# A note on adiabatic time evolution and quasi-static processes in
translation-invariant quantum systems

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 Avancé, UMR 7501 Université
de Strasbourg et CNRS, 7 rue René-Descartes, 67000 Strasbourg, France

Dedicated to the memory of Krzysztof Gawędzki

Abstract. We study the slowly varying, non-autonomous quantum dynamics of a
translation invariant spin or fermion system on the lattice $\mathbb{Z}^{d}$.
This system is assumed to be initially in thermal equilibrium, and we consider
realizations of quasi-static processes in the adiabatic limit. By combining
the Gibbs variational principle with the notion of quantum weak Gibbs states
introduced in [JPT23], we establish a number of general structural results
regarding such realizations. In particular, we show that such a quasi-static
process is incompatible with the property of approach to equilibrium studied
in this previous work.

##  1 Introduction

This paper is a direct continuation of [JPT23]. Although its topic and
implications are different, we will rely heavily on the technical tools and
conceptual framework introduced in this earlier work. In particular, in what
follows and without further saying, we will use terminology, notation, and
results stated in the introductory section of [JPT23].111The general
references in [JPT23] related to the mathematical theory of algebraic quantum
statistical mechanics apply to this work as well.

The set of quantum weak Gibbs states $\mathrm{WG}(\Phi)$ for the spin/fermion
interaction $\Phi\in\mathcal{B}^{r}$ is introduced in Section 2.2 of [JPT23].
An immediate consequence of our definition of weak Gibbs states is the
existence of the specific relative entropy and the validity of the formula

$s(\nu|\omega)\coloneqq\lim_{\Lambda\uparrow\mathbb{Z}^{d}}\frac{S(\nu_{\Lambda}|\omega_{\Lambda})}{|\Lambda|}=-s(\nu)+\nu(E_{\Phi})+P(\Phi),$
(1)

for any $\nu\in\mathcal{S}_{\mathrm{I}}(\mathfrak{A})$ and
$\omega\in\mathrm{WG}(\Phi)$. The current technology allowed us to prove the
identification $\mathcal{S}_{\mathrm{eq}}(\Phi)=\mathrm{WG}(\Phi)$ in
dimension $d=1$ for finite range interactions, and for any $d$ and
$\Phi\in\mathcal{B}^{r}$ in the high temperature regime. Since this
identification is expected to hold much more generally, we have introduced the
notion of regular pair
$(\omega,\Phi)\in\mathcal{S}_{\mathrm{I}}(\mathfrak{A})\times\mathcal{B}^{r}$
for which (1) holds for all $\nu\in\mathcal{S}_{\mathrm{I}}(\mathfrak{A})$.
This notion, combined with the Gibbs variational principle, has led to several
structural results about the fundamental problem of Approach to
Equilibrium—the zeroth law of thermodynamics—in quantum statistical mechanics.

In this work, starting with the same general ingredients, we study whether
quasi-static transitions of extended spin or fermion systems can be described
by slowly varying, non-autonomous translation invariant interactions. The
emerging structural theory in particular yields that a realization of such a
quasi-static transition is possible only if the specific entropy is constant
along the state trajectory. Combined with the results of [JPT23], this gives
that, in the adiabatic limit, quasi-static transitions are incompatible with
the approach to equilibrium in the translation invariant setting of algebraic
quantum statistical mechanics.

The paper is organized as follows. In Sections 2 and 3 we briefly review the
basic adiabatic theorems of quantum mechanics and quantum statistical
mechanics. For a more complete review, historical perspective, and additional
references, we refer the reader to [BFJP].222In particular, we have not
attempted here to formulate these results in a technically optimal setting. We
introduce our translation invariant framework in Section 4.1. Our main results
are stated and proved in Section 4.2. They are compared with the main result
of [JPT23] in Section 4.3. Similar to [JPT23], most of our proofs are
technically simple. The proofs of two technically more involved results,
Propositions 3.2 and 4.1, are postponed to Section 5. For the reader’s
convenience, Appendix A collects the notations inherited from [JPT23].

