# Approach to equilibrium in translation-invariant
quantum systems: some structural results

Vojkan Jakšić  Department of Mathematics and Statistics, McGill University,
805 Sherbrooke Street West, Montreal, QC, H3A 2K6, Canada  Claude-Alain Pillet
Aix Marseille Univ, Université de Toulon, CNRS, CPT, Marseille, France
Clément Tauber  Institut de Recherche Mathématique Avancée, UMR 7501
Université de Strasbourg et CNRS, 7 rue René-Descartes, 67000 Strasbourg,
France

Dedicated to the memory of Krzysztof Gawędzki

Abstract. We formulate the problem of approach to equilibrium in algebraic
quantum statistical mechanics and study some of its structural aspects,
focusing on the relation between the zeroth law of thermodynamics (approach to
equilibrium) and the second law (increase of entropy). Our main result is that
approach to equilibrium is necessarily accompanied by a strict increase of the
specific (mean) energy and entropy. In the course of our analysis, we
introduce the concept of quantum weak Gibbs state which is of independent
interest.

##  1 Introduction

Algebraic quantum statistical mechanics provides a general framework for
studying dynamical aspects of infinitely extended quantum systems. Lattice
spins and fermions are examples of such systems which, in the translation
invariant setting, are also the main focus of this work. Adapting the
algebraic framework, in this paper we initiate a research program dealing with
the fundamental problem of Boltzmann111For additional information and
references about this problem see for example [Leb93, UF63] which is
succinctly summarized in [Sim93, Appendix I.1] as follows:

> Approach to equilibrium: heuristic description.
>
> “As for the laws of thermodynamics, most textbooks primarily discuss the
> first and the second law. Actually, there is a fundamental experimental fact
> basic to all thermodynamic descriptions that is usually implicitly assumed:
> It is occasionally called the zeroth law222This terminology is unfortunate;
> see [BU01] […]. The zeroth law deals with the observed fact that a large
> system seem to normally have ‘states’ described by a few macroscopic
> parameters like a temperature and density, and that any system not in one of
> these states, left alone, rapidly approaches one of these states. When
> Boltzmann and Gibbs tried to find a macroscopic basis for thermodynamics,
> they realized that the approach to equilibrium was the most puzzling and
> deepest problem in such a formalism.”

Previous works on this topic in the algebraic formalism are scarce. Approach
to equilibrium was discussed in [HKK73, LR72, Suk83] in the context of quasi-
free fermionic dynamics, and more generally for interacting fermionic systems
in [ESY04, Hug83, Hug87]. The quantum Ising model was analyzed in [Rad70].

The goal of this work is to investigate some structural properties of the
quantum dynamical systems associated to interacting, infinitely extended
lattice spin systems and lattice Fermi gases. Besides setting the program, we
examine one important special case where the system, initially in thermal
equilibrium w.r.t. an interaction $\Psi$, evolves according to the dynamics
generated by some other interaction $\Phi$, focusing on the relation between
the zeroth and the second law of thermodynamics. Our main result states that,
in this setting, approach to equilibrium is accompanied by a strict increase
of the specific entropy and of the specific energy of the original interaction
$\Psi$. Let us emphasize the strict increase aspect of our result. That the
entropy can’t decrease is an easy consequence of the upper-semicontinuity of
the specific entropy and of the classical result of Lanford and Robinson
[LR68] that this specific entropy remain constant along the state trajectory.
That the specific energy of $\Psi$ cannot decrease is a consequence of the
Gibbs variational principle and a general estimate of Hiai and Petz [HP93,
Lemma 2.3]; see Theorem 2.18. One compelling aspect of our results is their
generality.

Our analysis builds on the foundational works on statistical mechanics of
quantum spin systems that go back to 1970’s (summarized in the monographs
[Rue69, BR81, Isr79, Sim93]), and their extensions to fermionic systems
[AM03]. The two additional ingredients are:

  1. ()

The notion of weak Gibbs states.

  2. ()

The set of ideas that emerged during the last twenty years in the studies of
entropy production in the algebraic framework of non-equilibrium quantum
statistical mechanics; see [JP01b, Rue00].

The notion of weak Gibbs states was implicit in the early works on
thermodynamic formalism of classical spin systems, but surprisingly it was
formalized only relatively late [Yur02]; see also [BFV19, FP97, KLNR04, JR11,
MRVML99, PS20, Var12, vEV04]. Similarly, the notion of quantum weak Gibbs
state has been implicit in a number of works in quantum statistical mechanics
[AI74, Ara76, HP93, HMO07, ORB11] but, to the best of our knowledge, it has
not been formalized before. Besides our work, the study of quantum weak Gibbs
states is of independent interest, and most of foundational questions remain
open.

Regarding (b), and more generally in order to put the results of this paper
and the research program it initiates in a proper perspective, the next
section provides a short account of dynamical aspects of algebraic quantum
statistical mechanics, assuming that the reader has had a previous exposure to
the subject. The problem of approach to equilibrium in algebraic quantum
statistical mechanics is also formulated in this section.

###  1.1 The algebraic approach to nonequilibrium quantum statistical
mechanics

Our starting point is a $C^{\ast}$-dynamical system $(\mathcal{O},\alpha)$,
where $\mathcal{O}$ is the unital333The unit will be denoted by $\mathbbm{1}$.
$C^{\ast}$-algebra of observables of the quantum mechanical system under
consideration. The Heisenberg dynamics of this system is described by a
strongly continuous one-parameter subgroup $\alpha=\\{\alpha^{t}\mid
t\in\mathbb{R}\\}$ of the group $\mathrm{Aut}(\mathcal{O})$ of
$\ast$-automorphisms of $\mathcal{O}$—we will often refer to such $\alpha$ as
a $C^{\ast}$-dynamics. States of the system are normalized positive linear
functionals on $\mathcal{O}$. The set $\mathcal{S}(\mathcal{O})$ of all states
is a convex $\mathrm{weak}^{\ast}$-compact subset of the dual Banach space
$\mathcal{O}^{\ast}$. In the specific case of a system of spins or fermions on
a lattice, we will denote the algebra of observables by $\mathfrak{A}$; see
Section 2.1. For some models the natural starting point is rather a
$W^{\ast}$-dynamical system; we will comment on such models in Section 2.4.

The seminal work [HHW67] introduced the KMS-condition as a characterization of
thermal equilibrium states of $(\mathcal{O},\alpha)$ at inverse temperature
$\beta>0$. A state $\omega\in\mathcal{S}(\mathcal{O})$ is $(\alpha,\beta)$-KMS
if, for all $A,B\in\mathcal{O}$, the function defined by

$\mathbb{R}\ni t\mapsto F_{A,B}(t)=\omega(A\alpha^{t}(B)),$

has an analytic continuation to the strip $0<\mathop{\mathrm{Im}}\nolimits
z<\beta$, that is bounded and continuous on its closure, and satisfies the
KMS-boundary condition

$F_{A,B}(t+\mathrm{i}\beta)=\omega(\alpha^{t}(B)A).$

Any $(\alpha,\beta)$-KMS state is $\alpha$-invariant. A quantum dynamical
system $(\mathcal{O},\alpha,\omega)$, where $\omega$ is a $(\alpha,\beta)$-KMS
state, describes a physical system in thermal equilibrium at temperature
$1/\beta$.

It is mathematically convenient to extend, in the obvious way, the definition
of KMS-states to all $\beta\in\mathbb{R}$. Of particular interest is the value
$\beta=-1$ which links the KMS-condition to the Tomita–Takesaki modular theory
of operator algebras.

The general theory of KMS states, developed in the early days of algebraic
quantum statistical mechanics, is summarized in [BR81, Sections 5.3 and 5.4].
To set up the notation to be used below, let us mention the following
remarkable stability property of a $(\alpha,\beta)$-KMS state
$\omega\in\mathcal{S}(\mathcal{O})$. Denoting by $\delta$ the generating
derivation $\alpha^{t}=\mathrm{e}^{t\delta}$, there is a norm-continuous map
$\mathcal{O}\ni V=V^{\ast}\mapsto\omega_{V}\in\mathcal{S}(\mathcal{O})$, such
that $\omega_{0}=\omega$ and $\omega_{V}$ is $(\alpha_{V},\beta)$-KMS, where
$\alpha_{V}$ is generated by $\delta+\mathrm{i}[V,\,\cdot\,]$. We shall say
that the $C^{\ast}$-dynamics $\alpha_{V}$ is a local perturbation of $\alpha$.

Dynamical characterizations of KMS states are of particular relevance in the
context of the present work; see [BR81, Section 5.4.2]. Indeed, the problem of
approach to equilibrium can be viewed as an ultimate step in such
characterizations. More generally, the foundations of algebraic quantum
statistical mechanics are described in the classical monographs [BR87, BR81,
Haa96, Isr79, Sim93, Rue69, Thi02]; for a more modern perspective see [DJP03,
JOPP10, Pil06]. From those foundations stem the two directions of research
that were actively pursued over the last twenty-five years.

> 1\. Return to Equilibrium. The formulation of this property of quantum
> dynamical systems and the first basic results go back to the seminal works
> of Robinson [Rob73, Rob76]; [BM83, AM03, JP96, JP97, BFS00, JP01a, BF02,
> DJ03, FMSU03, FM04, DRK11] is an incomplete list of references of the
> follow-up works on the subject. Among several equivalent formulations of
> return to equilibrium we choose here the following one. The quantum
> dynamical system $(\mathcal{O},\alpha,\omega)$, in thermal equilibrium at
> inverse temperature $\beta$, is said to have the property of return to
> equilibrium if, for all $V=V^{\ast}\in\mathcal{O}$ and all
> $A\in\mathcal{O}$,
>
>
> $\lim_{T\rightarrow\infty}\frac{1}{T}\int_{0}^{T}\omega_{V}\circ\alpha^{t}(A)\mathrm{d}t=\omega(A).$
>
> In the setting of classical dynamical systems, the property of return to
> equilibrium is equivalent to ergodicity, and its importance/relevance in
> quantum statistical mechanics parallels the classical one.

> 2\. Non-Equilibrium Steady States and Entropy Production. This topic was
> implicit in some early works in algebraic quantum statistical mechanics; see
> in particular [PW78]. Its modern formulation goes back to [Rue00, Rue01,
> JP01b] and builds on the notions of non-equilibrium steady states and
> entropy production in classical statistical mechanics introduced in early
> 1990’s; see [Rue99] for references and additional information. To describe
> this topic, consider a quantum dynamical system
> $(\mathcal{O},\alpha,\omega)$ where $\omega$ is $\alpha$-invariant but not
> $(\alpha,\beta)$-KMS for any $\beta\in\mathbb{R}$. The Non-Equilibrium
> Steady States (NESS) of $(\mathcal{O},\alpha,\omega)$ associated to
> $V=V^{\ast}\in\mathcal{O}$ are the $\mathrm{weak}^{\ast}$-limit points of
> the net
>
>
> $\left\\{\frac{1}{T}\int_{0}^{T}\omega\circ\alpha_{V}^{t}\mathrm{d}t\right\\}_{T>0}$
>
> as $T\uparrow\infty$. The set of NESS is non-empty and its elements are
> $\alpha_{V}$-invariant. To define the entropy production of a NESS, we
> suppose that there exists a reference $C^{\ast}$-dynamics
> $\varsigma_{\omega}$ such that $\omega$ is a $(\varsigma_{\omega},-1)$-KMS
> state. Let $\delta_{\omega}$ be the generator of $\varsigma_{\omega}$, and
> suppose that $V$ is in the domain of $\delta_{\omega}$. The associated
> entropy production observable is $\sigma_{V}=\delta_{\omega}(V)$. The
> entropy production rate of a NESS $\omega_{+}$ is the real number
> $\omega_{+}(\sigma_{V})$. The pertinence of this definition stems from the
> entropy balance equation of [PW78, Rue78, JP01b]:
>
>
> $S(\omega\circ\alpha_{V}^{T}|\omega)=\int_{0}^{T}\omega\circ\alpha_{V}^{t}(\sigma_{V})\mathrm{d}t,$
> (1)
>
> where
> $S:\mathcal{S}(\mathcal{O})\times\mathcal{S}(\mathcal{O})\rightarrow[0,\infty]$
> is Araki’s relative entropy functional.444Our sign and ordering conventions
> are such that, for density matrices $\rho_{1},\rho_{2}$,
> $S(\rho_{1}|\rho_{2})=\mathop{\mathrm{tr}}\nolimits(\rho_{1}(\log\rho_{1}-\log\rho_{2}))$.
> We refer the reader to [OP93] for an in depth discussion of relative
> entropy. In particular, the sign of $S$ and the definition of NESS
> immediately give that $\omega_{+}(\sigma_{V})\geq 0$. The works [HA00,
> Pil01, JP01b, JP02a, JP02b, FMU03, JP03, MO03, TM03, Oga04, AJPP06, JOP06a,
> JOP06c, JOP06b, Pil06, AJPP07, JOP07, MMS07a, MMS07b, DR09, JOPP10, JOPS12]
> build on these starting points and develop the structural theory of NESS and
> entropy production, including studies of several classes of concrete
> physically relevant models.

In this paper we initiate a third direction of research that also originates
in the early foundations of algebraic quantum statistical mechanics, and
closes the circle of ideas and techniques introduced in the study of the above
two directions.

