tilde {g}_{\epsilon}({\lambda})|\le 1,\quad {\rm Re}\,{\lambda}>0$$ and, as a result, $$\label{JG}
{\gamma}_1|\tilde {J}_{\epsilon}({\lambda})\tilde {g}_{\epsilon}({\lambda})|\le 2,\quad {\rm Re}\,{\lambda}>0.$$ The following result shows in particular that $\tilde
{g}_{\epsilon}({\epsilon}-i{\omega}(k))$ approximates in some sense $\nu(k)$, | 0 | non_member_986 |
as ${\epsilon}\to0+$ (see ).
\[cor010304-19\] Suppose that $K:{{\mathbb T}}\to{{\mathbb R}}_+$ is a uniformly continuous and bounded function satisfying $$\label{lim-is}
\inf_{k\in{{\mathbb T}}}K(k)>0.$$ Then, $$\label{010304-19}
\tilde g_{\epsilon}({\lambda})=\tilde g({\lambda})+{\epsilon}\tilde r_{\epsilon}({\lambda}),\quad
{\lambda}\in\mathbb C_+,$$ where $$\lim_{{\epsilon}\to0+}{\epsilon}^p\int_{{{\mathbb T}}}|\tilde r_{\epsilon}({\epsilon}K(k)-i{\omega}(k))|^pdk=0,$$ for any $p\in[1,+\infty)$.
The proof of Proposition \[cor010304-19\] is shown in Appendix \[appC\].
Define by $g_{\epsilon}(ds)$ the distribution such that $$\label{g-eps}
\tilde {g}_{\epsilon}({\lambda})=\int_0^{+\infty}e^{-{\lambda}t}g_{\epsilon}(ds).$$ From it satisfies $$\label{g-eps1}
g_{\epsilon}(ds)=\delta(ds)-{\gamma}_1 J_{\epsilon}\star g_{\epsilon}(s)ds.$$ The Volterra equation has a unique real-valued solution and ${\gamma}_1 J_{\epsilon}\star g_{\epsilon}(s)$ is a $C^\infty$ smooth function, see e.g. the argument made in Section 3 of [@kors]. The solution of can be then written as follows $$\begin{aligned}
\label{eq:10}
&
\hat\Phi(t,k) =U(t) \hat\Psi(k):=e_{{\Omega}_{\epsilon}}(k,t)\hat\Psi(k) - i\gamma_1 \int_0^t
e_{{\Omega}_{\epsilon}}(k,t-s){\frak f}p_0^0\star g_{\epsilon}(s) ds \\
&
=e_{{\Omega}_{\epsilon}}(k,t) \hat\Psi(k) - \frac{\gamma_1}{2} \int_0^tds\int_0^sg_{\epsilon}(ds_1)\int_{{{\mathbb T}}}
e_{{\Omega}_{\epsilon}}(k,t-s){\bf D}e_{{\Omega}_{\epsilon}}(k,s-s_1) \hat\Psi(\ell) d\ell .\nonumber
\end{aligned}$$
Dynamics of the energy density when $T=0$ {#sec4}
=========================================
Starting with the present section untill Section \[sec10\] we shall assume that the thermostat temperature $T=0$, see . We maintain this assumption untill Section \[sec10\]. Let $$\label{Psit}
\hat\Psi^{({\epsilon})}(t,k)=\left[
\begin{array}{c}
\hat\psi^{({\epsilon})}_+(t,k)\\
\hat\psi^{({\epsilon})}_-(t,k)
\end{array}\right],$$ where $\hat\psi^{({\epsilon})}_+(t,k):=\hat \psi^{({\epsilon})}(t,k)$ and $\hat\psi^{({\epsilon})}_-(t,k):=\left(\hat
\psi^{({\epsilon})}\right)^\star(t,-k)$ (cf ). From we get $$\begin{aligned}
\label{basic:sde:2av1}
&&
d \hat\Psi^{({\epsilon})}(t,k)=\frac{1}{{\epsilon}}{\Omega}_{\epsilon}(k) \hat\Psi^{({\epsilon})}(t,k) dt
+i\sqrt{{\gamma}_0}\int_{{{\mathbb T}}}r(k,k'){\bf
D} \hat\Psi^{({\epsilon})}\left(t,k-k'\right)B(dt,dk')-\frac{i{\gamma}_1}{{\epsilon}}{\frak p}_0(t){\frak
g}dt,\nonumber\\
&&
\Psi^{({\epsilon})}(0,k)= \hat\Psi(k).
\end{aligned}$$
With some abuse of notation | 0 | non_member_986 |
we denote by ${\cal A}$ the Banach space of all matrix valued functions obtained by the completion of functions of the form $$\begin{aligned}
\label{bFS}
{\bf F}(y,k)=\left[\begin{array}{ll}
G(y,k)&H(y,k)\\
H^\star(y,k)&G(y,-k)
\end{array}\right],\quad (y,k)\in{{\mathbb R}}\times {{\mathbb T}},\end{aligned}$$ with $C^\infty$ smooth entries satisfying $G$ is real valued and $H$ is even in $k$. The completion is taken in the norm given by the maximum of the ${\cal A}$ norms of the entries, see .