#### 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
the CY Initiative of Excellence through the grant Investissements d’Avenir
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 Martin Fraas for useful
discussions.

##  2 Adiabatic theorems in quantum mechanics

The adiabatic theorem with a gap. The formulation and the proof of the
adiabatic theorem in quantum mechanics go back to the seminal work of Born and
Fock [BF28]. The first modern proof, in functional analytic framework, is due
to Kato [Kat50]. The method of the proof introduced there played an important
role in all the future developments of the subject. For refinements of Kato’s
result under the gap assumption made in [Kat50], see [ASY87, JP91, Nen93].

Let $H$ be a (possibly unbounded) self-adjoint operator on the Hilbert space
$\mathcal{H}$ and

$V:[0,1]\to\mathcal{B}(\mathcal{H})$ (2)

a continuous map taking its values in the bounded self-adjoint operators on
$\mathcal{H}$. For $T>0$, we consider the non-autonomous Schrödinger equation

$\mathrm{i}\partial_{t}\psi(t)=(H+V(t/T))\psi(t),\qquad\psi(s)=f,\qquad
s,t\in[0,T].$

For any $f\in\mathop{\mathrm{Dom}}\nolimits(H)$ this equation has a unique
solution $[0,T]\ni t\mapsto\psi(t)\in\mathop{\mathrm{Dom}}\nolimits(H)$, which
can be written as $\psi(t)=U_{T}^{s/T\to t/T}f$, where
$[0,1]\times[0,1]\ni(\sigma,\tau)\mapsto
U_{T}^{\sigma\to\tau}\in\mathcal{B}(\mathcal{H})$ is a strongly continuous map
taking its values in the unitary operators on $\mathcal{H}$ and satisfying
$U_{T}^{\sigma\to\sigma}=I$,
$U_{T}^{\sigma\to\tau}\mathop{\mathrm{Dom}}\nolimits(H)\subset\mathop{\mathrm{Dom}}\nolimits(H)$,
and $U_{T}^{\tau\to\upsilon}U_{T}^{\sigma\to\tau}=U_{T}^{\sigma\to\upsilon}$
for all $\sigma,\tau,\upsilon\in[0,1]$. $U_{T}$ is the propagator associated
to the time-dependent Hamiltonian $[0,1]\ni\tau\mapsto T(H+V(\tau))$.

For each $\tau\in[0,1]$, let $E(\tau)$ be an eigenvalue of finite multiplicity
of $H(\tau)=H+V(\tau)$ which is uniformly separated from the remaining part of
its spectrum, i.e.,

$\inf_{\tau\in[0,1]}{\rm dist}(E(\tau),{\rm
sp}(H(\tau))\setminus\\{E(\tau)\\})>0,$ (3)

and denote by $P(\tau)$ the orthogonal projection onto the associated
eigenspace.

######  Theorem 2.1.

[Kat50] Suppose, in addition to (3), that the map (2) is $C^{2}$. Then, as
$T\uparrow\infty$,

$\sup_{\tau\in[0,1]}\|(I-P(\tau))U_{T}^{0\to\tau}P(0)\|=O(T^{-1}).$

The adiabatic theorem without a gap. This fundamental refinement of Theorem
2.1 goes back to Avron and Elgart [AE99], who dispense with the gap assumption
(3); for an important technical comment on their result see [Teu01]. The
setting is the same as in Theorem 2.1, except that (3) is replaced by the
following assumption:

> Assumption (AD) There exists a $C^{2}$ map $[0,1]\ni\tau\mapsto
> P(\tau)\in\mathcal{B}(\mathcal{H})$ such that, for Lebesgue a.e.
> $\tau\in[0,1]$, $P(\tau)$ is the orthogonal projection onto the eigenspace
> of a finite multiplicity eigenvalue of $H(\tau)$.