> 3\. The problem of approach to equilibrium in quantum statistical mechanics.
> Consider a quantum dynamical system $(\mathfrak{A},\alpha_{\Phi},\omega)$,
> where $\mathfrak{A}$ is the $C^{\ast}$-algebra of a system of lattice spins
> or fermions, $\alpha_{\Phi}$ is the $C^{\ast}$-dynamics generated by a
> sufficiently regular translation invariant interaction $\Phi$, and $\omega$
> is a translation invariant state. The Equilibrium Steady States (ESS) of
> $(\mathfrak{A},\alpha_{\Phi},\omega)$ are the $\mathrm{weak}^{\ast}$-limit
> points of the net
>
>
> $\left\\{\frac{1}{T}\int_{0}^{T}\omega\circ\alpha_{\Phi}^{t}\mathrm{d}t\right\\}_{T>0}$
>
> as $T\uparrow\infty$. The set of ESS is non-empty and its elements are
> $\alpha_{\Phi}$-invariant. The aim is to develop the structural theory of
> ESS and eventually determine the conditions under which ESS are KMS-states
> for $\alpha_{\Phi}$.

In this work we examine the structural aspects of the problem of approach to
equilibrium in the special case where $\omega$ is a KMS state for some
interaction $\Psi$. We show that, except in trivial cases, the process of
approach to equilibrium is accompanied by a strict increase of the specific
energy and entropy. To establish that, we introduce the concept of weak
Gibbsianity in quantum statistical mechanics and adapt the idea of entropy
balance to the approach to equilibrium setting.

The remaining parts of this article are organized as follows. In Section 2.1,
we give a telegraphic review of quantum spin and fermion systems. Weak Gibbs
states and the related entropy balance equation are discussed in Section 2.2.
Our main results are stated in Section 2.3 and discussed in Section 2.4.

The proofs are given in Section 3.

In the follow-up work [JPT22], we use the same general ingredients (a) and (b)
above to examine the status of the adiabatic theorem in the translation
invariant setting of infinitely extended lattice quantum spin and fermionic
systems.

#### Acknowledgments

This work was supported by the French Agence Nationale de la Recherche, grant
NONSTOPS (ANR-17-CE40-0006-01, ANR-17-CE40-0006-02, ANR-17-CE40-0006-03) and
CY Initiative of Excellence, Investissements d’Avenir program (grant
ANR-16-IDEX-0008). It was partly developed during VJ’s stay at the CY Advanced
Studies, whose support is gratefully acknowledged. VJ also acknowledges the
support of NSERC. The authors wish to thank Laurent Bruneau and Aernout van
Enter for useful discussions, and Luc Rey-Bellet for pointing them to the
reference [Mor20].

##  2 Setting and main results

###  2.1 Quantum spin and fermion systems

#### The spin $\boldsymbol{C^{\ast}}$-algebra.

Let $\mathcal{H}$ be the finite dimensional Hilbert space of a single spin and
consider the lattice $\mathbb{Z}^{d}$ equipped with the norm
$|x|=\sum_{j=1}^{d}|x_{j}|$. Let $\mathcal{F}$ be the collection of all finite
subsets of $\mathbb{Z}^{d}$. We denote by
$\mathop{\mathrm{diam}}\nolimits(X)=\max\\{|x-y|\mid x,y\in X\\}$ the diameter
of $X\in\mathcal{F}$, and write $\mathcal{H}_{X}=\bigotimes_{x\in
X}\mathcal{H}_{x}$, where $\mathcal{H}_{x}=\mathcal{H}$, and
$\mathfrak{A}_{X}=\mathcal{B}(\mathcal{H}_{X})$.555$\mathcal{B}(\mathcal{H})$
denotes the $C^{\ast}$-algebra of all linear operators on the Hilbert space
$\mathcal{H}$. If $Y\subset X$, we identify in a natural way
$\mathfrak{A}_{Y}$ with a subalgebra of $\mathfrak{A}_{X}$. For
$X\subset\mathbb{Z}^{d}$, we denote by ${\mathfrak{A}}_{X}$ the inductive
limit $C^{\ast}$-algebra over the family $\\{\mathfrak{A}_{Y}\\}_{X\supset
Y\in\mathcal{F}}$. The kinematical algebra of the spin system is
${\mathfrak{A}}_{\mathbb{Z}^{d}}$ and in the sequel we will omit the subscript
$\mathbb{Z}^{d}$. The algebra $\mathfrak{A}$ is simple, which implies in
particular that any KMS state on $\mathfrak{A}$ is faithful; see [Isr79, Lemma
III.3.3].

For any $X\subset\mathbb{Z}^{d}$, $X^{c}=\mathbb{Z}^{d}\setminus X$, we have
the identification $\mathfrak{A}=\mathfrak{A}_{X}\otimes\mathfrak{A}_{X^{c}}$,
and
$\mathop{\mathrm{tr}}\nolimits_{X}:\mathfrak{A}\mapsto\mathfrak{A}_{X^{c}}$
denotes the usual normalized partial trace. An element $A\in{\mathfrak{A}}$ is
called local if $A\in{\mathfrak{A}}_{X}$ for some $X\in\mathcal{F}$. The
minimal $X$ for which this hold is called the support of $A$ and is denoted by
$\mathop{\mathrm{supp}}\nolimits(A)$. The set ${\mathfrak{A}_{\mathrm{loc}}}$
of local elements is a dense $\ast$-subalgebra of $\mathfrak{A}$.

In the following $\Lambda$ will always denote a cube in $\mathbb{Z}^{d}$
centered at the origin and $\Lambda\uparrow\mathbb{Z}^{d}$ denotes the limit
over an increasing family of such cubes.

#### Translation invariant states.

The spin algebra $\mathfrak{A}$ is uniformly asymptotically Abelian w.r.t. the
natural group action $\varphi:\mathbb{Z}^{d}\ni
x\mapsto\varphi^{x}\in\mathrm{Aut}(\mathfrak{A})$, i.e.,

$\lim_{|x|\rightarrow\infty}\|[\varphi^{x}(A),B]\|=0$

for all $A,B\in\mathfrak{A}$. We denote by
$\mathcal{S}_{\mathrm{I}}(\mathfrak{A})$ the set of all translation invariant,
i.e., $\varphi$-invariant states on $\mathfrak{A}$.
$\mathcal{S}_{\mathrm{I}}(\mathfrak{A})$ is a $\mathrm{weak}^{\ast}$-compact
convex subset of $\mathcal{S}(\mathfrak{A})$ and a Choquet simplex.

#### Specific entropies.

The restriction $\nu_{\Lambda}$ of a state
$\nu\in\mathcal{S}_{\mathrm{I}}(\mathfrak{A})$ to $\mathfrak{A}_{\Lambda}$ is
represented by a density matrix in $\mathfrak{A}_{\Lambda}$ which we denote by
the same letter: For any $A\in\mathfrak{A}_{\Lambda}$,

$\nu(A)=\nu_{\Lambda}(A)=\mathop{\mathrm{tr}}\nolimits(\nu_{\Lambda}A).$

The von Neumann entropy of $\nu_{\Lambda}$ is defined by
$S(\nu_{\Lambda})=-\mathop{\mathrm{tr}}\nolimits(\nu_{\Lambda}\log\nu_{\Lambda})$
and is non-negative. The specific entropy of $\nu$ is given by the limit

$s(\nu)=\lim_{\Lambda\uparrow\mathbb{Z}^{d}}\frac{S(\nu_{\Lambda})}{|\Lambda|},$
(2)

which exists and defines an affine, $\mathrm{weak}^{\ast}$-upper-
semicontinuous map $\mathcal{S}_{\mathrm{I}}(\mathfrak{A})\ni\nu\mapsto
s(\nu)$, taking its values in $[0,\log\dim\mathcal{H}]$; see [BR81,
Proposition 6.2.38].

The specific relative entropy of $\nu$ w.r.t.
$\omega\in\mathcal{S}_{\mathrm{I}}(\mathfrak{A})$ is defined by

$s(\nu|\omega)=\lim_{\Lambda\uparrow\mathbb{Z}^{d}}\frac{S(\nu_{\Lambda}|\omega_{\Lambda})}{|\Lambda|},$

whenever the limit exists, in which case we have $s(\nu|\omega)\geq 0$.

#### Interactions.

An interaction is a family $\Phi=\\{\Phi(X)\\}_{X\in\mathcal{F}}$, where
$\Phi(X)\in\mathfrak{A}_{X}$ is self-adjoint. The interaction $\Phi$ is
translation invariant if

$\varphi^{x}(\Phi(X))=\Phi(X+x)$

holds for all $X\in\mathcal{F}$ and $x\in\mathbb{Z}^{d}$. In what follows we
consider only translation invariant interactions. The local Hamiltonians
associated to $\Phi$ are defined by

$H_{\Lambda}(\Phi)=\sum_{X\subset\Lambda}\Phi(X).$

For $r>0$ we denote by $\mathcal{B}^{r}$ the set of all translation invariant
interactions such that

$\|\Phi\|_{r}=\sum_{X\ni 0}\mathrm{e}^{r(|X|-1)}\|\Phi(X)\|<\infty,$ (3)

where $|X|$ denotes the cardinality of $X\in\mathcal{F}$. The pair
$(\mathcal{B}^{r},\|\cdot\|_{r})$ is a Banach space. An interaction $\Phi$ is
called finite range whenever, for some $m\in\mathbb{N}$, ${\rm diam}(X)>m$
implies $\Phi(X)=0$. The set $\mathcal{B}_{\mathrm{f}}$ of all finite range
interactions is a dense subspace of $\mathcal{B}^{r}$.

In what follows we restrict ourselves to interactions in $\mathcal{B}^{r}$
although many of the stated results hold in more general settings.

#### The Gibbs variational principle.

[BR81, Proposition 6.2.39 and Theorem 6.2.40] For any interaction
$\Phi\in\mathcal{B}^{r}$ the limit

$P(\Phi)=\lim_{\Lambda\uparrow\mathbb{Z}^{d}}\frac{1}{|\Lambda|}\log\mathop{\mathrm{tr}}\nolimits(\mathrm{e}^{-H_{\Lambda}(\Phi)})$
(4)

exists and is finite. The function $P$ defined in this way is called the
pressure. Setting

$E_{\Phi}=\sum_{X\ni 0}\frac{\Phi(X)}{|X|}\in\mathfrak{A},$

one has

$\lim_{\Lambda\uparrow\mathbb{Z}^{d}}\frac{1}{|\Lambda|}\left\|H_{\Lambda}(\Phi)-\sum_{x\in\Lambda}\varphi^{x}(E_{\Phi})\right\|=0,$
(5)

and in particular

$\lim_{\Lambda\uparrow\mathbb{Z}^{d}}\frac{1}{|\Lambda|}\omega\left(H_{\Lambda}(\Phi)\right)=\omega(E_{\Phi}),$
(6)

for any $\omega\in\mathcal{S}_{\mathrm{I}}(\mathfrak{A})$. Thus, $E_{\Phi}$
can be considered as the specific energy observable of the interaction $\Phi$.

######  Theorem 2.1 (Gibbs variational principle).

For any $\beta\in\mathbb{R}$ and $\Phi\in\mathcal{B}^{r}$,

$P(\beta\Phi)=\sup_{\nu\in\mathcal{S}_{\mathrm{I}}(\mathfrak{A})}\left(s(\nu)-\beta\nu(E_{\Phi})\right).$

Moreover,

$\mathcal{S}_{\mathrm{eq}}(\beta\Phi)=\left\\{\nu\in\mathcal{S}_{\mathrm{I}}(\mathfrak{A})\mid
P(\beta\Phi)=s(\nu)-\beta\nu(E_{\Phi})\right\\}$

is a non-empty convex compact subset of
$\mathcal{S}_{\mathrm{I}}(\mathfrak{A})$.

The elements of $\mathcal{S}_{\mathrm{eq}}(\beta\Phi)$ are called equilibrium
states for the interaction $\beta\Phi$.

#### Dynamics.

[Rue67, Theorem 7.6.2] For an interaction $\Phi\in\mathcal{B}^{r}$ and a
finite cube $\Lambda$, set

$\alpha_{\Phi,\Lambda}^{t}(A)=\mathrm{e}^{\mathrm{i}tH_{\Lambda}(\Phi)}A\mathrm{e}^{-\mathrm{i}tH_{\Lambda}(\Phi)}$

with $t\in\mathbb{R}$ and $A\in{\mathfrak{A}_{\mathrm{loc}}}$. Then, the limit

$\alpha_{\Phi}^{t}(A)=\lim_{\Lambda\uparrow\mathbb{Z}^{d}}\alpha_{\Phi,\Lambda}^{t}(A)$

exists and is uniform for $t$ in compact sets. Moreover, the family of maps
$\alpha_{\Phi}=\\{\alpha_{\Phi}^{t}\mid t\in\mathbb{R}\\}$, originally defined
on ${\mathfrak{A}_{\mathrm{loc}}}$, extends uniquely to a $C^{\ast}$-dynamics
on $\mathfrak{A}$. The group actions $\alpha_{\Phi}$ and $\varphi$ commute.

We recall the following basic result [BR81, Theorem 6.2.42].

######  Theorem 2.2.

Suppose that $\Phi\in\mathcal{B}^{r}$. Then, for any $\beta\in\mathbb{R}$,
$\mathcal{S}_{\mathrm{eq}}(\beta\Phi)$ coincides with the set of all
translation invariant $(\alpha_{\Phi},\beta)$-KMS states.