The Wigner distribution, corresponding to the wave function $\psi^{({\epsilon})}(t)$, is a $2\times 2$-matrix tensor ${\bf W}_{\epsilon}(t)$, whose entries are distributions, given by their respective Fourier transforms $$\begin{aligned}
\label{hbw}
&
\widehat {{\bf W}}_{\epsilon}(t,\eta,k):=\frac{{\epsilon}}{2}{\mathbb E}\left[\hat\Psi^{({\epsilon})}\left(t,k+\frac{{\epsilon}\eta}{2}\right)\otimes \left(\hat\Psi^{({\epsilon})}\right)^\star\left(t,k-\frac{{\epsilon}\eta}{2}\right)\right]
\\
&
=\left[\begin{array}{ll}
\widehat W_{{\epsilon},+}(t,\eta,k)&\widehat Y_{{\epsilon},+}(t,\eta,k)\\
\widehat Y_{{\epsilon},-}(t,\eta,k)&\widehat W_{{\epsilon},-}(t,\eta,k)
\end{array}\right], \quad (\eta,k)\in{{\mathbb T}}_{{\epsilon}}\times {{\mathbb T}},\end{aligned}$$ with $$\begin{aligned}
&
\widehat W_{{\epsilon},+}(t,\eta,k):=\widehat W_{{\epsilon}}(t,\eta,k)=\frac{{\epsilon}}{2}{\mathbb E}_{\epsilon}\left[\hat\psi^{({\epsilon})}\left(t,k+\frac{{\epsilon}\eta}{2}\right) \left(\hat\psi^{({\epsilon})}\right)^\star\left(t,k-\frac{{\epsilon}\eta}{2}\right)\right],
\\
&
\widehat Y_{{\epsilon},+}(t,\eta,k):=\frac{{\epsilon}}{2}{\mathbb E}_{\epsilon}\left[\hat\psi^{({\epsilon})}\left(t,k+\frac{{\epsilon}\eta}{2}\right) \hat\psi^{({\epsilon})}\left(t,-k+\frac{{\epsilon}\eta}{2}\right)\right],
\\
&
\widehat Y_{{\epsilon},-}(t,\eta,k):=\widehat
Y_{{\epsilon},+}^\star(t,-\eta,k),\quad
\widehat W_{{\epsilon},-}(t,\eta,k):=\widehat W_{{\epsilon},+}(t,\eta,-k).\end{aligned}$$ Then ${\bf W}_{\epsilon}(t)$ belongs to $ {\cal A}'$ - the dual to ${\cal A}$ that is made of all distributions ${\bf W}$, whose Fourier transform in the first variable equals $$\begin{aligned}
\label{bW}
\widehat{\bf W}(\eta,k)=\left[\begin{array}{ll}
\widehat W_{+}(\eta,k)&\widehat Y_{+}(\eta,k)\\
\widehat Y_{-}(\eta,k)&\widehat W_{-}(\eta,k)
\end{array}\right],\quad (\eta,k)\in{{\mathbb T}}_{{\epsilon}}\times {{\mathbb T}},\end{aligned}$$ whose entries belong to ${\cal A}'$ and satisfy $$\begin{aligned}
&\widehat W_{+}^\star(\eta,k)=\widehat W_{+}(-\eta,k),\quad \widehat
Y_{+}(\eta,k)=\widehat Y_{+}(\eta,-k),\\
| 0 | non_member_986 |
&
\widehat W_{-}(\eta,k)=\widehat W_{+}(\eta,-k),\quad \widehat Y_{-}(\eta,k)=\widehat
Y_{+}^\star(-\eta,k).\end{aligned}$$ The duality pairing between ${\cal A}'$ and ${\cal A}$ is determined by the relation $$\begin{aligned}
\label{pairing}
&
\left\langle {\bf F},{\bf
W}\right\rangle:=\int_{{{\mathbb T}}_{\epsilon}\times{{\mathbb T}}}\widehat {\bf
F}(\eta,k)\cdot {\bf
W}(\eta,k)d\eta dk\\
&
=2\int_{{{\mathbb T}}_{\epsilon}\times{{\mathbb T}}}\left\{\widehat F_{+}(\eta,k)\widehat
W_{+}^\star(\eta,k)+{\rm Re}\left(\widehat H_{+}(\eta,k)\widehat
Y_{+}^\star(\eta,k)\right)\right\},\nonumber\end{aligned}$$ wher the scalar product of two matrices is given by $$\label{scalar}
\widehat {\bf
F}\cdot \widehat {\bf
W}=\sum_{\iota=\pm}\left(\widehat {
F}_\iota \widehat {
W}^\star_\iota+\widehat {
H}_\iota \widehat {
Y}^\star_\iota\right).$$ The norm $\|{\bf
W}\|_{{\cal A}'}$ is therefore the sum of the norm of its entries.
Thanks to and we conclude that $$\label{030406-19}
\sup_{{\epsilon}\in(0,1]}\sup_{t\ge0}\|{\bf W}_{{\epsilon}}(t)\|_{{\cal A}'}=:A_*'<+\infty.$$ Therefore $\left({\bf W}_{{\epsilon}}(\cdot)\right)$ is bounded in $L^\infty([0,+\infty),{\cal A}')$. In consequence, from any sequence ${\epsilon}_n\to0$ we can choose a subsequence, that we still denote by the same symbol, such that $\left({\bf W}_{{\epsilon}_n}(\cdot)\right)$ is $\vphantom{1}^\star$-weakly convergent in $\left(L^1([0,+\infty),{\cal A})\right)'$.