######  Theorem 2.2.

[AE99, Teu01] Suppose that (AD) holds. Then

$\lim_{T\rightarrow\infty}\,\sup_{\tau\in[0,1]}\|(I-P(\tau))U_{T}^{0\to\tau}P(0)\|=0.$

##  3 Adiabatic theorems in quantum statistical mechanics

The isothermal adiabatic theorem for local perturbations. Let
$(\mathcal{O},\alpha)$ be a $C^{\ast}$-dynamical system. Denote by $\delta$
its generator, $\alpha^{t}=\mathrm{e}^{t\delta}$, and by $\omega$ an
$\alpha$-KMS state at inverse temperature $\beta>0$. To a self-adjoint
$V\in\mathcal{O}$ we associate the perturbed dynamics
$\alpha_{V}^{t}=\mathrm{e}^{t\delta_{V}}$, where
$\delta_{V}=\delta+\mathrm{i}[V,\,\cdot\,]$, and the perturbed
$(\alpha_{V},\beta)$-KMS state $\omega_{V}$.

Let

$[0,1]\ni\tau\mapsto V(\tau)\in\mathcal{O}$ (4)

be a continuous map such that $V(\tau)$ is self-adjoint for all $\tau$. For
$T>0$, the unique solution of the Cauchy problem for the non-autonomous
Heisenberg equation

$\partial_{t}\gamma^{t}(A)=\gamma^{t}\circ\delta_{V(t/T)}(A),\qquad\gamma^{s}(A)=A\in\mathop{\mathrm{Dom}}\nolimits(\delta),\qquad
s,t\in[0,T],$

is given by $\gamma^{t}=\alpha_{T}^{s/T\to t/T}$, where
$[0,1]\times[0,1]\ni(\sigma,\tau)\mapsto\alpha_{T}^{\sigma\to\tau}$ is a
strongly continuous two-parameter family of $\ast$-automorphisms of
$\mathcal{O}$ satisfying the relation
$\alpha_{T}^{\sigma\to\tau}\circ\alpha_{T}^{\tau\to\upsilon}=\alpha_{T}^{\sigma\to\upsilon}$
for any $\sigma,\tau,\upsilon\in[0,1]$. For $0\leq\sigma\leq\tau\leq 1$, it
has the norm-convergent expansion333It is understood that the zeroth term in
this expansion is $\alpha^{(\tau-\sigma)T}(A)$. A similar expression holds for
$0\leq\tau\leq\sigma\leq 1$.

$\alpha_{T}^{\sigma\to\tau}(A)=\sum_{n=0}^{\infty}\hskip
6.0ptT^{n}\hskip-16.0pt\int\limits_{\sigma\leq\sigma_{1}\leq\cdots\leq\sigma_{n}\leq\tau}\hskip-14.0pt\mathrm{i}[\alpha^{(\sigma_{1}-\sigma)T}(V(\sigma_{1})),\mathrm{i}[\cdots,\mathrm{i}[\alpha^{(\sigma_{n}-\sigma)T}(V(\sigma_{n})),\alpha^{(\tau-\sigma)T}(A)]\cdots]]\mathrm{d}\sigma_{1}\cdots\mathrm{d}\sigma_{n}.$

Recall the property of Return to Equilibrium reviewed in [JPT23, Section 1.1],
where the reader can also find references to this well-studied topic.

######  Theorem 3.1.

Suppose that the map (4) is $C^{2}$. If, for Lebesgue a.e. $\tau\in[0,1]$, the
quantum dynamical system $(\mathcal{O},\alpha_{V(\tau)},\omega_{V(\tau)})$ has
the property of return to equilibrium, then

$\lim_{T\to\infty}\,\sup_{\tau\in[0,1]}\|\omega_{V(0)}\circ\alpha_{T}^{0\to\tau}-\omega_{V(\tau)}\|=0.$
(5)

This result goes back to [ASF07, JP]; see also [JP14]. The proof is a simple
combination of the Avron–Elgart gapless adiabatic theorem 2.2 and Araki’s
perturbation theory of the modular structure.

For an in depth discussion of Theorem 3.1 and its thermodynamical implications
we refer the reader to [BFJP]. Here we recall three results that will be of
relevance in what follows; see also [JP14, Section 5] for a related
discussion.