The proofs of the above results give additional information. In particular,
the following result follows from the proofs of [Rue69, Lemma 7.6.1 and
Theorem 7.6.2] or [BR81, Theorem 6.2.4]:

######  Theorem 2.3.

Suppose that $\Phi\in\mathcal{B}^{r}$. Then, for all
$A\in{\mathfrak{A}_{\mathrm{loc}}}$, the map

$\mathbb{R}\ni t\mapsto\alpha_{\Phi}^{t}(A)\in\mathfrak{A}$

has an analytic extension to the strip

$|\mathop{\mathrm{Im}}\nolimits z|<\frac{r}{2\|\Phi\|_{r}},$

and for $z$ in this strip

$\|\alpha_{\Phi}^{z}(A)\|\leq\|A\|\mathrm{e}^{r|\mathop{\mathrm{supp}}\nolimits(A)|}\frac{r}{r-2\|\Phi\|_{r}|\mathop{\mathrm{Im}}\nolimits
z|}.$

Remark. According to [BR81, Theorem 6.2.4], the conclusions of Theorem 2.3
still hold for interactions $\Phi$ which lack translation invariance, provided
the norm (3) is replaced by

$\|\Phi\|_{r}=\sum_{N\geq 1}\sup_{x\in\mathbb{Z}^{d}}\sum_{X\ni
x\atop|X|=N}\|\Phi(X)\|\mathrm{e}^{r(|X|-1)}<\infty.$

#### Setting $\boldsymbol{\beta=1}$.

To reduce the number of parameters, it is convenient to absorb $\beta$ into
the interaction $\Phi$. With this convention small interactions correspond to
high temperatures. We will comment further on this point at the end of Section
2.3.

In the remaining part of the paper we set $\beta=1$. $(\alpha_{\Phi},1)$-KMS
state will be abbreviated as $\alpha_{\Phi}$-KMS state.

#### Uniqueness of equilibrium state.

######  Theorem 2.4.

Suppose that $\Phi\in\mathcal{B}^{r}$ and that either:

  1. (1)

$d=1$ and $\displaystyle\sum_{X\ni 0}{\rm diam}(X)\|\Phi(X)\|<\infty$.

  2. (2)

$d\geq 1$, $r>\log\dim\mathcal{H}$, and
$\displaystyle\|\Phi\|_{r}<\frac{1}{2\dim\mathcal{H}}$.

Then $\mathcal{S}_{\mathrm{eq}}(\Phi)$ is a singleton.

Part (1) is the classical result of Araki; see [Sim93, Theorem IV.6.1].
Regarding Part (2), among the many results in the literature that establish
the uniqueness of the equilibrium state in the high temperature regime, we
have quoted here the one obtained relatively recently in [FU15].

#### The Gibbs condition.

[BR81, Proposition 6.2.17] We will only introduce the part of the Gibbs
condition that will be needed in the sequel. For an interaction
$\Phi\in\mathcal{B}^{r}$, the surface energies

$W_{\Lambda}(\Phi)=\sum_{X\cap\Lambda\not=\emptyset\atop
X\cap\Lambda^{c}\not=\emptyset}\Phi(X)$ (7)

are well defined and satisfy

$\lim_{\Lambda\uparrow\mathbb{Z}^{d}}\frac{\|W_{\Lambda}(\Phi)\|}{|\Lambda|}=0.$
(8)

######  Theorem 2.5.

Let $\omega$ be a $\alpha_{\Phi}$-KMS state, and $\omega_{-W}$ the perturbed
KMS-state with $W=W_{\Lambda}(\Phi)$. Then for any $\Lambda$ and any
$A\in\mathfrak{A}_{\Lambda}$ one has

$\omega_{-W}(A)=\frac{\mathop{\mathrm{tr}}\nolimits(\mathrm{e}^{-H_{\Lambda}(\Phi)}A)}{\mathop{\mathrm{tr}}\nolimits(\mathrm{e}^{-H_{\Lambda}(\Phi)})}.$

#### Physical equivalence.

The concept of physical equivalence was first introduced in [Roo74] as a
physical interpretation of a technical assumption from [GR71], and was further
developed in [Isr79]. Our definition below differs from the original one and
is based on [Rue78, Section 4.7]; for additional information see [vEFS93,
Section 2.4.6].

######  Definition 2.6.

Two interactions $\Phi,\Psi\in\mathcal{B}^{r}$ are called physically
equivalent, denoted $\Phi\sim\Psi$, if

$\lim_{\Lambda\uparrow\mathbb{Z}^{d}}\frac{1}{|\Lambda|}\sum_{x\in\Lambda}\varphi^{x}(E_{\Phi-\Psi})=\mathop{\mathrm{tr}}\nolimits(E_{\Phi-\Psi})\mathbbm{1}.$

Notice that $E_{\Phi-\Psi}=E_{\Phi}-E_{\Psi}$. Physical equivalence is
obviously an equivalence relation on $\mathcal{B}^{r}$. Moreover, if
$\Phi_{1}\sim\Psi_{1}$ and $\Phi_{2}\sim\Psi_{2}$, then
$a\Phi_{1}+b\Phi_{2}\sim a\Psi_{1}+b\Psi_{2}$ for any $a,b\in\mathbb{R}$.

######  Theorem 2.7.

Let $\Phi,\Psi\in\mathcal{B}^{r}$. The following statements are equivalent:

  1. (a)

$\Phi\sim\Psi$.

  2. (b)

$\displaystyle\lim_{\Lambda\uparrow\mathbb{Z}^{d}}\frac{1}{|\Lambda|}\big{(}H_{\Lambda}(\Phi)-H_{\Lambda}(\Psi)\big{)}=\mathop{\mathrm{tr}}\nolimits(E_{\Phi-\Psi})\mathbbm{1}$.

  3. (c)

The pressure $P$ is linear on the line segment connecting $\Phi$ and $\Psi$.

  4. (d)

$\mathcal{S}_{\mathrm{eq}}(\Phi)\cap\mathcal{S}_{\mathrm{eq}}(\Psi)\not=\emptyset$.

  5. (e)

$\mathcal{S}_{\mathrm{eq}}(\Phi)=\mathcal{S}_{\mathrm{eq}}(\Psi)$.

  6. (f)

$\alpha_{\Phi}=\alpha_{\Psi}$.

  7. (g)

$\displaystyle\sum_{X\in\mathcal{F}}[\Phi(X)-\Psi(X),A]=0$ for all
$A\in{\mathfrak{A}_{\mathrm{loc}}}$.

The equivalence between (c), (d), (e), and (f) is proved in [Isr79, Theorem
III.4.2]. We prove the remaining parts in Section 3.1 below.

#### The case of fermions.

We denote by $\mathcal{A}=\mathop{\mathrm{CAR}}\nolimits(\mathfrak{h})$ the
CAR algebra over the Hilbert space $\mathfrak{h}$. As usual,
$a^{\ast}(f)/a(f)$ are the creation/annihilation operators associated to
$f\in\mathfrak{h}$, and $a^{\\#}$ stands for either $a$ or $a^{\ast}$. Given a
unitary operator $u$ on $\mathfrak{h}$, the map $a^{\\#}(f)\mapsto
a^{\\#}(uf)$ uniquely extends to a $\ast$-automorphism $\alpha_{u}$ of
$\mathcal{A}$, the Bogoliubov automorphism generated by $u$. A strongly
continuous one-parameter group of unitaries $t\mapsto u^{t}$ on $\mathfrak{h}$
generates a strongly continuous one-parameter subgroup of
$\mathrm{Aut}(\mathcal{A})$. The gauge group of $\mathcal{A}$ is the group of
Bogoliubov automorphisms $\theta\mapsto\vartheta^{\theta}$ generated by
$\mathrm{e}^{\mathrm{i}\theta\mathbbm{1}}$. Physical observables are gauge
invariant, i.e., elements of

$\mathfrak{A}(\mathfrak{h})=\\{A\in\mathcal{A}\mid\vartheta^{\theta}(A)=A\text{
for all }\theta\in\mathbb{R}\\},$

which coincides with the $C^{\ast}$-algebra generated by
$\\{a^{\ast}(f)a(g)\mid f,g\in\mathfrak{h}\\}\cup\\{\mathbbm{1}\\}$. We will
only consider the case $\mathfrak{h}=\ell^{2}(\mathbb{Z}^{d})$ and write
$\mathfrak{A}$ for $\mathfrak{A}(\ell^{2}(\mathbb{Z}^{d}))$. For
$X\subset\mathbb{Z}^{d}$ we set $\mathfrak{A}_{X}=\mathfrak{A}(\ell^{2}(X))$
and identify $\mathfrak{A}_{X}$ with the $C^{*}$-subalgebra of $\mathfrak{A}$
generated by

$\\{a^{\ast}(f)a(g)\mid
f,g\in\ell^{2}(\mathbb{Z}^{d}),\mathop{\mathrm{supp}}\nolimits(f)\cup\mathop{\mathrm{supp}}\nolimits(g)\subset
X\\}\cup\\{\mathbbm{1}\\}.$

Note that if $X\cap Y=\emptyset$, then $\mathfrak{A}_{X}$ and
$\mathfrak{A}_{Y}$ commute. The natural unitary group action of
$\mathbb{Z}^{d}$ on $\ell^{2}(\mathbb{Z}^{d})$ gives rise to a group of
Bogoliubov automorphisms $\varphi^{x}$ such that
$\varphi^{x}(\mathfrak{A}_{X})=\mathfrak{A}_{X+x}$.

For $X\in\mathcal{F}$,
$\mathcal{A}_{X}=\mathop{\mathrm{CAR}}\nolimits(\ell^{2}(X))$ is isomorphic to
the full matrix-algebra $\mathcal{B}(\mathbb{C}^{2^{|X|}})$, so that
$\mathcal{A}=\mathop{\mathrm{CAR}}\nolimits(\ell^{2}(\mathbb{Z}^{d}))$ has a
unique tracial state $\mathop{\mathrm{tr}}\nolimits$ as the extension of the
unique tracial states of $\mathcal{A}_{X}$, $X\in\mathcal{F}$. $\mathcal{A}$
is isomorphic to the spin algebra, with the main difference that the
identification of $\mathcal{A}$ with
$\mathcal{A}_{X}\otimes\mathcal{A}_{X^{c}}$ does not hold anymore.
Nevertheless, there exists an analogue of partial trace for fermions called
conditional expectation in [AM03, Theorem 4.7]: for $X\subset\mathbb{Z}^{d}$,
there exists a projection $\mathcal{T}_{X}:\mathcal{A}\to\mathcal{A}_{X}$ such
that
$\mathop{\mathrm{tr}}\nolimits(AB)=\mathop{\mathrm{tr}}\nolimits(\mathcal{T}_{X}(A)B)$
for any $B\in\mathcal{A}_{X}$.

All the above definitions and results stated for the spin algebra extend to
the gauge-invariant sector $\mathfrak{A}$ of the fermionic algebra
$\mathcal{A}$. Most, but not all, of these extensions are straightforward: for
the Gibbs variational principle and the Gibbs condition see [AM03, Theorems
11.4 and 7.5]. In the fermionic case there is also a special one-body
interaction $N\in\mathcal{B}_{\mathrm{f}}$, related to gauge invariance, and
given by

$N(X)=\begin{cases}a_{x}^{\ast}a_{x}&\text{ if }X=\\{x\\};\\\ 0&\text{
otherwise}.\end{cases}$

We denote $N_{\Lambda}=\sum_{X\subset\Lambda}N(X)$ the local Hamiltonians
associated to $N$.666The interaction $N$ generates the gauge group,
$\vartheta^{\theta}=\alpha_{N}^{\theta}$. For $\omega\in S_{I}(\mathfrak{A})$,

$\lim_{\Lambda\uparrow\mathbb{Z}^{d}}\dfrac{1}{|\Lambda|}\omega(N_{\Lambda})=\omega(E_{N}),$

where $E_{N}=a^{\ast}_{0}a_{0}$. The perturbed interactions $\Phi-\mu N$
introduces a chemical potential $\mu$.

To summarize, all results of this paper apply to both spin and fermions, and
so in the following $\mathfrak{A}$ refers equivalently to the spin or the
gauge-invariant sector of the CAR algebra, unless stated explicitly.

We finish this section with:

#### Conservation laws.

######  Theorem 2.8.

For any $\omega\in\mathcal{S}_{\mathrm{I}}(\mathfrak{A})$ and any
$\Phi\in\mathcal{B}^{r}$ the following hold for all $t\in\mathbb{R}$:

  1. (1)

$s(\omega)=s(\omega\circ\alpha_{\Phi}^{t})$.

  2. (2)

$\omega(E_{\Phi})=\omega\circ\alpha_{\Phi}^{t}(E_{\Phi})$.

  3. (3)

In the fermionic case, $\omega(E_{N})=\omega\circ\alpha_{\Phi}^{t}(E_{N})$.

Part (1) is due to [LR68]. The proofs of the remaining statements are given in
Section 3.2.

###  2.2 Weak Gibbs states and regularity

We set

$\omega_{\Phi,\Lambda}^{\mathrm{c}}=\mathrm{e}^{-H_{\Lambda}(\Phi)-P_{\Lambda}(\Phi)},\qquad
P_{\Lambda}(\Phi)=\log\mathop{\mathrm{tr}}\nolimits(\mathrm{e}^{-H_{\Lambda}(\Phi)}).$

######  Definition 2.9.

A state $\omega\in\mathcal{S}_{\mathrm{I}}(\mathfrak{A})$ is called weak Gibbs
for the interaction $\Phi\in\mathcal{B}^{r}$ if there exist constants
$c_{\Lambda}>0$ satisfying

$\lim_{\Lambda\uparrow\mathbb{Z}^{d}}\frac{1}{|\Lambda|}c_{\Lambda}=0,$ (9)

and such that, for all $\Lambda$,

$\mathrm{e}^{-c_{\Lambda}}\omega_{\Phi,\Lambda}^{\mathrm{c}}\leq\omega_{\Lambda}\leq\mathrm{e}^{c_{\Lambda}}\omega_{\Phi,\Lambda}^{\mathrm{c}}.$
(10)

$\omega$ is called log weak Gibbs for $\Phi$ if there exist constants
$c_{\Lambda}>0$ satisfying (9) and such that, for all $\Lambda$,

$-c_{\Lambda}+\log\omega_{\Phi,\Lambda}^{\mathrm{c}}\leq\log\omega_{\Lambda}\leq
c_{\Lambda}+\log\omega_{\Phi,\Lambda}^{\mathrm{c}}.$ (11)

We denote by $\mathrm{WG}(\Phi)$ the set of all weak Gibbs states for $\Phi$,
and by $\mathrm{LWG}(\Phi)$ the set of all log weak Gibbs states.