In what follows we shall also consider the Hilbert spaces ${\cal
L}_{2,{\epsilon}}$ with the scalar product $\langle\cdot,\cdot\rangle_{{\cal L}_{2,{\epsilon}}}$ given by the formula . The respective Hilbert space norms are $$\|{\bf W}\|_{{\cal L}_{2,{\epsilon}}}:=\left\{2\left(\| W_{+}\|_{{\cal
L}_{2,{\epsilon}}}^2+\| Y_{+}\|_{{\cal
L}_{2,{\epsilon}}}^2\right)\right\}^{1/2}. $$ We introduce the following notation, given a function $f:{{\mathbb T}}\to\mathbb C$, we let $$\label{bar-f}
\bar f(k,\eta):=\frac{1}{2}\left[f\left(k+\frac{\eta}{2}\right)+f\left(k-\frac{ \eta}{2}\right)\right]$$ and the difference quotient for the dispersion relation $$\label{d-om}
\delta_{{\epsilon}}{\omega}(k,\eta):=\frac{1}{{\epsilon}}\left[{\omega}\left(k+\frac{{\epsilon}\eta}{2}\right)-{\omega}\left(k-\frac{{\epsilon}\eta}{2}\right)\right].$$ Equipped with this | 0 | non_member_986 |
notation we introduce $$\begin{aligned}
\label{bH}
\widehat{\bf H}_{\epsilon}(\eta,k)=\left[\begin{array}{cc}
-i\delta_{{\epsilon}}{\omega}(k;\eta)&-\dfrac{2i}{{\epsilon}} \bar{\omega}(k,{\epsilon}\eta)\\
&\\
\dfrac{2i}{{\epsilon}} \bar{\omega}(k,{\epsilon}\eta)&i\delta_{{\epsilon}}{\omega}(k;\eta)
\end{array}\right],\quad (\eta,k)\in{{\mathbb T}}_{{\epsilon}}\times {{\mathbb T}}\end{aligned}$$ and $$\begin{aligned}
\label{021605-19}
&&
{L}_\eta f(k):=2{\cal R}_{ \eta} f(k) - 2\bar R (k,\eta) f(k),\qquad
{L}_{\eta}^\pm f(k):=2{\cal R}_{\eta} f(k) -2R\left(k\pm\frac
\eta2\right) f(k),\nonumber\\
&&
{\cal R}_\eta f(k):=\int_{{{\mathbb T}}} R(k,k',\eta)f(k')dk',
\\
&&
R(k,k',\ell):=\frac12\sum_{\iota=\pm1}r\left(k-\frac{ \ell}{2},k-\iota
k'\right)r\left(k+\frac{ \ell}{2},k-\iota k'\right),\quad
k,k'\in{{\mathbb T}},\,\ell\in 2{{\mathbb T}}. \nonumber\end{aligned}$$ We denote by ${\frak
L}_{{\epsilon}\eta}$, ${\frak H}_{\epsilon}$, ${\frak T}_{\epsilon}$ the operators, acting on ${\cal L}_{2,{\epsilon}}$, defined by $$\label{011906-19}
\widehat{{\frak
L}_{{\epsilon}\eta}{\bf W}}=\widehat{\frak
L}_{{\epsilon}\eta}\widehat{\bf W},\quad \widehat{{\frak
H}_{{\epsilon}}{\bf W}}=\widehat{\frak
H}_{{\epsilon}}\widehat{\bf W},\quad \widehat{{\frak
T}_{{\epsilon}}{\bf W}}=\widehat{\frak
T}_{{\epsilon}}\widehat{\bf W}.$$ Here $\widehat{\bf W}$ is the Fourier transform of ${\bf W}\in {\cal L}_{2,{\epsilon}}$, given by . Operator $\hat{\frak H}_{\epsilon}$, acting on $L^2({{\mathbb T}}_{\epsilon}\times{{\mathbb T}})$, is given by $$\begin{aligned}
\label{fH}
\widehat{\frak H}_{\epsilon}\widehat {{\bf W}}(\eta,k):=\widehat{\bf H}_{\epsilon}(\eta,k) \circ\widehat {{\bf W}}(\eta,k)=\left[\begin{array}{cc}
-i\delta_{{\epsilon}}{\omega}(k;\eta)\widehat W_{+}(\eta,k)&-\dfrac{2i}{{\epsilon}} \bar{\omega}(k,{\epsilon}\eta)\widehat Y_{+}(\eta,k)\\
&\\
\dfrac{2i}{{\epsilon}} \bar{\omega}(k,{\epsilon}\eta)\widehat Y_{-}(\eta,k)&i\delta_{{\epsilon}}{\omega}(k;\eta)\widehat W_{-}(\eta,k)
\end{array}\right],\end{aligned}$$ with $\circ$ denoting the Hadamard’s product of $2\times 2$ matrices. Moreover, $\widehat{\frak
L}_{{\epsilon}\eta}$ and $\widehat{\frak
T}_{{\epsilon}}$ act on $L^2({{\mathbb T}}_{\epsilon}\times{{\mathbb T}})$ via the formulas $$\begin{aligned}
\label{LHT}
\widehat{\frak
L}_{{\epsilon}\eta}\widehat {{\bf W}}(\eta,k):=\left[\begin{array}{cc}
\widehat W'_{+}(\eta,k)&\widehat Y'_{+}(\eta,k)\\
\widehat Y'_{-}(\eta,k)&\widehat W'_{-}(\eta,k)
\end{array}\right],\qquad \widehat{\frak
T}_{{\epsilon}}\widehat {{\bf W}}(\eta,k):=\left[\begin{array}{cc}
\widehat W''_{+}(\eta,k)&\widehat Y''_{+}(\eta,k)\\
\widehat Y''_{-}(\eta,k)&\widehat W''_{-}(\eta,k)
\end{array}\right],\end{aligned}$$ with $$\begin{aligned}
\label{011605-19z}
&&\widehat W_{\pm}'(\eta,k)={L}_{{\epsilon}\eta}\widehat W_{\pm} (\eta,k)-\frac{1}{2}\sum_{\iota=\pm}{L}^{\pm}_{\iota{\epsilon}\eta}\widehat Y_{-\iota}(\eta,k),
\\
&&
\widehat Y_{\pm}'(\eta,k)={L}_{{\epsilon}\eta}\widehat Y_{\pm}(\eta,k)
+ {\cal R}_{{\epsilon}\eta}(\widehat
Y_{\mp}-\widehat | 0 | non_member_986 |
---
abstract: 'The current fleet of space-based solar observatories offers us a wealth of opportunities to study solar flares over a range of wavelengths. Significant advances in our understanding of flare physics often come from coordinated observations between multiple instruments. Consequently, considerable efforts have been, and continue to be made to coordinate observations among instruments (*e.g.* through the *Max Millennium Program of Solar Flare Research*). However, there has been no study to date that quantifies how many flares have been observed by combinations of various instruments. Here we describe a technique that retrospectively searches archival databases for flares jointly observed by the *Ramaty High Energy Solar Spectroscopic Imager* (RHESSI), *Solar Dynamics Observatory* (SDO)/*EUV Variability Experiment* (EVE) (*Multiple EUV Grating Spectrograph* (MEGS)-A and MEGS-B), *Hinode*/(EUV Imaging Spectrometer, *Solar Optical Telescope*, and *X-Ray Telescope*), and *Interface Region Imaging Spectrograph* (IRIS). Out of the 6953 flares of GOES magnitude C1 or greater that we consider over the 6.5 years after the launch of SDO, 40 have been observed by six or more instruments simultaneously. Using each instrument’s individual rate of success in observing flares, we show that the numbers of flares co-observed by three or more instruments are higher than the number expected | 0 | non_member_987 |
under the assumption that the instruments operated independently of one another. In particular, the number of flares observed by larger numbers of instruments is much higher than expected. Our study illustrates that these missions often acted in cooperation, or at least had aligned goals. We also provide details on an interactive widget () now available in that allows a user to search for flaring events that have been observed by a chosen set of instruments. This provides access to a broader range of events in order to answer specific science questions. The difficulty in scheduling coordinated observations for solar-flare research is discussed with respect to instruments projected to begin operations during Solar Cycle 25, such as the *Daniel K. Inouye Solar Telescope*, *Solar Orbiter*, and *Parker Solar Probe*.'