The first is the following entropy balance equation:

######  Proposition 3.2.

Suppose that the map (4) is $C^{1}$ and denote by $\dot{V}$ its derivative.
Then444 $S(\,\cdot\,|\,\cdot\,)$ is the relative entropy functional, with the
sign and ordering convention of [JPT23, Section 1.1].

$S\left(\omega_{V(0)}\circ\alpha_{T}^{0\to\tau}\bigg{|}\omega_{V(\tau)}\right)=\beta\int_{0}^{\tau}\left(\omega_{V(0)}\circ\alpha_{T}^{0\to\sigma}-\omega_{V(\sigma)}\right)(\dot{V}(\sigma))\mathrm{d}\sigma$

holds for all $\tau\in[0,1]$.

For the reader’s convenience, a proof of this proposition is given in Section
5.1.

The entropy balance equation yields the estimate

$\sup_{\tau\in[0,1]}S\left(\omega_{V(0)}\circ\alpha_{T}^{0\to\tau}\bigg{|}\omega_{V(\tau)}\right)\leq\beta\sup_{\tau\in[0,1]}\|\omega_{V(0)}\circ\alpha_{T}^{0\to\tau}-\omega_{V(\tau)}\|\,\sup_{\tau\in[0,1]}\|\dot{V}(\tau)\|,$

and the validity of (5) implies that the adiabatic theorem for local
perturbations also holds in the entropic sense:

$\lim_{T\to\infty}\,\sup_{\tau\in[0,1]}S\left(\omega_{V(0)}\circ\alpha_{T}^{0\to\tau}\bigg{|}\omega_{V(\tau)}\right)=0.$
(6)

On the other hand, the Pinsker–Csiszàr inequality [OP93, Proposition 5.23],
$\|\omega-\nu\|^{2}\leq 2S(\omega|\nu)$, gives that (6) implies (5). In
summary:

######  Theorem 3.3.

Suppose that the map (4) is $C^{1}$. Then, the following statements are
equivalent:

  1. (i)

$\lim_{T\to\infty}\,\sup_{\tau\in[0,1]}S\left(\omega_{V(0)}\circ\alpha_{T}^{0\to\tau}\bigg{|}\omega_{V(\tau)}\right)=0.$

  2. (ii)

$\lim_{T\to\infty}\,\sup_{\tau\in[0,1]}\|\omega_{V(0)}\circ\alpha_{T}^{0\to\tau}-\omega_{V(\tau)}\|=0.$

Starting with Proposition 3.2, a similar argument yields the third result
which is also of independent interest.

######  Theorem 3.4.

Suppose that the map (4) is $C^{1}$. Then, the following statements are
equivalent:

  1. (i)

For Lebesgue a.e. $\tau\in[0,1]$,

$\lim_{T\to\infty}\omega_{V(0)}\circ\alpha_{T}^{0\to\tau}(\dot{V}(\tau))=\omega_{V(\tau)}(\dot{V}(\tau)).$

  2. (ii)

For all $\tau\in[0,1]$,

$\lim_{T\to\infty}S\left(\omega_{V(0)}\circ\alpha_{T}^{0\to\tau}\bigg{|}\omega_{V(\tau)}\right)=0.$

  3. (iii)

For all $\tau\in[0,1]$,

$\lim_{T\to\infty}\|\omega_{V(0)}\circ\alpha_{T}^{0\to\tau}-\omega_{V(\tau)}\|=0.$

In Section 4.2 we extend Theorem 3.4 to translation invariant many body
systems.

The adiabatic theorem for gapped ground states of quantum spin systems. A very
general extension of Theorem 2.1 to many body quantum spin systems was
established in [BDRF17, BDRF18]. Since this result plays no role in our work,
we will not discuss it further. The low temperature adiabatic theory of
lattice fermion systems was studied in the recent work [GLMP22]. The relation
between our work and [GLMP22] remains to be studied further.

##  4 Structural theory

###  4.1 Setup

As in [JPT23], we will work with quantum spin systems on the lattice
$\mathbb{Z}^{d}$ and set the inverse temperature to $\beta=1$. All the results
and proofs directly extend to the fermionic case. The notation and terminology
are the same as in [JPT23]; see Appendix A below for a brief summary.