The next proposition, whose proof is given in Section 3.3, collects a few
elementary properties of the sets $\mathrm{WG}(\Phi)$ and
$\mathrm{LWG}(\Phi)$.

######  Proposition 2.10.

For $\Phi,\Psi\in\mathcal{B}^{r}$, the following hold:

  1. (1)

$\omega\in\mathrm{LWG}(\Phi)$ $\Longleftrightarrow$
$\displaystyle\limsup_{\Lambda\uparrow\mathbb{Z}^{d}}|\Lambda|^{-1}\|\log\omega_{\Lambda}-\log\omega_{\Phi,\Lambda}^{c}\|=0$.

  2. (2)

$\mathrm{WG}(\Phi)\subset\mathrm{LWG}(\Phi)\subset\mathcal{S}_{\mathrm{eq}}(\Phi)$.

  3. (3)

Either $\mathrm{LWG}(\Phi)=\mathrm{LWG}(\Psi)$ or
$\mathrm{LWG}(\Phi)\cap\mathrm{LWG}(\Psi)=\emptyset$.

  4. (4)

$\Phi\sim\Psi$ $\Longrightarrow$ $\mathrm{LWG}(\Phi)=\mathrm{LWG}(\Psi)$.
Moreover, if $\mathrm{LWG}(\Phi)=\mathrm{LWG}(\Psi)\neq\emptyset$, then
$\Phi\sim\Psi$.

Regarding possible equalities in Part (2), one has the following results.

######  Theorem 2.11.

Suppose that either:

  1. (a)

$d=1$ and $\Phi\in\mathcal{B}_{\mathrm{f}}$;

  2. (b)

$d\geq 1$, $\Phi\in\mathcal{B}^{r}$, and $\|\Phi\|_{r}<r$.

Then $\mathrm{WG}(\Phi)=\mathrm{LWG}(\Phi)=\mathcal{S}_{\mathrm{eq}}(\Phi)$.

Section 3.4 is devoted to the proof which is based on general results from
[Ara69] and [AI74, Section 3.1 and proof of Theorem 2], see also [Ara76] and
[LRB05].

The following characterization of $\mathrm{WG}(\Phi)$, based on the Gibbs
condition, is proved in Section 3.5.

######  Proposition 2.12.

Suppose that $\Phi\in\mathcal{B}^{r}$ and let
$\omega\in\mathcal{S}_{\mathrm{eq}}(\Phi)$. Set

$\underline{d}_{\Lambda}=\inf_{A\in\mathfrak{A}_{\Lambda}\atop
A>0}\frac{\omega(A)}{\omega_{-W_{\Lambda}(\Phi)}(A)},\qquad\overline{d}_{\Lambda}=\sup_{A\in\mathfrak{A}_{\Lambda}\atop
A>0}\frac{\omega(A)}{\omega_{-W_{\Lambda}(\Phi)}(A)}.$

Then $\omega\in\mathrm{WG}(\Phi)$ iff

$\lim_{\Lambda\uparrow\mathbb{Z}^{d}}\frac{1}{|\Lambda|}\log\underline{d}_{\Lambda}=\lim_{\Lambda\uparrow\mathbb{Z}^{d}}\frac{1}{|\Lambda|}\log\overline{d}_{\Lambda}=0.$
(12)

We finish this brief discussion of quantum weak Gibbs states with a
characterization of $\mathrm{LWG}(\Phi)$.

######  Proposition 2.13.

Suppose that $\Phi\in\mathcal{B}^{r}$ and let
$\omega\in\mathcal{S}_{\mathrm{eq}}(\Phi)$.

  1. (1)

For any $\nu\in\mathcal{S}(\mathfrak{A})$ and all $\Lambda\in\mathcal{F}$,

$S(\nu_{\Lambda}|\omega_{\Lambda})-S(\nu_{\Lambda}|\omega_{\Phi,\Lambda}^{\mathrm{c}})\leq
2\|W_{\Lambda}(\Phi)\|.$

  2. (2)

For $\Lambda\in\mathcal{F}$, set

$D_{\Lambda}=\inf_{\nu\in\mathcal{S}(\mathfrak{A})}\left(S(\nu_{\Lambda}|\omega_{\Lambda})-S(\nu_{\Lambda}|\omega_{\Phi,\Lambda}^{\mathrm{c}})\right).$

Then $\omega\in\mathrm{LWG}(\Phi)$ iff

$\lim_{\Lambda\uparrow\mathbb{Z}^{d}}\frac{D_{\Lambda}}{|\Lambda|}=0.$ (13)

Part (1) is easily seen to be equivalent to the statement that for all
$\Lambda\in\mathcal{F}$,

$\log\omega_{\Phi,\Lambda}^{\mathrm{c}}-\log\omega_{\Lambda}\leq
2\|W_{\Lambda}(\Phi)\|,$

which is proven in [HP93, Lemma 2.3]. With $D_{\Lambda}$ as defined in Part
(2), one has

$D_{\Lambda}\leq\log\omega_{\Phi,\Lambda}^{\mathrm{c}}-\log\omega_{\Lambda},$

so that Part (2) is an immediate consequence of Part (1), Relation (8) and the
definition of log weak Gibbs states.

For our purposes, the importance of weak Gibbs states stems from the following
result that is an immediate consequence of Definition 2.9.

######  Proposition 2.14.

Let $\Phi\in\mathcal{B}^{r}$ and $\omega\in\mathrm{LWG}(\Phi)$. Then for any
$\nu\in\mathcal{S}_{\mathrm{I}}(\mathfrak{A})$,

$s(\nu|\omega)=-s(\nu)+\nu(E_{\Phi})+P(\Phi).$ (14)

In particular, $s(\nu|\omega)=0$ iff $\nu\in\mathcal{S}_{\mathrm{eq}}(\Phi)$.

We will comment further on weak Gibbsianity and regularity in Section 2.4.

Since the validity of (14) plays a central role in our arguments, we single it
out in:

######  Definition 2.15.

A pair
$(\omega,\Phi)\in\mathcal{S}_{\mathrm{I}}(\mathfrak{A})\times\mathcal{B}^{r}$
is called regular whenever Relation (14) holds for any
$\nu\in\mathcal{S}_{\mathrm{I}}(\mathfrak{A})$.

Remarks. 1\. Obviously, if the pair $(\omega,\Phi)$ is regular, then
$\omega\in\mathcal{S}_{\mathrm{eq}}(\Phi)$. If, in addition,
$\mathcal{S}_{\mathrm{eq}}(\Phi)=\\{\omega\\}$, then $\omega$ is
$\varphi$-ergodic since it is extreme point of the simplex
$\mathcal{S}_{\mathrm{I}}(\mathfrak{A})$ (see [BR81, Theorem 6.2.44] and its
proof). Using the regularity assumption, one can also argue in the following
way. Suppose that $\omega=\alpha\mu+(1-\alpha)\nu$ for some
$\mu,\nu\in\mathcal{S}_{\mathrm{I}}(\mathfrak{A})$ and $\alpha\in]0,1[$. The
joint convexity of the relative entropy [OP93, Theorem 1.4] yields

$\displaystyle 0=s(\omega|\omega)$ $\displaystyle\leq\alpha
s(\mu|\omega)+(1-\alpha)s(\nu|\omega)$
$\displaystyle=\alpha(-s(\mu)+\mu(E_{\Phi})+P(\Phi))+(1-\alpha)(-s(\nu)+\nu(E_{\Phi})+P(\Phi))$
$\displaystyle=-s(\alpha\mu+(1-\alpha)\nu)+(\alpha\mu+(1-\alpha)\nu)(E_{\Phi})+P(\Phi)=s(\omega|\omega)=0,$

from which we conclude that $s(\mu|\omega)=s(\nu|\omega)=0$ and hence
$\mu,\nu\in\mathcal{S}_{\mathrm{eq}}(\Phi)=\\{\omega\\}$.

2\. For $\nu\in\mathcal{S}_{\mathrm{I}}(\mathfrak{A})$ and
$\omega\in\mathcal{S}_{\mathrm{eq}}(\Phi)$, $\Phi\in\mathcal{B}^{r}$, we set

$\bar{s}(\nu|\omega)=\limsup_{\Lambda\uparrow\mathbb{Z}^{d}}\frac{S(\nu_{\Lambda}|\omega_{\Lambda})}{|\Lambda|}.$

By Part (1) of Proposition 2.13 and Relations (2), (4) and (6),

$\bar{s}(\nu|\omega)\leq-s(\nu)+\nu(E_{\Phi})+P(\Phi).$ (15)

Hence, proving regularity reduces to establishing the lower bound

$\liminf_{\Lambda\uparrow\mathbb{Z}^{d}}\frac{S(\nu_{\Lambda}|\omega_{\Lambda})}{|\Lambda|}\geq-s(\nu)+\nu(E_{\Phi})+P(\Phi).$

Remarkably, Theorem 2.11 remains the best existing result about this estimate.

3\. In [HMO07, ORB11], the authors consider asymptotically decoupled states
$\omega\in\mathcal{S}(\mathfrak{A})$ defined by the property

$\mathrm{e}^{-c_{\Lambda}}\omega_{\Lambda}\otimes\omega_{\Lambda^{c}}\leq\omega\leq\mathrm{e}^{c_{\Lambda}}\omega_{\Lambda}\otimes\omega_{\Lambda^{c}},$

where the $c_{\Lambda}$ satisfy (9). If in addition
$\omega\in\mathcal{S}_{\mathrm{eq}}(\Phi)$, $\Phi\in\mathcal{B}^{r}$, the
proof of [HP93, Lemma 2.3] gives that $\omega\in\mathrm{LWG}(\Phi)$.

###  2.3 Main results

For $\omega\in\mathcal{S}_{\mathrm{I}}(\mathfrak{A})$ and
$\Phi\in\mathcal{B}^{r}$, we set

$\bar{\omega}_{T}=\frac{1}{T}\int_{0}^{T}\omega\circ\alpha_{\Phi}^{t}\mathrm{d}t,$
(16)

and denote by $\mathcal{S}_{+}(\omega,\Phi)$ the set of
$\mathrm{weak}^{\ast}$-limit points of the net $\\{\bar{\omega}_{T}\\}_{T>0}$
as $T\rightarrow\infty$. We will call the elements of
$\mathcal{S}_{+}(\omega,\Phi)$ Equilibrium Steady States (ESS) of
$(\mathfrak{A},\alpha_{\Phi},\omega)$. The set $\mathcal{S}_{+}(\omega,\Phi)$
is obviously non-empty. Since $\mathfrak{A}$ is separable,
$\omega_{+}\in\mathcal{S}_{+}(\omega,\Phi)$ iff there exists sequence
$T_{n}\uparrow\infty$ such that
$\omega_{+}=\lim_{n\rightarrow\infty}\bar{\omega}_{T_{n}}$. The invariance
properties stated in Theorem 2.8 survive time-averaging, see Section 3.6.

######  Proposition 2.16.

For any $\omega\in\mathcal{S}_{\mathrm{I}}(\mathfrak{A})$ and any
$\Phi\in\mathcal{B}^{r}$ the following hold for all $T\in\mathbb{R}$:

  1. (1)

$s(\omega)=s(\bar{\omega}_{T})$.

  2. (2)

$\omega(E_{\Phi})=\bar{\omega}_{T}(E_{\Phi})$.

  3. (3)

In the fermionic case, $\omega(E_{N})=\bar{\omega}_{T}(E_{N})$.

The elements of $\mathcal{S}_{+}(\omega,\Phi)$ have the following basic
properties:

######  Proposition 2.17.

Let $\omega_{+}\in\mathcal{S}_{+}(\omega,\Phi)$. Then:

  1. (1)

$\omega_{+}\in\mathcal{S}_{\mathrm{I}}(\mathfrak{A})$ and
$\omega_{+}\circ\alpha_{\Phi}^{t}=\omega_{+}$ for all $t\in\mathbb{R}$.

  2. (2)

$s(\omega)\leq s(\omega_{+})$.

  3. (3)

$\omega(E_{\Phi})=\omega_{+}(E_{\Phi})$.

  4. (4)

In the fermionic case, $\omega(E_{N})=\omega_{+}(E_{N})$.

Part (1) follows from the translation invariance of the time averaged states
$\bar{\omega}_{T}$ and a well known property on Cesàro averages. Part (2)
follows from Proposition 2.16-(2) and the upper semi-continuity of the
specific entropy. Parts (3) and (4) follows from Parts (2) and (3) of
Proposition 2.16.

We now turn to our main results.

######  Theorem 2.18.

Let $\Phi,\Psi\in\mathcal{B}^{r}$, $\omega\in\mathcal{S}_{\mathrm{eq}}(\Psi)$,
and $\omega_{+}\in\mathcal{S}_{+}(\omega,\Phi)$. Then
$\omega(E_{\Psi})\leq\omega_{+}(E_{\Psi})$.