author:
- 'Ryan O. $^{1,2,3,4}$, Jack $^{3,5}$'
title: 'On the Performance of Multi-Instrument Solar Flare Observations During Solar Cycle 24'
---
Introduction {#s:intro}
============
The study of solar flares is a high-priority research area in the international heliophysics community. Understanding the physics of these energetic events is crucial, not only for the field of space weather, but also in the broader scope of astrophysics where similar processes are believed to occur in | 0 | non_member_987 |
stellar flares, black-hole accretion disks, and in the Earth’s magnetotail. Observations of solar flares are made by many different instruments, both in space and on the ground. These instruments provide imaging, photometric, and spectroscopic data over a range of wavelengths, from radio waves through the optical and EUV to X-rays and $\gamma$-rays: often the greatest advances in our understanding of solar flares come through various combinations of these datasets. From @flet11 [Section 7.2]:
> The multifarious observations across the broad spectrum of phenomena each help us to characterize the equilibrium change in the corona and chromosphere that we call a flare, and it should be clear that the multiwavelength approach is crucial in flare studies. It tells us where the flare energy starts and where it ends up, and something about the intermediate steps. It also provides some geometrical and diagnostic information about the flare magnetic environment, at different levels in the atmosphere, and how and when this changes as the flare proceeds. This big picture cannot be reached using one spectral region on its own. The multiwavelength observations have many detailed applications as we try to understand specific mechanisms that are at work in various phases and regions of | 0 | non_member_987 |
the flare development.
However, it is difficult to keep track of which flares have been observed by which instruments. While most currently operational missions have their own individual flare lists (*e.g.* Hinode Flare Catalog: @wata12 or *Solar Dynamics Observatory* (SDO)/*EUV Variability Experiment* (EVE): @hock12), it was only recently that the first inter-instrument catalog became available, hosted by New Jersey Institute of Technology (NJIT)[^1] [@sady17]. The *Max Millennium Program for Solar Flare Research* (see @bloo16 for a recent review) and others have aimed to coordinate ground- and space-based instrumentation to observe a flaring active region simultaneously in order to optimize the scientific return. However, this can be difficult due to factors such as coordinating across multiple time zones, planning schedules being uploaded days in advance, ground-based seeing conditions, competing scientific priorities, and so on.
Therefore when a solar flare is known to have been observed by a combination of instruments, the event can receive considerable attention as a consequence. A notable recent example of this is the 29 March 2014 X-class flare, which was observed by four space-based observatories and one ground based telescope[^2]. Consequently there have been 23 refereed publications that discuss this flare, according to a NASA ADS fulltext | 0 | non_member_987 |
search on the Solar Object Locator keyword . Similarly, the first X-class flare of Solar Cycle 24 () was simultaneously observed by multiple instruments at high cadence, resulting in 42 refereed publications to date. The exceptional data coverage of each of these events allowed [@klei16] and [@mill14], respectively, to investigate the redistribution of nonthermal electron energy. They were both able to compare radiative losses in the chromosphere across a range of wavelengths with the energy injected by nonthermal particles from hard X-ray observations. In both cases only 15–20% of the nonthermal energy could be accounted for from longer-wavelength measurements. Understanding where this “missing energy’ went to can only be answered by even better data coverage. Clearly there is great scientific merit in multi-instrument observations of the same event.
Likewise for other astronomical research areas where coordinated observations of transient objects (“Targets Of Opportunity”) at different wavelengths are highly desirable. The study of supernovae, for example, is facilitated by the availability of both lightcurves (to understand the evolution) and spectra (to understand the composition and dynamics). The *Open Supernova Catalog* [@guil17] acts as a central repository providing access to data for over 42,000 known supernova events. According to the statistics page | 0 | non_member_987 |
of the website[^3] only 6% of these events have *both* photometric and spectroscopic data available (34% only have lightcurves, 7% only have spectra, and 53% have neither).