Given an interaction $\Phi\in\mathcal{B}^{r}$, the associated local
Hamiltonian in a finite cube $\Lambda$ is

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

The $C^{\ast}$-dynamics it generates on $\mathfrak{A}$ is

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

and its thermodynamic limit

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

yields a $C^{\ast}$-dynamics commuting with the natural action of the
translation group $\mathbb{Z}^{d}\ni x\mapsto\varphi^{x}$.

We consider time-dependent translation invariant interactions described by
continuous maps

$\Phi:[0,1]\ni\tau\mapsto\Phi_{\tau}\in\mathcal{B}^{r}.$

In analogy with Section 3, the local time-dependent Hamiltonians
$H_{\Lambda}(\Phi_{\tau})$ generate a non-autonomous $C^{\ast}$-dynamics
$\alpha_{\Phi,\Lambda}^{\sigma\to\tau}$ on $\mathfrak{A}$, uniquely
characterized by the Cauchy problem for the Heisenberg equation of motion

$\partial_{\tau}\alpha_{\Phi,\Lambda}^{\sigma\to\tau}(A)=\alpha_{\Phi,\Lambda}^{\sigma\to\tau}(\mathrm{i}[H_{\Lambda}(\Phi_{\tau}),A]),\qquad\alpha_{\Phi,\Lambda}^{\sigma\to\sigma}(A)=A\in\mathfrak{A},\qquad\sigma,\tau\in[0,1].$

The following proposition summarizes the basic properties of its thermodynamic
limit needed in this paper. The reader should consult [NSY19] for a more
detailed and more general discussion of non-autonomous $C^{\ast}$-dynamics.

######  Proposition 4.1.

Let $\Phi:[0,1]\to\mathcal{B}^{r}$ be a continuous, time-dependent,
translation invariant interaction.

  1. (1)

For all $\sigma,\tau\in[0,1]$ and $A\in\mathfrak{A}$, the limit

$\alpha_{\Phi}^{\sigma\to\tau}(A)=\lim_{\Lambda\uparrow\mathbb{Z}^{d}}\alpha_{\Phi,\Lambda}^{\sigma\to\tau}(A)$
(7)

exists and defines a two-parameter family of $\ast$-automorphisms of
$\mathfrak{A}$ such that

$\alpha_{\Phi}^{\sigma\to\tau}\circ\alpha_{\Phi}^{\tau\to\upsilon}=\alpha_{\Phi}^{\sigma\to\upsilon},\qquad\alpha_{\Phi}^{\sigma\to\sigma}=\mathrm{Id},$

for all $\sigma,\tau,\upsilon\in[0,1]$.

  2. (2)

$\alpha_{\Phi}$ commutes with the group action of $\mathbb{Z}^{d}$, i.e.,

$\alpha_{\Phi}^{\sigma\to\tau}\circ\varphi^{x}=\varphi^{x}\circ\alpha_{\Phi}^{\sigma\to\tau},$

for all $\sigma,\tau\in[0,1]$ and $x\in\mathbb{Z}^{d}$.

  3. (3)

The map
$[0,1]\ni\sigma,\tau\mapsto\alpha_{\Phi}^{\sigma\to\tau}\in\mathrm{Aut}(\mathfrak{A})$
is strongly continuous. Moreover, for all $\sigma,\tau\in{]}0,1{[}$ and any
$A\in{\mathfrak{A}_{\mathrm{loc}}}$,

$\partial_{\tau}\alpha_{\Phi}^{\sigma\to\tau}(A)=\alpha_{\Phi}^{\sigma\to\tau}\circ\delta_{\Phi_{\tau}}(A),$

where $\delta_{\Phi_{\tau}}$ denotes the map defined by

$\delta_{\Phi_{\tau}}(A)=\sum_{X\cap\mathop{\mathrm{supp}}\nolimits(A)\neq\emptyset}\mathrm{i}[\Phi_{\tau}(X),A].$

  4. (4)

For $\sigma,\tau\in[0,1]$ and integer $N\geq 1$, set $\xi_{N}=(\tau-\sigma)/N$
and $\upsilon_{k}=\sigma+k\xi_{N}$. Then, for $A\in\mathfrak{A}$, one has