The remaining results depend critically on the regularity assumption.

######  Theorem 2.19.

Let $\Phi,\Psi\in\mathcal{B}^{r}$. Suppose that $(\omega,\Psi)$ is a regular
pair and that $\omega_{+}\in\mathcal{S}_{+}(\omega,\Phi)$. Then the following
statements are equivalent:

  1. (a)

$\omega(E_{\Psi})=\omega_{+}(E_{\Psi})$.

  2. (b)

$\omega_{+}\in\mathcal{S}_{\mathrm{eq}}(\Psi)$ and $s(\omega)=s(\omega_{+})$.

Suppose in addition that $\omega_{+}\in\mathcal{S}_{\mathrm{eq}}(\Phi)$. Then
(a) and (b) are further equivalent to:

  1. (c)

$\Phi\sim\Psi$.

  2. (d)

$\omega=\omega_{+}$.

######  Theorem 2.20.

Let $\omega\in\mathcal{S}_{\mathrm{I}}(\mathfrak{A})$,
$\Phi\in\mathcal{B}^{r}$. For any
$\omega_{+}\in\mathcal{S}_{+}(\omega,\Phi)\cap\mathcal{S}_{\mathrm{eq}}(\Phi)$
one has

$s(\omega)=s(\omega_{+})\ \Longleftrightarrow\ \omega=\omega_{+}.$

An immediate consequence of these two theorems is:

######  Corollary 2.21.

Let $\omega\in\mathcal{S}_{\mathrm{I}}(\mathfrak{A})$,
$\Phi,\Psi\in\mathcal{B}^{r}$, $\omega_{+}\in\mathcal{S}_{+}(\omega,\Phi)$,
and suppose that $\omega$ is not $\alpha_{\Phi}$-invariant.

  1. (1)

If the pair $(\omega,\Psi)$ is regular and
$\mathcal{S}_{\mathrm{eq}}(\Psi)=\\{\omega\\}$, then
$\omega(E_{\Psi})<\omega_{+}(E_{\Psi})$.

  2. (2)

If $\omega_{+}\in\mathcal{S}_{\mathrm{eq}}(\Phi)$, then
$s(\omega)<s(\omega_{+})$.

The proofs of Theorems 2.18, 2.19, and 2.20 are given in Sections 3.7, 3.8 and
3.9. Although they are technically not difficult, to the best of our knowledge
they are the first general results about the strict increase of specific
energy/entropy in quantum statistical mechanics. Central to the proof of the
inequality $\omega(E_{\Psi})<\omega_{+}(E_{\Psi})$ is the properly identified
concept of regularity. We emphasize that the latter property is quite
universal: recall that according to Theorem 2.11, every equilibrium state at
high temperature is weak Gibbs and hence is regular. The inequality
$s(\omega)<s(\omega_{+})$ is an immediate consequence of energy conservation
$\omega(E_{\Phi})=\omega_{+}(E_{\Phi})$ and the Gibbs variational principle
applied to $\omega_{+}\in\mathcal{S}_{\mathrm{eq}}(\Phi)$.

Remark. Fixing $\beta=1$ arises differently in the contexts of the initial
state $\omega\in\mathcal{S}_{\mathrm{eq}}(\Psi)$ and the EES
$\omega_{+}\in\mathcal{S}_{+}(\omega,\Phi)$. In the case of $\omega$, $\beta$
is simply absorbed in $\Psi$. On the other hand, rescaling $\Phi$ as
$\gamma\Phi$, $\gamma>0$, does not change EES, and instead gives a free
parameter $\gamma$. Since $\omega_{+}$ can be in at most one of the sets
$\mathcal{S}_{\mathrm{eq}}(\gamma\Phi)$, $\gamma>0$,777These sets are disjoint
if it happens that we have approach to equilibrium and that
$\omega_{+}\in\mathcal{S}_{\mathrm{eq}}(\bar{\gamma}\Phi)$ for some
$\bar{\gamma}>0$, rescaling $\Phi$ as $\bar{\gamma}\Phi$ sets the inverse
temperature of $\omega_{+}$ to $1$. In the next section we will comment on the
choice of sign $\gamma>0$ in this remark.

###  2.4 Remarks

#### Approach to equilibrium in algebraic quantum statistical mechanics.

As we have already remarked, algebraic quantum statistical mechanics provides
a natural mathematical framework for the study of dynamical aspects of
important classes of infinitely extended quantum systems. In spite of its many
successes, it remains unknown whether this framework, even in principle, can
account for the zeroth law of thermodynamics. To the best of our knowledge,
the works [LR68, Hug83, Hug87] are the only attempts at a formulation of the
problem. Guided by more recent works on foundations of non-equilibrium quantum
statistical mechanics in the algebraic framework, we have presented here what
we believe to be the minimal formulation of the problem of approach to
equilibrium, and have obtained several structural results that are physically
natural. Much remains to be done in developing a general structural theory and
studying concrete physically relevant models in the context of this proposal.

We emphasize that Theorem 2.20 and the second parts of Theorem 2.19 and
Corollary 2.21 are conditional results. The basic questions when the set
$\mathcal{S}_{+}(\omega,\Phi)$ of ESS contains $\omega_{+}$ such that
$\omega_{+}\in\mathcal{S}_{\mathrm{eq}}(\Phi)$ remains open; no non-trivial
example is known.

Theorem 2.18 and the first part of Theorem 2.19 and Corollary 2.21 are
unconditional results. Their basic consequence is the inherent irreversibility
of approach to equilibrium. To elucidate this point, let
$\Phi,\Psi\in\mathcal{B}^{r}$, and suppose that $(\omega,\Psi)$ is a regular
pair satisfying $\mathcal{S}_{\mathrm{eq}}(\Psi)=\\{\omega\\}$. Let
$\omega_{+}\in\mathcal{S}_{+}(\omega,\Phi)$ and assume reversibility in the
sense that $\omega\in\mathcal{S}_{+}(\omega_{+},\Psi)$. Then, by Part (3) of
Proposition 2.17, $\omega(E_{\Psi})=\omega_{+}(E_{\Psi})$, and the implication
(a) $\Rightarrow$ (b) of Theorem 2.19 gives that
$\omega_{+}\in\mathcal{S}_{\mathrm{eq}}(\Psi)=\\{\omega\\}$. Hence,
$\omega=\omega_{+}$, and the setting is trivial in the sense that $\omega$ is
$\alpha_{\Phi}$-invariant. This irreversibility is in sharp contrast with the
reversibility of return to equilibrium which we discuss below.

#### Direction of time.

Changing the direction of time amounts to replacing the average (16) with

$\bar{\omega}_{T}=\frac{1}{T}\int_{0}^{T}\omega\circ\alpha_{\Phi}^{-t}\mathrm{d}t.$

We denote by $\mathcal{S}_{-}(\omega,\Phi)$ the corresponding set of ESS.
Since $\mathcal{S}_{-}(\omega,\Phi)=\mathcal{S}_{+}(\omega,-\Phi)$, none of
results discussed in this paper is affected by such a change, as expected; see
[Pei79, Section 3.8] for a lucid discussion of this point. If the quantum
dynamical system $(\mathfrak{A},\alpha_{\Phi},\omega)$ is time-reversal
invariant with time-reversal $\Theta$888$\Theta$ is an anti-linear
$\ast$-automorphism of $\mathfrak{A}$ satisfying
$\Theta\circ\alpha_{\Phi}^{t}=\alpha_{\Phi}^{-t}\circ\Theta$ for all
$t\in\mathbb{R}$ and $\omega(\Theta(A))=\omega(A^{\ast})$ for all
$A\in\mathfrak{A}$., then $\nu\in\mathcal{S}_{-}(\omega,\Phi)$ iff
$\overline{\nu\circ\Theta}\in\mathcal{S}_{+}(\omega,\Phi)$ and
$\mathcal{S}_{-}(\omega,\Phi)\cap\mathcal{S}_{\mathrm{eq}}(\Phi)=\mathcal{S}_{+}(\omega,\Phi)\cap\mathcal{S}_{\mathrm{eq}}(\Phi)$.

#### Question of Ruelle.

To the best of our knowledge, the question about the increase of specific
entropy with time was first raised in 1967 by David Ruelle. In [Rue67, p.
1666] Ruelle writes:

> It is unclear to the author whether the evolution of an infinite system
> should increase its entropy per unit volume. Another possibility is that,
> when the time tends to $+\infty$, a state has a limit with strictly larger
> entropy.

In the context of quantum spin systems, the first part of the question was
answered a year later by Lanford and Robinson [LR68]—the specific entropy
remains constant for finite times. One of our main results sheds a light on
the second part of Ruelle’s question by establishing the conditional result:
if $\omega_{+}\in\mathcal{S}_{\mathrm{eq}}(\Phi)$ and $\omega\not=\omega_{+}$,
then $s(\omega_{+})>s(\omega)$. As discussed above, a parallel to Ruelle’s
question and another signature of irreversibility is the inequality
$\omega_{+}(E_{\Psi})>\omega(E_{\Psi})$. Our other main result is that this
inequality unconditionally holds in the high temperature regime and that its
general validity is linked to the concept of regularity.

#### Comparison with return to equilibrium.

In spite of a formal similarity, it is important to emphasize that the
properties of return to equilibrium and approach to equilibrium are very
different. Return to equilibrium is an ergodic property asserting that thermal
equilibrium states are stable under local perturbations. Approach to
equilibrium asserts that, under a global perturbation, the system approaches a
new equilibrium state. The mechanisms leading to the respective phenomena are
completely different. In return to equilibrium, a local perturbation $V$
disperses to spatial infinity while the specific entropy remains constant in
the large time limit. It is perfectly possible, and in fact quite common, that
$\omega\not=\omega_{V}$ and that both dynamical systems
$(\mathcal{O},\alpha,\omega)$ and $(\mathcal{O},\alpha_{V},\omega_{V})$ have
the property of return to equilibrium, in which case return to equilibrium is
non-trivially reversible in the sense that

$\lim_{T\rightarrow\infty}\frac{1}{T}\int_{0}^{t}\omega\circ\alpha_{V}^{t}\mathrm{d}t=\omega_{V}\qquad\hbox{and}\qquad\lim_{T\rightarrow\infty}\frac{1}{T}\int_{0}^{t}\omega_{V}\circ\alpha^{t}\mathrm{d}t=\omega.$

In comparison, approach to equilibrium is inherently irreversible, as
discussed in the previous remark.

The formal similarity between the return and approach to equilibrium requires
a further comment. Consider a triple $(\mathfrak{A},\alpha_{\Psi},\omega)$,
where $\Psi\in\mathcal{B}^{r}$ and $\omega$ is a $\alpha_{\Psi}$-KMS state.
Let $\Phi\in\mathcal{B}^{r}$ and consider the local perturbation

$V_{\Lambda}=\sum_{X\subset\Lambda}(\Phi(X)-\Psi(X)).$

Denoting by $\alpha_{\Lambda}$ the perturbed $C^{\ast}$-dynamics, and by
$\omega_{\Lambda}$ the perturbed KMS-state, and assuming that
$(\mathfrak{A},\alpha_{\Lambda},\omega_{\Lambda})$ has the property of return
to equilibrium, one has

$\lim_{T\rightarrow\infty}\frac{1}{T}\int_{0}^{T}\omega\circ\alpha_{\Lambda}^{t}\mathrm{d}t=\omega_{\Lambda}.$

It is not difficult to show that
$\lim_{\Lambda\uparrow\mathbb{Z}^{d}}\alpha_{\Lambda}^{t}=\alpha_{\Phi}^{t}$
strongly, and that for sufficiently small interactions
$\lim_{\Lambda\uparrow\mathbb{Z}^{d}}\omega_{\Lambda}=\omega_{+}$, where
$\omega_{+}$ is the unique $\alpha_{\Phi}$-KMS state. Under these
circumstances999The principal of which is that
$(\mathfrak{A},\alpha_{\Lambda},\omega_{\Lambda})$ has the property of return
to equilibrium for all large enough $\Lambda$. one has

$\lim_{\Lambda\uparrow\mathbb{Z}^{d}}\lim_{T\rightarrow\infty}\frac{1}{T}\int_{0}^{T}\omega\circ\alpha_{\Lambda}^{t}\mathrm{d}t=\omega_{+}.$
(17)

If the two limits in (17) can be interchanged, then approach to equilibrium
holds. Our main results imply that this strategy cannot be reduced to
technical improvements of the existing spectral/scattering estimates used in
the literature to establish return to equilibrium and that novel ideas are
needed.

#### Comparison with the non-equilibrium theory.

The mathematical theory of non-equilibrium quantum statistical mechanics
developed over the last twenty years has played an important role in the
genesis of this work. In fact, a strong similarity exists between the
conceptual framework of the present paper and the one developed in [JP01b,
Rue99], in the context of a now mature non-equilibrium quantum statistical
program.

The entropy balance equation, Identity (1), is the central starting point of
the non-equilibrium theory. The regularity condition (14) plays a similar role
in the structural theory of approach to equilibrium developed in this work.