This article presents an analysis of flare statistics by retrospectively cross-referencing metadata from a suite of instruments that take flare-relevant observations – the *Ramaty High Energy Solar Spectroscopic Imager* (RHESSI: @lin02), the *Multiple EUV Grating Spectrograph* (MEGS; @crot04) -A and -B components of EVE [@wood12], the *EUV Imaging Spectrometer* (EIS: @culh07), the *Solar Optical Telescope* (SOT: @tsun08), the *X-Ray Telescope* (XRT: @golu07), and the *Interface Region Imaging Spectrometer* (IRIS: @depo14) – to search for flaring events observed simultaneously, either intentionally or serendipitously. The purpose of this article is to present an overview of how successful the solar community has been in capturing flare data through coordinated efforts. We also describe a database of these events that give researchers access to multi-wavelength datasets with which to address a given science question. Section \[s:data\_anal\] describes how the various archives from each instrument were exploited. Section \[s:results\] presents the findings. The conclusions and a discussion are presented in Sections \[s:conc\] and \[s:disc\], respectively.
Data Analysis {#s:data_anal}
=============
In order to cross-reference datasets from different instruments to | 0 | non_member_987 |
infer which observed a given solar flare simultaneously, it is important to define what exactly constitutes a flare. The most commonly accepted catalog is that of the *Geostationary Operational Environmental Satellite* (GOES) event list provided by NOAA/SWPC. This defines a solar flare as a continuous increase in the one-minute averaged X-ray flux in the long-wavelength channel (1–8Å) of the GOES *X-ray Sensor* (XRS: @hans96) for the first four minutes of the event. The flux in the fourth minute must be at least 1.4 times the initial flux. The start time of the event is then defined as the first of these four minutes. The peak time is when the long-wavelength channel flux reaches a maximum, thus defining its class. The end of an event is defined as the time when the long channel flux reaches a level halfway between the peak and initial values[^4]. However in the vast majority of instances the NOAA catalog does not provide information on the location of a flare on the solar disk. As this is necessary for cross-referencing with the pointing information for reduced field-of-view instruments, the location of each flare was determined from the SSW Latest Events list, which is accessible through the | 0 | non_member_987 |
Heliophysics Events Knowledgebase[^5] (HEK). Flare locations are determined by subtracting the SDO/*Atmospheric Imaging Assembly* (AIA: @leme12) 131Å image closest to the GOES start time, from that image closest to the GOES peak time. The flare location is then extracted from the peak intensity of this difference image (S. Freeland; private communication, 2017). Knowing the timing and position of each event then allowed this information to be cross-referenced with the metadata from other instruments to determine whether or not they observed the same location at the same time. *Note that this does not guarantee that a given instrument actually detected flaring emission, but only that the timing and pointing of a given dataset were consistent with the timing and location of the flare.* B-class flares were not included in this study due to discrepancies between flare locations derived from RHESSI and SDO/AIA and were therefore deemed unreliable. The SSW Latest Events list also has several months of data missing[^6]. Nevertheless, out of the 8090 flares of GOES class C1 or greater that appear in the NOAA event list, 6953 (86%) are also in the SSW Latest Events list and include location information.
![Solar Cycles 23 and 24 (average monthly sunspot number) | 0 | non_member_987 |
with mission durations overplotted. The two vertical-dotted lines denote the 6.5-year time range considered for this study. Note that SDO/EVE MEGS-A and IRIS only overlapped for $\approx$11 months.[]{data-label="solar_cycle"}](solar_cycle_monthly_ssn-eps-converted-to.pdf){width="\textwidth"}
For the purposes of this study, only flares greater than GOES C1 class that occurred over the 6.5 years of Solar Cycle 24 observed by SDO [@pesn12] were considered. This defines the date range 1 May 2010 to 31 October 2016, as denoted by the vertical-dotted lines in Figure \[solar\_cycle\]. Also shown are the durations of the missions considered in this study. Note that EVE MEGS-A and IRIS were only operational together for around 11 months after IRIS was launched, and before MEGS-A suffered a power anomaly on 26 May 2014[^7].
Figure \[sff\_plot\] shows a sample plot from the widget, which was developed in tandem with this study (see Appendix \[appendixa\]). Plots such as this have been generated for every SSW event since the launch of SDO, and they are being continuously updated. These plots allow the user to readily view the timing and pointing of each instrument during a chosen event. This particular plot shows one flare from this study that was found to have been observed by all seven instruments: | 0 | non_member_987 |
an M1.5 flare that occurred on 4 February 2014. The upper-left panel shows the GOES X-ray lightcurves with the start, peak, and end times overlaid (vertical gray-dotted, solid, and dashed lines, respectively). Note that for completeness, the time profiles of GOES EUVS-E (centered on the Lyman-$\alpha$ – [Ly$\alpha$]{} – line of hydrogen at 1216Å; @vier07) are also shown in gray. [@mill16] recently showed that these data are more reliable for flare studies than the EVE MEGS-P data given that the GOES/EUVS-E data exhibit a more impulsive profile – as one would expect for chromospheric emission – whereas current EVE MEGS-P data erroneously show a more gradually varying behavior.