$\alpha_{\Phi}^{\sigma\to\tau}(A)=\lim_{N\to\infty}\alpha_{\Phi_{\upsilon_{0}}}^{\xi_{N}}\circ\cdots\circ\alpha_{\Phi_{\upsilon_{N-1}}}^{\xi_{N}}(A).$

A proof is given in Section 5.2. The family
$\\{\alpha_{\Phi}^{\sigma\to\tau}\\}_{\sigma,\tau\in[0,1]}$ defines the non-
autonomous $C^{\ast}$-dynamics generated by the time-dependent interaction
$\Phi$.

The basic result on the constancy of specific entropy along the state
trajectory, [LR68, Theorem 5], extends to time-dependent interactions.

######  Proposition 4.2.

For any $\nu\in\mathcal{S}_{\mathrm{I}}(\mathfrak{A})$ and
$\sigma,\tau\in[0,1]$,

$s(\nu)=s(\nu\circ\alpha_{\Phi}^{\sigma\to\tau}).$

###### Proof.

Invoking reversibility, it suffices to prove that, for all
$\sigma,\tau\in[0,1]$, one has

$s(\nu)\leq s(\nu\circ\alpha_{\Phi}^{\sigma\to\tau}).$ (8)

Part (4) of Proposition 4.1 gives

$s(\nu\circ\alpha_{\Phi}^{\sigma\to\tau})=s\left(\mathop{\mathrm{w}^{\ast}-\mathrm{lim}}\limits_{N\to\infty}\nu\circ\alpha_{\Phi_{\upsilon_{0}}}^{\xi_{N}}\circ\cdots\circ\alpha_{\Phi_{\upsilon_{N-1}}}^{\xi_{N}}\right),$

and by [LR68, Theorem 5], we have

$s(\nu\circ\alpha_{\Phi_{\upsilon_{0}}}^{\xi_{N}}\circ\cdots\circ\alpha_{\Phi_{\upsilon_{N-1}}}^{\xi_{N}})=s(\nu).$

The upper-semicontinuity of specific entropy yields (8). ∎

###  4.2 Main results

In this section we associate to a time-dependent interaction $\Psi$ a family
of instantaneous equilibrium states. More precisely, we will denote by
$\llbracket\nu,\Psi\rrbracket$ a continuous map

$[0,1]\ni\tau\mapsto(\nu_{\tau},\Psi_{\tau})\in\mathcal{S}_{\mathrm{I}}(\mathfrak{A})\times\mathcal{B}^{r}$

such that $\nu_{\tau}\in\mathcal{S}_{\mathrm{eq}}(\Psi_{\tau})$ for all
$\tau\in[0,1]$. We will consider the non-autonomous time evolution on
$\mathfrak{A}$ defined by the Cauchy problem

$\partial_{t}\gamma^{t}(A)=\gamma^{t}\circ\delta_{\Psi_{t/T}}(A),\qquad\gamma^{s}(A)=A\in{\mathfrak{A}_{\mathrm{loc}}},\qquad
s,t\in[0,T],$

in the adiabatic limit $T\to\infty$. A rescaling of the time variables $s$ and
$t$ gives

$\gamma^{t}=\alpha_{T\Psi}^{s/T\to t/T},$

where $\\{\alpha_{T\Psi}^{\sigma\to\tau}\\}_{\sigma,\tau\in[0,1]}$ is the two-
parameter family of $\ast$-automorphisms generated by the rescaled interaction
$T\Psi$, as described in Proposition 4.1.

######  Definition 4.3.

  1. (1)

A regular pair
$(\nu,\Psi)\in\mathcal{S}_{\mathrm{I}}(\mathfrak{A})\times\mathcal{B}^{r}$ is
uniquely regular whenever $\mathcal{S}_{\mathrm{eq}}(\Psi)=\\{\nu\\}$. A path
$\llbracket\nu,\Psi\rrbracket$ is uniquely regular whenever
$(\nu_{\tau},\Psi_{\tau})$ is uniquely regular for all $\tau\in[0,1]$.