In the construction of Non-Equilibrium Steady States (NESS) proposed in the
above mentioned program, the state of the system at time $t$, given by
$\omega\circ\alpha_{V}^{t}$, is normal with respect to the initial state
$\omega$. The NESS, however, are typically singular with respect to $\omega$,
and in fact under suitable regularity assumptions, the strict positivity of
entropy production—a signature of nonequilibrium— is equivalent to this
singularity. These regularity assumptions can be either verified generically
or by a detailed study of concrete models; for discussion of these points we
refer the reader to [AJPP06, JP02a, JP07]. On the other hand, in the context
of Theorems 2.19 and 2.20, and recalling the remark after Definition 2.15, if
$(\omega,\Psi)$ is a regular pair such that
$\mathcal{S}_{\mathrm{eq}}(\Psi)=\\{\omega\\}$, then, for any $t$, the state
$\omega\circ\alpha_{\Phi}^{t}$ is $\varphi$-ergodic, and so either
$\omega\circ\alpha_{\Phi}^{t}=\omega$ or the states
$\omega\circ\alpha_{\Phi}^{t}$ and $\omega$ are mutually singular. The same
holds for the ESS
$\omega_{+}\in\mathcal{S}_{+}(\omega,\Phi)\cap\mathcal{S}_{\mathrm{eq}}(\Phi)$:
either $\omega=\omega_{+}$ or the states $\omega$ and $\omega_{+}$ are
mutually singular, and in the later case $s(\omega_{+})>s(\omega)$. To
summarize, the singularity of the NESS in the non-equilibrium setting is an
essential and non-trivial dynamical problem. The singularity of ESS in the
approach of equilibrium setting is a direct consequence of the translation
invariance and the general structure of algebraic quantum statistical
mechanics. The generality of our results, in comparison with related results
in non-equilibrium statistical mechanics, can be understood in these terms.

#### Weak coupling limit.

The weak coupling (or van Hove) limit sheds further light on the above
remarks. In the context of return to equilibrium, NESS, and Entropy
Production, open quantum systems consisting of small quantum system with
finite dimensional Hilbert space locally coupled to finitely many free
fermionic or bosonic reservoirs—such models are often called Pauli–Fierz
systems—play a privileged role. The weak coupling limit of such models is
described by Pauli’s master equation and the respective theory was developed
in seminal works [Dav74, SL78]; for modern expositions of this theory see
[DF06, JPW14]. The weak coupling limit theory played a very important role in
more recent studies of return of equilibrium. In a similar spirit, the
interacting fermion systems of [ESY04, Hug83, Hug87] play a privileged role in
study of approach to equilibrium. The weak coupling limit of such models is
formally described by the quantum Boltzmann equation whose mathematically
rigorous derivation remains an outstanding open problem; see [BCEP07, LS11,
HLP21] for a review and important progresses in this direction.

#### Weak Gibbsianity and regularity.

In spite of its relative late appearance, the concept of weak Gibbs state
plays an important role in the study of invariant measures of classical
dynamical systems, as a natural boundary for the validity of the thermodynamic
formalism. In the framework of statistical mechanics it is known that, under
very general conditions, the equilibrium states of classical spin systems with
summable interactions are weak Gibbs101010The converse does not hold. [PS20].
Theorem 2.11 is a much weaker result. Its proof relies on the Gibbs condition
(see Proposition 2.12) and Theorem 3.1, the restrictions coming from the
latter result. Regarding these restrictions, we mention the article [Mor20]
where the proof of regularity of any pair
$(\omega,\Phi)\in\mathcal{S}_{\mathrm{eq}}(\Phi)\times\mathcal{B}^{r}$ is
announced. Unfortunately, this proof is incomplete and cannot be fixed along
the proposed lines.111111The error concerns the identity (32) in [Mor20],
which implies that the KMS vector representative constructed by Ejima and
Ogata in [EO19] is in the natural cone. Unfortunately, a simple argument shows
that this is so only in the commutative case.

A better understanding of the status of weak Gibbs states in equilibrium
quantum statistical mechanics, and more generally quantum spin dynamics, is
central to the success of the proposed research program. We also believe that,
similarly to the classical setting, quantum weak Gibbs states will find many
other applications in quantum statistical mechanics.

####  $\boldsymbol{W^{\ast}}$-dynamical systems.

In many important physical examples, and in particular those involving bosons,
it may be more convenient to work in a $W^{\ast}$-framework. Similar to the
non-equilibrium quantum statistical mechanics121212See [MMS07a, MMS07b]., it
appears difficult to develop a general structural theory of approach to
equilibrium in the $W^{\ast}$-setting. In this case one is naturally limited
to the study of concrete models.

##  3 Proofs

###  3.1 Proof of Theorem 2.7

In this section we prove the parts of Theorem 2.7 that are not in [Isr79,
Theorem III.4.2].

(a) $\Leftrightarrow$ (b). This equivalence is a direct consequence of
Relation (5) and the fact that

$H_{\Lambda}(\Phi)-H_{\Lambda}(\Psi)=H_{\Lambda}(\Phi-\Psi).$

(f) $\Leftrightarrow$ (g). By [BR81, Theorem 6.2.4], if
$\Phi\in\mathcal{B}^{r}$, then $\alpha_{\Phi}^{t}=\mathrm{e}^{t\delta}$, where
the generating derivation $\delta$ is the closure of the map

${\mathfrak{A}_{\mathrm{loc}}}\ni
A\mapsto\sum_{Y\cap\mathop{\mathrm{supp}}\nolimits(A)\neq\emptyset}\mathrm{i}[\Phi(Y),A].$

Thus, $\alpha_{\Phi}=\alpha_{\Psi}$ iff, for any
$A\in{\mathfrak{A}_{\mathrm{loc}}}$,

$\sum_{Y\cap\mathop{\mathrm{supp}}\nolimits(A)\neq\emptyset}\mathrm{i}[\Phi(Y)-\Psi(Y),A]=0.$

Since $\mathrm{i}[\Phi(Y)-\Psi(Y),A]=0$ whenever
$Y\cap\mathop{\mathrm{supp}}\nolimits(A)=\emptyset$, this condition is
equivalent to (g).

(g) $\Rightarrow$ (b). By the simplicity of $\mathfrak{A}$, (g) implies that
for any $\Lambda$ there exists $C_{\Lambda}\in\mathbb{R}$ such that

$\sum_{X\cap\Lambda\neq\emptyset}(\Phi(X)-\Psi(X))=C_{\Lambda}\mathbbm{1}.$

Recalling (7), we deduce

$\frac{1}{|\Lambda|}\left(H_{\Lambda}(\Phi)-H_{\Lambda}(\Psi)\right)=\frac{C_{\Lambda}}{|\Lambda|}\mathbbm{1}-\frac{1}{|\Lambda|}W_{\Lambda}(\Phi-\Psi),$
(18)

while (5) further yields

$\lim_{\Lambda\uparrow\mathbb{Z}^{d}}\frac{1}{|\Lambda|}\mathop{\mathrm{tr}}\nolimits\left(H_{\Lambda}(\Phi)-H_{\Lambda}(\Psi)\right)=\mathop{\mathrm{tr}}\nolimits(E_{\Phi-\Psi}).$

Taking the trace on both sides of (18) and taking (8) into account yields

$\lim_{\Lambda\uparrow\mathbb{Z}^{d}}\frac{C_{\Lambda}}{|\Lambda|}=\mathop{\mathrm{tr}}\nolimits(E_{\Phi-\Psi}).$

Finally, letting $\Lambda\uparrow\mathbb{Z}^{d}$ in (18) gives (b).

(b) $\Rightarrow$ (c). For $t\in[0,1]$ set $\Phi_{t}=(1-t)\Psi+t\Phi$ and note
that

$H_{\Lambda}(\Phi_{t})-H_{\Lambda}(\Psi)=H_{\Lambda}(\Phi_{t}-\Psi)=t\left(H_{\Lambda}(\Phi)-H_{\Lambda}(\Psi)\right)$

and (b) imply

$\lim_{\Lambda\uparrow\mathbb{Z}^{d}}\frac{1}{|\Lambda|}\left(H_{\Lambda}(\Phi_{t})-H_{\Lambda}(\Psi)\right)=t\mathop{\mathrm{tr}}\nolimits(E_{\Phi-\Psi})\mathbbm{1}.$

Thus, given $\delta>0$, and for large enough $\Lambda$,

$H_{\Lambda}(\Psi)+|\Lambda|(t\mathop{\mathrm{tr}}\nolimits(E_{\Phi-\Psi})-\delta)\mathbbm{1}\leq
H_{\Lambda}(\Phi_{t})\leq
H_{\Lambda}(\Psi)+|\Lambda|(t\mathop{\mathrm{tr}}\nolimits(E_{\Phi-\Psi})+\delta)\mathbbm{1}.$

Invoking the finite-volume Gibbs variational principle [OP93, Proposition
1.10]

$P_{\Lambda}(\Phi)=\log\,\mathop{\mathrm{tr}}\nolimits\mathrm{e}^{-H_{\Lambda}(\Phi)}=\max_{\rho\in\mathcal{S}(\mathfrak{A}_{\Lambda})}\left(S(\rho)-\rho\left(H_{\Lambda}(\Phi)\right)\right),$
(19)

the previous inequalities yield

$\left|P_{\Lambda}(\Phi_{t})-P_{\Lambda}(\Psi)+|\Lambda|t\mathop{\mathrm{tr}}\nolimits(E_{\Phi-\Psi})\right|\leq\delta|\Lambda|.$

In view of (4), dividing both sides of this estimate by $|\Lambda|$, taking
the thermodynamic limit $\Lambda\uparrow\mathbb{Z}^{d}$, and finally letting
$\delta\downarrow 0$, we deduce

$P(\Phi_{t})=P(\Psi)-t\mathop{\mathrm{tr}}\nolimits(E_{\Phi-\Psi}),$

which is (c).

###  3.2 Proof of Theorem 2.8

(1) Invariance of the specific entropy goes back to [LR68, Theorem 5]. Even
though it may look surprising, its proof can be understood as follows.
Consider a tiling of $\mathbb{Z}^{d}$ with a large cube $\Lambda$, and perturb
the interaction $\Phi$ so that all cubes are disconnected. The entropy of
$\omega$ restricted to $\Lambda$ is constant under the perturbed dynamics
because it is a closed finite system by construction. Finally, the entropy of
the perturbed dynamics coincides with the original one up to a $o(|\Lambda|)$
boundary term whose contribution to the specific entropy vanishes in the
$\Lambda\uparrow\mathbb{Z}^{d}$-limit.

(2) We recall again that, by [BR81, Theorem 6.2.4],
$\alpha_{\Phi}^{t}=\mathrm{e}^{t\delta}$, where the generator $\delta$ is the
closure of the map

${\mathfrak{A}_{\mathrm{loc}}}\ni
A\mapsto\sum_{Y\cap\mathop{\mathrm{supp}}\nolimits(A)\neq\emptyset}\mathrm{i}[\Phi(Y),A].$

Since $\omega\circ\alpha_{\Phi}^{t}\in\mathcal{S}_{\mathrm{I}}(\mathfrak{A})$,
it suffices to show that $E_{\Phi}\in\mathop{\mathrm{Dom}}\nolimits(\delta)$
and $\omega(\delta(E_{\Phi}))=0$ for all
$\omega\in\mathcal{S}_{\mathrm{I}}(\mathfrak{A})$.

For latter reference, we will show that
$E_{\Psi}\in\mathop{\mathrm{Dom}}\nolimits(\delta)$ holds for all
$\Psi\in\mathcal{B}^{r}$. Setting

$E_{\Psi,n}=\sum_{X\ni 0\atop\mathop{\mathrm{diam}}\nolimits(X)\leq
n}\frac{\Psi(X)}{|X|}\in{\mathfrak{A}_{\mathrm{loc}}},$

and observing that

$\sum_{n=0}^{\infty}\sum_{X\ni
0\atop\mathop{\mathrm{diam}}\nolimits(X)=n}\frac{\|\Psi(X)\|}{|X|}\leq\|\Psi\|_{r}<\infty,$

we have

$\lim_{n\to\infty}\|E_{\Psi}-E_{\Psi,n}\|\leq\lim_{n\to\infty}\sum_{X\ni
0\atop\mathop{\mathrm{diam}}\nolimits(X)>n}\frac{\|\Psi(X)\|}{|X|}=0.$

Moreover, for $m\leq n$, the formula

$\delta(E_{\Psi,n})-\delta(E_{\Psi,m})=\sum_{X\ni 0\atop
m<\mathop{\mathrm{diam}}\nolimits(X)\leq n}\frac{1}{|X|}\sum_{Y\cap
X\neq\emptyset}\mathrm{i}[\Phi(Y),\Psi(X)],$

gives the estimate

$\displaystyle\|\delta(E_{\Psi,n})-\delta(E_{\Psi,m})\|$ $\displaystyle\leq
2\sum_{X\ni 0\atop m<\mathop{\mathrm{diam}}\nolimits(X)\leq
n}\frac{\|\Psi(X)\|}{|X|}\sum_{Y\cap X\neq\emptyset}\|\Phi(Y)\|$
$\displaystyle\leq 2\sum_{X\ni
0\atop\mathop{\mathrm{diam}}\nolimits(X)>m}\|\Psi(X)\|\frac{1}{|X|}\sum_{x\in
X}\sum_{Y\ni x}\|\Phi(Y)\|\leq 2\|\Phi\|_{r}\sum_{X\ni
0\atop\mathop{\mathrm{diam}}\nolimits(X)>m}\|\Psi(X)\|,$

which shows that the sequence $(\delta(E_{\Psi,n}))_{n\in\mathbb{N}}$ is
Cauchy. It follows that $E_{\Psi}\in\mathop{\mathrm{Dom}}\nolimits(\delta)$
and