![Sample event from this study that was observed by all instruments; an M1.5 flare that occurred on 4 February 2014. Upper left panel: GOES/XRS lightcurves in 1–8Å (solid black curve) and 0.5–4Å (dotted-black curve), along with the GOES/EUVS-E ([Ly$\alpha$]{}) profile in gray. Vertical dotted, solid, and dashed grey lines denote the start, peak, and end times of the GOES event, respectively. Dotted- and dashed-green ticks mark the start and end times of each *Hinode*/EIS raster, respectively, while red and yellow ticks mark the times of each SOT and XRT image, respectively. Horizontal | 0 | non_member_987 |
blue and cyan lines illustrate the times which MEGS-A and MEGS-B were exposed, respectively, while the horizontal purple line shows the time of the corresponding IRIS study. Lower-left panel: RHESSI lightcurves up to the maximum energy detected, with GOES start, peak, and end times overlaid. Right panel: A PROBA2/SWAP 174Å image taken near the peak of the flare. The white circle is 100$''$ wide centered on the location derived from AIA 131Å images, while the black contours mark out the 6–25keV emission observed by RHESSI. The fields of view of EIS, SOT, XRT, and IRIS are overplotted in green, yellow, red, and purple, respectively.[]{data-label="sff_plot"}](20140204_152500_M1_hsi100_megsab_eis_sot_xrt_iris.png){width="\textwidth"}
The *Ramaty High Energy Solar Spectroscopic Imager* {#ss:rhessi}
---------------------------------------------------
RHESSI, launched on 5 February 2002[^8], observes the full disk of the Sun in X-rays and $\gamma$-rays. It orbits the Earth at an inclination angle of 38$^{\circ}$, at an altitude of $\approx$600 km, and as such suffers from eclipse passes and transits through the South Atlantic Anomaly. In order to determine whether or not RHESSI observed a given GOES flare event, the IDL routine was run between the start and end times of each flare. This searches the RHESSI flare catalog[^9] for the largest event detected in | 0 | non_member_987 |
the time range of interest. If a RHESSI flare is detected, the fraction of the rise time (GOES start $\rightarrow$ GOES peak) that the RHESSI flare flag was active was also calculated. While RHESSI is a full-disk instrument, its orbit implies that it may have captured anywhere from a few seconds of a given flare up to around an hour (note that some long-duration flares are detectable over several RHESSI orbits).
From the lightcurves presented in the lower-left panel of Figure \[sff\_plot\] it can be seen that RHESSI captured the peak of the M1.5 flare up to an energy of 50–100keV. The contours of the RHESSI quicklook image (6–25keV; black contours overlaid on the EUV image) agree with the flare location computed from the AIA 131Å data (white circle).
The *EUV Variability Experiment* {#ss:sdo_eve}
--------------------------------
The SDO spacecraft is in a geosynchronous orbit allowing it to observe the full disk of the Sun continuously without interruption (except for the occasional lunar and terrestrial eclipses). For simplicity, it was assumed that both AIA and the *Helioseismic and Magnetic Imager* (HMI: @sche12) were observing continuously throughout each event. The EVE instrument, however, is less straightforward. While MEGS-A, which provides spatially integrated Sun-as-a-star | 0 | non_member_987 |
spectra over the 60–370Å range every ten seconds, and was exposed to the Sun continuously from launch until it ceased operations on 26 May 2014, the MEGS-B (370–1050Å) and MEGS-P ([Ly$\alpha$]{}) exposure times have been much more erratic due to unforeseen degradation soon after launch. For much of the mission MEGS-B has only been exposed for three hours per day in order to limit degradation, as well as five minutes per hour for the consistency of long term variability studies. During periods of substantial solar activity it would observe continuously for 24–48 hours. Recently the flight software was changed to allow MEGS-B to respond to a flare trigger based on the EVE EUV Solar Photometer (ESP) flux for events $>$M1. Although there is an EVE flare catalog online[^10], this includes events for which MEGS-B may have only been exposed for five minutes. Therefore for the purposes of this study, MEGS-B was considered to have observed a flare if it was exposed to the Sun continuously between the GOES start and GOES peak times as determined from the daily exposure times[^11]. However, this does not necessarily mean that the flare itself will show up in the data, as EVE is often | 0 | non_member_987 |
only sensitive to flares $\gtrsim$C5 level. The times at which MEGS-A and MEGS-B were exposed to the Sun around the time of a given flare are illustrated by the horizontal blue and cyan lines, respectively, as shown in the top-left panel of Figure \[sff\_plot\].
*Hinode* {#ss:hinode}
--------
The *Hinode* spacecraft [@kosu07] was launched into a Sun-synchronous orbit on 22 September 2006 and comprises three instruments: EIS, SOT, and XRT. They were designed to study the interplay between the photosphere and the corona by working in unison. However, by January 2008 *Hinode* had lost the use of its X-band transmitter, resulting in a dramatic reduction in the amount of data being transmitted to the ground.
### The Extreme-ultraviolet Imaging Spectrometer {#sss:eis}
EIS is a two-channel, normal-incidence EUV spectrometer. Its two channels cover the wavelength ranges 170–210Å and 250–290Å, selected to cover coronal emission lines with formation temperatures ranging from 8000 K (He [ii]{}) to 16 MK (Fe [xxiv]{}). It has a mirror that is tiltable in the solar X-direction, and is used to build up rastered spectral images of portions of the Sun in up to 25 spectral ranges. Additionally, EIS has both narrow (1$''$ and 2$''$ wide) slits, and wider | 0 | non_member_987 |
(40$''$ and 266$''$ wide) imaging slots, with up to 512$''$ in the solar Y-direction.
For this study, a flare successfully observed by EIS must have had at least one raster begin, end, or straddle the GOES start and end times as determined from the routine. If such a raster exists, then all rasters within -30 minutes and +60 minutes of the GOES start and end times, respectively, are returned. The flare location as projected from AIA must have also lain within the EIS field of view. This does not imply that EIS captured any flaring emission; due to the rastering nature of the instrument, the slit may not have been over the flare site at the opportune time. In the example shown in Figure \[sff\_plot\], EIS was running a sequence of $\approx$three-minute rasters (denoted by the vertical green-dotted and dashed ticks) around the peak of the M1.5 flare. The associated regions of the Sun corresponding to each raster are also overlaid on the EUV image as green boxes.
### The Solar Optical Telescope {#sss:sot}
SOT is the first large optical telescope flown in space to observe the Sun. It images sub-full-disk portions of the Sun. Its aperture is 50 cm | 0 | non_member_987 |
in diameter, the angular resolution is 0.25$''$ (corresponding to 175 km on the Sun), and the wavelengths covered extend from 4800 to 6500Å. SOT also includes the *Focal Plane Package*, which consists of a vector magnetograph and a spectrograph. The vector magnetograph provides time series of photospheric vector magnetograms, Doppler velocity and photospheric intensity.