$\delta(E_{\Psi})=\sum_{X\ni 0}\frac{1}{|X|}\sum_{Y\cap
X\neq\emptyset}\mathrm{i}[\Phi(Y),\Psi(X)].$ (20)

To proceed, we equip $X\in\mathcal{F}$ with the lexicographic order observing
that translation invariance implies $\min(X+x)=\min(X)+x$ for any
$X\in\mathcal{F}$ and $x\in\mathbb{Z}^{d}$. Set

$\mathcal{F}_{0}=\\{X\in\mathcal{F}\mid\min(X)=0\\},$

and note that the set of translates of $X\in\mathcal{F}_{0}$ containing $0$ is

$\mathcal{F}_{X}=\\{X-x\mid x\in X\\}.$

Finally, for $X,Y\in\mathcal{F}_{0}$, let

$Z(X,Y)=\\{z\in\mathbb{Z}^{d}\mid X\cap(Y+z)\neq\emptyset\\}.$

Relation (20) thus writes

$\delta(E_{\Phi})=\sum_{X\in\mathcal{F}_{0}}\frac{1}{|X|}\sum_{x\in
X}\sum_{Y\cap(X-x)\neq\emptyset}\mathrm{i}[\Phi(Y),\varphi^{-x}(\Phi(X))].$

Now for $Y\in\mathcal{F}$ one has $Y=Y_{0}+y$, with $Y_{0}\in\mathcal{F}_{0}$
and $y=\min(Y)$. Moreover,

$\emptyset\neq Y\cap(X-x)=(Y_{0}+y)\cap(X-x)=((Y_{0}+x+y)\cap X)-x$

iff $x+y\in Z(X,Y_{0})$, so that

$\delta(E_{\Phi})=\sum_{X\in\mathcal{F}_{0}}\frac{1}{|X|}\sum_{x\in
X}\sum_{Y\in\mathcal{F}_{0}}\sum_{x+y\in
Z(X,Y)}\mathrm{i}[\varphi^{y}(\Phi(Y)),\varphi^{-x}(\Phi(X))].$

It follows that, for $\omega\in\mathcal{S}_{\mathrm{I}}(\mathfrak{A})$,

$\displaystyle\omega(\delta(E_{\Phi}))$
$\displaystyle=\sum_{X\in\mathcal{F}_{0}}\frac{1}{|X|}\sum_{x\in
X}\sum_{Y\in\mathcal{F}_{0}}\sum_{z\in
Z(X,Y)}\omega(\mathrm{i}[\varphi^{z}(\Phi(Y)),\Phi(X)])$
$\displaystyle=\sum_{X,Y\in\mathcal{F}_{0}}\sum_{z\in
Z(X,Y)}\omega(\mathrm{i}[\varphi^{z}(\Phi(Y)),\Phi(X)]).$

Symmetrizing the inner sum and observing that $Z(Y,X)=-Z(X,Y)$ yields

$\displaystyle\omega(\delta(E_{\Phi}))$
$\displaystyle=\frac{1}{2}\sum_{X,Y\in\mathcal{F}_{0}}\sum_{z\in
Z(X,Y)}\omega(\mathrm{i}[\varphi^{z}(\Phi(Y)),\Phi(X)]+\mathrm{i}[\varphi^{-z}(\Phi(X)),\Phi(Y)])$
$\displaystyle=\frac{1}{2}\sum_{X,Y\in\mathcal{F}_{0}}\sum_{z\in
Z(X,Y)}\omega(\mathrm{i}[\varphi^{z}(\Phi(Y)),\Phi(X)]-\mathrm{i}[\varphi^{z}(\Phi(Y)),\Phi(X)])=0.$

(3) In a similar spirit, it suffices to show that
$E_{N}\in\mathop{\mathrm{Dom}}\nolimits(\delta)$, with
$\omega(\delta(E_{N}))=0$ for all
$\omega\in\mathcal{S}_{\mathrm{I}}(\mathfrak{A})$. The generator $\gamma$ of
the gauge group is the closure of the map

${\mathfrak{A}_{\mathrm{loc}}}\ni
A\mapsto\sum_{x\in\mathop{\mathrm{supp}}\nolimits(A)}\mathrm{i}[a_{x}^{\ast}a_{x},A]=\sum_{x\in\mathop{\mathrm{supp}}\nolimits(A)}\mathrm{i}[\varphi^{x}(E_{N}),A].$

Since any state $\omega\in\mathcal{S}_{\mathrm{I}}(\mathfrak{A})$ is gauge-
invariant131313Any state $\omega\in\mathcal{S}_{\mathrm{I}}(\mathcal{A})$
restricts to the gauge-invariant state
$\bar{\omega}=\int_{0}^{2\pi}\omega\circ\vartheta^{\theta}\frac{\mathrm{d}\theta}{2\pi}\in\mathcal{S}_{\mathrm{I}}(\mathfrak{A})$.,
one has $\omega(\gamma(A))=0$ for any
$A\in\mathop{\mathrm{Dom}}\nolimits(\gamma)$, and in particular

$0=\omega(\gamma(\Phi(X)))=\sum_{x\in
X}\omega(\mathrm{i}[\varphi^{x}(E_{N}),\Phi(X)])=\sum_{x\in
X}\omega(\mathrm{i}[E_{N},\varphi^{-x}(\Phi(X))])=-\sum_{Y\in\mathcal{F}_{X}}\omega(\mathrm{i}[\Phi(Y),E_{N}]).$

for all $X\in\mathcal{F}$. We conclude that

$\omega(\delta(E_{N}))=\sum_{X\ni
0}\omega(\mathrm{i}[\Phi(X),E_{N}])=\sum_{X\in\mathcal{F}_{0}}\sum_{Y\in\mathcal{F}_{X}}\omega(\mathrm{i}[\Phi(Y),E_{N}])=0.$

###  3.3 Proof of Proposition 2.10

(1) Is merely a reformulation of the defining relations (11) and (9) which
will be convenient in the proofs of the remaining statements.

(2) By the operator monotonicity of the logarithm [Pet08, Example 11.16], the
inequalities (11) follow from (10), which proves the first inclusion. To deal
with the second one, we note that

$\log\omega_{\Phi,\Lambda}^{c}=-H_{\Lambda}(\Phi)-P_{\Lambda}(\Phi),$

so that any $\omega\in\mathrm{LWG}(\Phi)$ satisfies

$-c_{\Lambda}\leq\log\omega_{\Lambda}+H_{\Lambda}(\Phi)+P_{\Lambda}(\Phi)\leq
c_{\Lambda},$

where $c_{\Lambda}$ is such that (9) holds. Multiplying with
$\omega_{\Lambda}$ and taking the trace yields

$|S(\omega_{\Lambda})-\omega(H_{\Lambda}(\Phi))-P_{\Lambda}(\Phi)|\leq
c_{\Lambda}.$

Dividing by $|\Lambda|$, letting $\Lambda\uparrow\mathbb{Z}^{d}$ and invoking
(2), (4) and (6) gives

$P(\Phi)=s(\omega)-\omega(E_{\Phi}),$

and hence $\omega\in\mathcal{S}_{\mathrm{eq}}(\Phi)$.

(3) If $\omega\in\mathrm{LWG}(\Phi)\cap\mathrm{LWG}(\Psi)$, then

$\limsup_{\Lambda\uparrow\mathbb{Z}^{d}}|\Lambda|^{-1}\|\log\omega_{\Lambda}-\log\omega_{\Phi,\Lambda}^{c}\|=\limsup_{\Lambda\uparrow\mathbb{Z}^{d}}|\Lambda|^{-1}\|\log\omega_{\Lambda}-\log\omega_{\Psi,\Lambda}^{c}\|=0.$

It follows that

$\limsup_{\Lambda\uparrow\mathbb{Z}^{d}}|\Lambda|^{-1}\|\log\omega^{\mathrm{c}}_{\Phi,\Lambda}-\log\omega^{\mathrm{c}}_{\Psi,\Lambda}\|=0,$

which, in view of (1), immediately leads to
$\mathrm{LWG}(\Phi)=\mathrm{LWG}(\Psi)$.

(4) Let $\omega\in\mathrm{LWG}(\Phi)$, and suppose that $\Psi\sim\Phi$. By
definition one has

$\limsup_{\Lambda\uparrow\mathbb{Z}^{d}}|\Lambda|^{-1}\|\log\omega_{\Lambda}-\log\omega_{\Phi,\Lambda}^{\mathrm{c}}\|=0.$
(21)

Theorem 2.7-(b) further implies that, for any $\delta>0$,

$|\Lambda|(\mathop{\mathrm{tr}}\nolimits(E_{\Phi-\Psi})-\delta)\mathbbm{1}\leq
H_{\Lambda}(\Phi)-H_{\Lambda}(\Psi)\leq|\Lambda|(\mathop{\mathrm{tr}}\nolimits(E_{\Phi-\Psi})+\delta)\mathbbm{1}$
(22)

provided $\Lambda$ is large enough. Writing the variational formula (19) as

$P_{\Lambda}(\Psi)=\max_{\rho\in\mathcal{S}(\mathfrak{A}_{\Lambda})}\left(S(\rho)-\rho\left(H_{\Lambda}(\Phi)\right)+\rho\left(H_{\Lambda}(\Phi)-H_{\Lambda}(\Psi)\right)\right),$

the inequalities (22) allow us to deduce

$|\Lambda|(-\mathop{\mathrm{tr}}\nolimits(E_{\Phi-\Psi})-\delta)\leq
P_{\Lambda}(\Phi)-P_{\Lambda}(\Psi)\leq|\Lambda|(-\mathop{\mathrm{tr}}\nolimits(E_{\Phi-\Psi})+\delta).$

Adding the last inequalities to (22) gives

$\|\log\omega_{\Phi,\Lambda}^{\mathrm{c}}-\log\omega_{\Psi,\Lambda}^{\mathrm{c}}\|\leq
2\delta|\Lambda|.$

Combining this estimate with (21) yields

$\limsup_{\Lambda\uparrow\mathbb{Z}^{d}}|\Lambda|^{-1}\|\log\omega_{\Lambda}-\log\omega_{\Psi,\Lambda}^{\mathrm{c}}\|\leq
2\delta,$

and taking $\delta\downarrow 0$ allows us to conclude that
$\omega\in\mathrm{LWG}(\Psi)$.

Reciprocally, if $\omega\in\mathrm{LWG}(\Phi)=\mathrm{LWG}(\Psi)$, then it
follows from (1) that
$\omega\in\mathcal{S}_{\mathrm{eq}}(\Phi)\cap\mathcal{S}_{\mathrm{eq}}(\Psi)$
and Theorem 2.7-(d) yields $\Phi\sim\Psi$.

###  3.4 Proof of Theorem 2.11

We will make use of [LRB05, Theorem 3.7], which we state and prove in the
general setting of Section 1.1.

######  Theorem 3.1.

Let $\omega$ be a faithful $(\alpha,1)$-KMS state and $\omega_{V}$ the
perturbed $(\alpha_{V},1)$-KMS state induced by $V=V^{\ast}\in\mathcal{O}$.

  1. (i)

If the map

$\mathbb{R}\ni t\mapsto\alpha^{t}(V)\in\mathcal{O}$ (23)

has an analytic extension to the strip $0<\mathop{\mathrm{Im}}\nolimits z<1/2$
which is bounded and continuous on its closure, then

$\omega_{V}\leq C_{V}\omega,$

where $C_{V}=\mathrm{e}^{\|V\|+\|\alpha^{\mathrm{i}/2}(V)\|}$.

  2. (ii)

If the map (23) has an analytic extension to the strip
$|\mathop{\mathrm{Im}}\nolimits z|<1/2$ which is bounded and continuous on its
closure, then we also have a lower bound

$\omega_{V}\geq D_{V}\omega,$

where $D_{V}=\mathrm{e}^{-\|V\|-\|\alpha_{-V}^{\mathrm{i}/2}(V)\|}$.

Proof. For later reference, we give here a proof that is different from the
original argument in [LRB05] and emphasizes the role of the modular structure.
We will freely use the notation and results of modular theory and perturbation
of the KMS-structure discussed in [DJP03]. In particular,
$\mathrm{E}_{V}^{\alpha}$ denotes the Araki–Dyson expansional associated to
$V$,

$\mathrm{E}_{V}^{\alpha}(t)=\sum_{n\geq 0}(\mathrm{i}t)^{n}\int\limits_{0\leq
s_{n}\leq\cdots\leq s_{1}\leq
1}\alpha^{ts_{n}}(V)\cdots\alpha^{ts_{1}}(V)\mathrm{d}s_{1}\cdots\mathrm{d}s_{n},$
(24)

see [DJP03, Section 3.1].

(i) Passing to the GNS representation $(\mathcal{H},\pi,\Omega)$ induced by
$\omega$, we need to prove that for any $A>0$ in $\mathcal{O}$,

$\omega_{V}(A)=\frac{\langle\Omega_{V},\pi(A)\Omega_{V}\rangle}{\|\Omega_{V}\|^{2}}\leq
C_{V}\langle\Omega,\pi(A)\Omega\rangle,$

where $\Omega_{V}=\pi(\mathrm{E}_{V}^{\alpha}(\mathrm{i}/2))\Omega$. Since
$\|\Omega_{V}\|\geq\mathrm{e}^{-\|V\|/2}$ by the Peierls–Bogoliubov
inequality, it suffices to prove that

$\sup_{\mathcal{O}\ni
A>0}\frac{\langle\Omega_{V},\pi(A)\Omega_{V}\rangle}{\langle\Omega,\pi(A)\Omega\rangle}\leq\mathrm{e}^{\|\alpha^{\mathrm{i}/2}(V)\|}.$
(25)

From the fact that $J\Omega_{V}=\Omega_{V}$ we get

$\langle\Omega_{V},\pi(A)\Omega_{V}\rangle=\|\pi(A^{1/2})J\pi(\mathrm{E}_{V}^{\alpha}(\mathrm{i}/2))\Omega\|^{2}=\|\pi({\rm
E}_{V}^{\alpha}(\mathrm{i}/2))J\pi(A^{1/2})\Omega\|^{2},$

and $\langle\Omega,\pi(A)\Omega\rangle=\|J\pi(A^{1/2})\Omega\|^{2}$ yields

$\sup_{\mathcal{O}\ni
A>0}\frac{\langle\Omega_{V},\pi(A)\Omega_{V}\rangle}{\langle\Omega,\pi(A)\Omega\rangle}=\|\pi({\rm
E}_{V}^{\alpha}(\mathrm{i}/2))\|^{2},$

so that (25) easily follows from (24).