In order to determine whether SOT observed a given flare, the routine was run between the GOES start and end times. If the routine returned at least one image, and the flare location fell within the SOT field of view, then all corresponding images between -30 and +60 minutes of the GOES start and end times, respectively, were returned and plotted over the GOES X-ray lightcurves as shown in Figure \[sff\_plot\] (vertical yellow ticks). The associated regions of the Sun corresponding to each SOT image are also overlaid on the EUV image as yellow boxes.
### The X-Ray Telescope {#sss:xrt}
XRT is a high-resolution (1$''$) grazing-incidence Wolter telescope that obtains high-resolution soft X-ray images covering the energy range 0.2 to 2 keV. This reveals magnetic-field configurations and their evolution, allowing the observation of energy buildup, storage, and release process in the corona for any transient event. | 0 | non_member_987 |
XRT covers a wide temperature range from 0.5 to 10 million Kelvin allowing it to see coronal features that are not visible with a normal incidence telescope. XRT can observe the full disk of the Sun, but can also return sub-full-disk images, depending on the science goal of the observation.
In order to determine whether XRT observed a given flare, the routine was run between the GOES start and end times. If the routine returned at least one image, and the flare location fell within the XRT field of view, then all corresponding images between -30 and +60 minutes of the GOES start and end times, respectively, were returned and plotted over the GOES X-ray lightcurves as shown in Figure \[sff\_plot\] (vertical red ticks). The associated regions of the Sun corresponding to each XRT image are also overlaid on the EUV image as red boxes.
The *Interface Region Imaging Spectrometer* {#ss:iris}
-------------------------------------------
Launched on 27 June 2013 into a Sun-synchronous polar orbit, IRIS obtains UV spectra and images with high spatial (1/3$''$) and temporal resolution (one-second) focused on the solar chromosphere and transition region. The instrument comprises an ultraviolet telescope combined with an imaging spectrograph. IRIS records observations of material | 0 | non_member_987 |
at specific temperatures, ranging from 5000 K and 65,000 K, and up to 10 MK during solar flares. IRIS is a sub-full-disk instrument, imaging portions of the solar disk and limb.
The timing and pointing of IRIS observation studies that were run during the start and end times of a given GOES event were obtained using the routine. This searches the Heliophysics Coverage Registry for the [^12] corresponding to the time of the flare, as shown by the horizontal purple line in the upper-left panel for Figure \[sff\_plot\]. Similar to the previously mentioned instruments with limited fields of view, the pointing information obtained from the was cross-referenced with the flare location to determine if IRIS was pointed at the required location (purple box overlaid on the EUV image in Figure \[sff\_plot\]).
[max width=]{}
------------------- ------------ ---------- --------- ------------ ----------------
Instrument/ Success Rate
Database C-class M-class X-class Total Over 6.5 Years
NOAA/GOES 7360 685 45 8090 100%
SSW Latest Events 6339 581 33 6953 86%
RHESSI 3673 370 23 4066 58%
SDO/EVE MEGS-A 3825 343 19 4187 100%
SDO/EVE MEGS-B 787 97 8 892 12%
*Hinode*/EIS 496 54 6 556 8%
*Hinode*/SOT 1167 177 15 1359 20%
*Hinode*/XRT 3739 357 26 | 0 | non_member_987 |
4122 59%
IRIS 523 (3349) 76 (335) 5 (16) 604 (3700) 16%
------------------- ------------ ---------- --------- ------------ ----------------
: Distribution of how many solar flares – and of which class – were observed by individual instruments between 1 May 2010 and 31 October 2016 based on the timing and pointing information available (where applicable). The percentage of SSW Latest Events found is calculated relative to the number of NOAA/GOES events. Percentage of flares captured by each instrument during their respective missions are calculated against the total number of events found via SSW Latest Events.
[MEGS-A was assumed to have observed all flares from launch until it ceased operations on 26 May 2014]{}
[The total number of flares listed in the HEK between the launch of IRIS and 31 October 2016 are given in parentheses]{}
\[tab:instr\_flares\]
Results {#s:results}
=======
[max width=]{}
Degree Number of flares observed % of potentially observable flares
---------------------- --------------------------- ------------------------------------
No instruments 127 1.8%
Exactly 1 instrument 1432 20.6%
Any 2 instruments 2371 34.1%
Any 3 instruments 2035 29.2%
Any 4 instruments 720 10.3%
Any 5 instruments 228 3.3%
Any 6 instruments 37 0.5%
All 7 instruments 3 0.3%
: Number and percentage of total flares observed | 0 | non_member_987 |
by different combinations of instruments. Note that there were 6953 flare events that were potentially observable by six or fewer instruments. Only 934 events were potentially observable by all seven of the instruments considered in this study.
[A total of 934 flares were recorded during the 11 months when both MEGS-A and IRIS were operational together.]{}
\[tab:joint\_flares\]
Based on the search criteria defined in Section \[s:data\_anal\], the number of flares, and their percentages of the total number of SSW Latest Events (which itself is a subset – 86% – of the available NOAA/GOES events) that were considered to have been observed by each of the instruments are listed in Table \[tab:instr\_flares\]. The instruments with full-disk capability and high duty cycles (RHESSI, MEGS-A, and *Hinode*/XRT) unsurprisingly were able to capture more than half of the total flares considered. The remaining instruments – which have either limited duty cycles and/or limited fields of view – were only able to capture around 20% or less of all flares during Solar Cycle 24. Similarly, the number of flares and their percentages that were observed by different combinations of instruments are listed in Table \[tab:joint\_flares\]. Around 84% of all flares were observed by between one | 0 | non_member_987 |
and three instruments. Most of the remaining 16% were observed by either four or five instruments, while a total of 37 flares were observed by different combinations of six instruments and only three out of 934 were observed by all seven instruments during the 11 months that they were simultaneously operating. Interestingly, 127 flares (1.8%) were not observed at all by *any* of the seven instruments considered.