(ii) We infer from the relation
$\alpha_{-V}^{t}(V)=\mathrm{E}_{-V}^{\alpha}(t)\alpha^{t}(V)\mathrm{E}_{-V}^{\alpha}(t)^{\ast}$
that
$\alpha_{-V}^{z}(V)=\mathrm{E}_{-V}^{\alpha}(z)\alpha^{z}(V)\mathrm{E}_{-V}^{\alpha}(\bar{z})^{\ast}$
is analytic in the strip $0<\mathop{\mathrm{Im}}\nolimits z<1/2$ and bounded
and continuous on its closure. Part (i) and the relation
$\omega=(\omega_{-V})_{V}$ yield the claim. ∎

We now proceed with the proof of Theorem 2.11. To simplify the notation, we
write $\alpha$ and $W_{\Lambda}$ for $\alpha_{\Phi}$ and $W_{\Lambda}(\Phi)$.

We only need to prove that
$\mathcal{S}_{\mathrm{eq}}(\Phi)\subset\mathrm{WG}(\Phi)$. To this end, let
$\omega\in\mathcal{S}_{\mathrm{eq}}(\Phi)$. By Theorem 3.1, if the surface
energy $W_{\Lambda}$ is $\alpha$-analytic in the strip
$|\mathop{\mathrm{Im}}\nolimits z|<a$ for some $a>1/2$, then

$\mathrm{e}^{-\|W_{\Lambda}\|-\|\alpha^{\mathrm{i}/2}(W_{\Lambda})\|}\leq\frac{\omega(A)}{\omega_{-W_{\Lambda}}(A)}\leq\mathrm{e}^{\|W_{\Lambda}\|+\|\alpha_{-W_{\Lambda}}^{\mathrm{i}/2}(W_{\Lambda})\|}$

holds for any $A\in\mathfrak{A}_{\Lambda}$ such that $A>0$. By (8) and
Proposition 2.12, to prove that $\omega\in\mathrm{WG}(\Phi)$ it is then
sufficient to show that

$\lim_{\Lambda\uparrow\mathbb{Z}^{d}}\frac{1}{|\Lambda|}\|\alpha^{\mathrm{i}/2}(W_{\Lambda})\|=0,\qquad\lim_{\Lambda\uparrow\mathbb{Z}^{d}}\frac{1}{|\Lambda|}\|\alpha_{-W_{\Lambda}}^{\mathrm{i}/2}(W_{\Lambda})\|=0.$
(26)

In the case (a), this follows from the celebrated estimate of Araki [Ara69],
see also [LRB05, Proposition 3.9] and discussion after it, and [Mat03,
Proposition 2.4]. In the case (b) we argue as follows. Let

$a=\frac{r}{2\|\Phi\|_{r}}>1/2.$

Then, by Theorem 2.3, for any $X\in\mathcal{F}$ the map $\mathbb{R}\ni
t\mapsto\alpha^{t}(\Phi(X))$ has analytic extension to the strip
$|\mathop{\mathrm{Im}}\nolimits z|<a$ such that

$\|\alpha^{\mathrm{i}/2}(\Phi(X))\|\leq\|\Phi(X)\|\frac{\mathrm{e}^{r|X|}}{1-\|\Phi\|_{r}/r}.$
(27)

This estimate gives that the map $\mathbb{R}\ni
t\mapsto\alpha^{t}(W_{\Lambda})$ also has an analytic extension to the strip
$|\mathop{\mathrm{Im}}\nolimits z|<a$ and that

$\|\alpha^{\mathrm{i}/2}(W_{\Lambda})\|\leq\sum_{X\cap\Lambda\not=\emptyset\atop
X\cap\Lambda^{c}\not=\emptyset}\|\Phi(X)\|\frac{\mathrm{e}^{r|X|}}{1-\|\Phi\|_{r}/r}.$

Thus, to establish the first limit in (26) it suffices to show that

$\lim_{\Lambda\uparrow\mathbb{Z}^{d}}\frac{1}{|\Lambda|}\sum_{X\cap\Lambda\not=\emptyset\atop
X\cap\Lambda^{c}\not=\emptyset}\|\Phi(X)\|\mathrm{e}^{r|X|}=0.$

This relation is immediate for $\Phi\in\mathcal{B}_{\mathrm{f}}$. The general
case follows from the density of $\mathcal{B}_{\mathrm{f}}$ in
$\mathcal{B}^{r}$ and the bound

$\frac{1}{|\Lambda|}\sum_{X\cap\Lambda\not=\emptyset}\|\Psi(X)\|\mathrm{e}^{r|X|}\leq\mathrm{e}^{r}\|\Psi\|_{r}$

that holds for all $\Psi\in\mathcal{B}^{r}$.

To deal with the second limit in (26) we observe that the perturbed dynamics
$\alpha_{-W_{\Lambda}}$ is associated with the non-translation-invariant
interaction $\Phi_{\Lambda}$ given by

$\Phi_{\Lambda}(X)=\begin{cases}0&\text{if }X\cap\Lambda\neq\emptyset\text{
and }X\cap\Lambda^{c}\neq\emptyset;\\\ \Phi(X)&\text{otherwise}.\end{cases}$

By the Remark after Theorem 2.3, and in view of the obvious fact that
$\|\Phi_{\Lambda}\|_{r}\leq\|\Phi\|_{r}$, the estimate (27) holds with
$\alpha_{-W_{\Lambda}}$ replacing $\alpha$. We can then argue as before.

###  3.5 Proof of Proposition 2.12

Theorem 2.5 gives

$\omega_{-W_{\Lambda}(\Phi)}=\omega_{\Phi,\Lambda}^{\mathrm{c}}.$ (28)

If $\omega\in\mathrm{WG}(\Phi)$, then for any $A\in\mathfrak{A}_{\Lambda}$
satisfying $A>0$ one has

$\mathrm{e}^{-c_{\Lambda}}\leq\frac{\omega(A)}{\omega_{-W_{\Lambda}}(A)}\leq\mathrm{e}^{c_{\Lambda}},$

and (12) follows.

To prove the converse statement, for $A\in\mathfrak{A}_{\Lambda}$ satisfying
$A>0$, the definitions of $\underline{d}_{\Lambda}$ and
$\overline{d}_{\Lambda}$ and Relation (28) give

$\underline{d}_{\Lambda}\omega_{\Phi,\Lambda}^{\mathrm{c}}(A)\leq\omega(A)\leq\overline{d}_{\Lambda}\omega_{\Phi,\Lambda}^{\mathrm{c}}(A).$

Setting
$c_{\Lambda}=\log\max(\overline{d}_{\Lambda},1/\underline{d}_{\Lambda})$, one
deduces from (12) that $\omega\in\mathrm{WG}(\Phi)$.

###  3.6 Proof of Proposition 2.16

Parts (2) and (3) are immediate consequences of the corresponding statements
of Theorem 2.8.

To prove Part (1), writing the integral (16) as a
$\mathrm{weak}^{\ast}$-limit of Riemann sums and using that the entropy map
is affine and upper-semicontinuous, we derive the inequality

$s(\bar{\omega}_{T})\geq\limsup_{N\to\infty}\sum_{k=0}^{N-1}\frac{1}{N}s(\omega\circ\alpha_{\Phi}^{kt/N}).$

Part (1) of Theorem 2.8 further gives
$s(\omega\circ\alpha_{\Phi}^{kt/N})=s(\omega)$, so that
$s(\bar{\omega}_{T})\geq s(\omega)$. To prove the reverse inequality, note
first that the identity,

$\bar{\omega}_{T}=\frac{1}{2}\bar{\omega}_{T/2}+\frac{1}{2}\bar{\omega}_{T/2}\circ\alpha_{\Phi}^{T/2},$

the affine property of the specific entropy, and Part (1) of Theorem 2.8 give
that $s(\bar{\omega}_{T})=s(\bar{\omega}_{T/2})$. Consequently,
$s(\bar{\omega}_{T})=s(\bar{\omega}_{T/2^{n}})$ for any $n\in\mathbb{N}$, and
invoking again the upper-semicontinuity, we conclude

$s(\bar{\omega}_{T})=\lim_{n\rightarrow\infty}s(\bar{\omega}_{T/2^{n}})\leq
s(\omega).$

###  3.7 Proof of Theorem 2.18

By Part (1) of Proposition 2.16, $s(\bar{\omega}_{T})=s(\omega)$. Relation
(15) and the non-negativity of relative entropy give

$\displaystyle 0\leq\bar{s}(\bar{\omega}_{T}|\omega)$
$\displaystyle\leq-s(\bar{\omega}_{T})+\bar{\omega}_{T}(E_{\Psi})+P(\Psi)$
$\displaystyle=-s(\omega)+\bar{\omega}_{T}(E_{\Psi})+P(\Psi)$ (29)
$\displaystyle=\bar{\omega}_{T}(E_{\Psi})-\omega(E_{\Psi}).$

Pick a sequence $(T_{n})$ such that
$\bar{\omega}_{T_{n}}\rightarrow\omega_{+}$. Taking the limit along this
sequence in (29), we derive that $\omega_{+}(E_{\Psi})\geq\omega(E_{\Psi})$.

###  3.8 Proof of Theorem 2.19

Since our hypothesis implies that $\omega\in\mathcal{S}_{\mathrm{eq}}(\Psi)$,
we can write

$\displaystyle 0\leq s(\bar{\omega}_{T}|\omega)$
$\displaystyle=-s(\bar{\omega}_{T})+\bar{\omega}_{T}(E_{\Psi})+P(\Psi)$
$\displaystyle=-s(\omega)+\bar{\omega}_{T}(E_{\Psi})+P(\Psi)$ (30)
$\displaystyle=\bar{\omega}_{T}(E_{\Psi})-\omega(E_{\Psi}).$

Pick again a sequence $(T_{n})$ such that
$\bar{\omega}_{T_{n}}\rightarrow\omega_{+}$ and note that Relation (14) gives
that the map $\mathcal{S}_{\mathrm{I}}(\mathfrak{A})\ni\nu\mapsto
s(\nu|\omega)$ is lower-semicontinuous. Hence, Relation (30) further yields

$0\leq
s(\omega_{+}|\omega)\leq\liminf_{n\to\infty}s(\bar{\omega}_{T_{n}}|\omega)=\omega_{+}(E_{\Psi})-\omega(E_{\Psi}).$

Thus, if $\omega_{+}(E_{\Psi})=\omega(E_{\Psi})$, then
$s(\omega_{+}|\omega)=0$ and, in view of the fact that
$\omega_{+}\in\mathcal{S}_{\mathrm{I}}(\mathfrak{A})$, (14) gives

$0=s(\omega_{+}|\omega)=-s(\omega_{+})+\omega_{+}(E_{\Psi})+P(\Psi).$ (31)

This allows us to conclude that
$\omega_{+}\in\mathcal{S}_{\mathrm{eq}}(\Psi)$. Since
$\omega\in\mathcal{S}_{\mathrm{eq}}(\Psi)$, we also have

$0=-s(\omega)+\omega(E_{\Psi})+P(\Psi)=-s(\omega)+\omega_{+}(E_{\Psi})+P(\Psi)$

which, upon comparison with (31), shows that $s(\omega)=s(\omega_{+})$, and so
(a) $\Rightarrow$ (b). The implication (b) $\Rightarrow$ (a) is obvious.

If in addition $\omega_{+}\in\mathcal{S}_{\mathrm{eq}}(\Phi)$, then (b) gives
that
$\omega_{+}\in\mathcal{S}_{\mathrm{eq}}(\Phi)\cap\mathcal{S}_{\mathrm{eq}}(\Psi)$
and (c) follows from Theorem 2.7-(d).

Finally, if $\Phi\sim\Psi$, then $\alpha_{\Phi}=\alpha_{\Psi}$ by Theorem
2.7-(f), and so $\omega$ is $\alpha_{\Phi}$-invariant. Hence,
$\omega=\omega_{+}$, and so (c) $\Rightarrow$ (d). Obviously (d) $\Rightarrow$
(a).

###  3.9 Proof of Theorem 2.20

Suppose that $s(\omega)=s(\omega_{+})$. Then, since
$\omega_{+}(E_{\Phi})=\omega(E_{\Phi})$,

$\displaystyle 0$ $\displaystyle=-s(\omega_{+})+\omega_{+}(E_{\Phi})+P(\Phi)$
$\displaystyle=-s(\omega)+\omega(E_{\Phi})+P(\Phi),$

and so $\omega\in\mathcal{S}_{\mathrm{eq}}(\Phi)$. This gives that $\omega$ is
$\alpha_{\Phi}$-invariant and that $\omega=\omega_{+}$.

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