The findings of how many solar flares were observed by different combinations of instruments are displayed in Figures \[f:upset\] as [^13] plots [@lex14]. This type of plot enables the efficient visualization of common elements of a large number of sets (the more common and familiar Venn diagram approach produces ineffective visualizations for more than $\approx$five sets). The top panel of Figure \[f:upset\] shows the intersections of the various combinations of datasets ordered by decreasing frequency (i.e. the most common combinations are on the left and decrease towards the right), while the bottom panel shows the same information only now ordered by increasing number of instruments (i.e. flares observed by individual instruments alone come first, with flares observed by all seven on the far right). In each plot, the total number of flares observed | 0 | non_member_987 |
by each instrument are given by the horizontal black bars in the bottom-left corner. The dots connected by lines at the bottom of each figure denote the combinations of instruments considered, while the histograms above give the number of events corresponding to a given combination. The most common combination of flare datasets was RHESSI+MEGS-A+*Hinode*/XRT (930 flares), due to their large fields of view and high duty cycles as mentioned above.\
![ plots of the intersection of flare datasets from each instrument as ordered by decreasing frequency (top panel) and increasing number of instruments (bottom panel). Zero-element sets are not included in either plot.[]{data-label="f:upset"}](UpSetR_dec_freq_cmx_nozero.pdf "fig:"){width="\textwidth"} ![ plots of the intersection of flare datasets from each instrument as ordered by decreasing frequency (top panel) and increasing number of instruments (bottom panel). Zero-element sets are not included in either plot.[]{data-label="f:upset"}](UpSetR_inc_deg_cmx_nozero.pdf "fig:"){width="\textwidth"}
Evaluation of Measured Versus Expected Success Rates
----------------------------------------------------
It is difficult to give a good estimate of how many flares one would *expect* to see with each instrument, given their different science goals and operational constraints. Table \[tab:estimated\_flares\] summarizes an attempt to estimate this expectation value \[${e}$\] for each instrument considered in this article. The estimates are based on the average field-of-views | 0 | non_member_987 |
times the duty cycles of each instrument. The area of consideration is estimated in two ways. The first estimate is simply the area of the full disk of the Sun. The second estimate assumes that there are four active regions on the Sun each with an area of $240''\times240''$, and that the majority of the duty cycle is spent examining the active-region areas. These two estimates give an upper and lower range to the percentage field of view. The percentage field of view is calculated as the percentage of the area of consideration covered by the average field-of-view of the instrument disk. The duty cycle is estimated as the percentage of the time that the instrument could have observed a flare. Crucially, the estimates assume that a *random* location within the area of consideration (either the full disk of the Sun, or an estimated average area of active regions that the instrument could point to, assuming that active regions form the majority of target areas during the duty cycle).
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---------------- ------------ --------- -------------- -------------- --
Instrument “Expected” Measured
Duty cycle %FOV Success Rate Success Rate
${e}$ ${m}$
RHESSI 50% 100% 50% 58%
SDO/EVE MEGS-A 100% 100% 100% 100%
| 0 | non_member_987 |
SDO/EVE MEGS-B 12.5% 100% 12.5% 12%
*Hinode*/EIS 25% 2–25% 0.5–6% 6%
*Hinode*/SOT 50% 1–17% 0.5–8% 13%
*Hinode*/XRT 100% 25–100% 25–100% 57%
IRIS 100% 0.5–3% 0.5–3% 11%
---------------- ------------ --------- -------------- -------------- --
: Estimates of the percentage of flares expected to be observed \[${e}$\] by each instrument based on the product of their duty cycles and field-of-view. The calculation assumes that each instrument points randomly in the area of consideration. The percentage of flares that were actually observed \[${m}$\] is also presented.
[Duty cycle estimated at approximately three hours per day (see text).]{}
[Duty cycle estimated by examining recent EIS planning notes. Field of view estimated at $240''\times 240''$, one quarter the full FOV of EIS.]{}
[Field of view estimated at $200''\times200''$, one quarter the full FOV of SOT.]{}
[Field of view estimated at $1024''\times1024''$, one quarter the full FOV of XRT.]{}
[Field of view estimated at $85''\times85''$, one quarter the full FOV of IRIS.]{}
[Out of the 934 flares listed in the HEK over the 11-month period that all seven instruments were operational. This is around half of all the flares listed in the NOAA/GOES event list (1774).]{}
\[tab:estimated\_flares\]
A very crude estimate of the “expected” success rate \[${e}$\] | 0 | non_member_987 |
is therefore the product of the duty cycle and the FOV. This can be readily compared to the measured success rate \[${m}$\] for each instrument individually from the 11-month period during which all seven instruments were operational together. This time period also happened to coincide with the peak of the solar cycle as illustrated in Figure \[solar\_cycle\]. These expected and measured values are presented in the last two columns of Table \[tab:estimated\_flares\]. The success rates over this 11-month period bear a reasonable agreement with the values measured over the entire 6.5 years under study that are presented in the final column of Table \[tab:instr\_flares\], and they can therefore be considered as characteristic of each instrument. They are also consistent with or better than the individual expected value implying that each pointing instrument is performing well. This reflects the fact that solar flares are a high priority science goal for these instruments, and that operators of course do not point their instruments randomly.
The measured success rates of each individual instrument in Table \[tab:estimated\_flares\] can be used to predict the number of flares expected to be seen by different combinations of instruments as follows. The measured success rate of each instrument | 0 | non_member_987 |