  2. (2)

We say that $\llbracket\nu,\Psi\rrbracket$ satisfies the _adiabatic theorem_
if

$\mathop{\mathrm{w}^{\ast}-\mathrm{lim}}\limits_{T\to\infty}\nu_{0}\circ\alpha_{T\Psi}^{0\to
1}=\nu_{1},$

and the _path adiabatic theorem_ if, for all $\tau\in[0,1]$,

$\mathop{\mathrm{w}^{\ast}-\mathrm{lim}}\limits_{T\to\infty}\nu_{0}\circ\alpha_{T\Psi}^{0\to\tau}=\nu_{\tau}.$

  3. (3)

We say that $\llbracket\nu,\Psi\rrbracket$ satisfies the _entropic adiabatic
theorem_ if

$\lim_{T\to\infty}s(\nu_{0}\circ\alpha_{T\Psi}^{0\to 1}|\nu_{1})=0,$

and the _entropic path adiabatic theorem_ if, for all $\tau\in[0,1]$,

$\lim_{T\to\infty}s(\nu_{0}\circ\alpha_{T\Psi}^{0\to\tau}|\nu_{\tau})=0.$

In the fermionic case, gauge-invariance implies that the adiabatic theorem can
hold only if $\nu_{0}(E_{N})=\nu_{1}(E_{N})$. Similarly, the path adiabatic
theorem can hold only if $\nu_{0}(E_{N})=\nu_{\tau}(E_{N})$ for all
$\tau\in[0,1]$. In what follows we will discuss only the quantum spin case,
and leave the elementary reformulations needed to accommodate the fermionic
setting to the interested reader.

######  Proposition 4.4.

Suppose that $(\nu_{1},\Psi_{1})$ is a uniquely regular pair. Then the
following statements are equivalent:

  1. (i)

$s(\nu_{0})=s(\nu_{1})$ and the adiabatic theorem holds for
$\llbracket\nu,\Psi\rrbracket$.

  2. (ii)

$s(\nu_{0})=s(\nu_{1})$ and
$\displaystyle\lim_{T\to\infty}\nu_{0}\circ\alpha_{T\Psi}^{0\to
1}(E_{\Psi_{1}})=\nu_{1}(E_{\Psi_{1}})$.

  3. (iii)

The entropic adiabatic theorem holds for $\llbracket\nu,\Psi\rrbracket$.

###### Proof.

Note that $\nu_{0}\circ\alpha_{T\Psi}^{0\to
1}\in\mathcal{S}_{\mathrm{I}}(\mathfrak{A})$ by Part (2) of Proposition 4.1.

(i)$\Rightarrow$(ii). Follows directly from Definition 4.3.

(ii)$\Rightarrow$(iii). The regularity of $(\nu_{1},\Psi_{1})$ gives

$s(\nu_{0}\circ\alpha_{T\Psi}^{0\to
1}|\nu_{1})=-s(\nu_{0}\circ\alpha_{T\Psi}^{0\to
1})+\nu_{0}\circ\alpha_{T\Psi}^{0\to 1}(E_{\Psi_{1}})+P(\Psi_{1}).$ (9)

The assumption $s(\nu_{0})=s(\nu_{1})$ and Proposition 4.2 yield that
$s(\nu_{0}\circ\alpha_{T\Psi}^{0\to 1})=s(\nu_{1})$, and so

$\lim_{T\to\infty}s(\nu_{0}\circ\alpha_{T\Psi}^{0\to
1}|\nu_{1})=-s(\nu_{1})+\nu_{1}(E_{\Psi_{1}})+P(\Psi_{1})=0,$

where we used that $\nu_{1}\in\mathcal{S}_{\mathrm{eq}}(\Psi_{1})$.

(iii)$\Rightarrow$(i). Let $\nu_{+}$ be a $\mathrm{weak}^{\ast}$-limit point
of the net $(\nu_{0}\circ\alpha_{T\Psi}^{0\to 1})_{T>0}$ as $T\to\infty$. The
regularity of $(\nu_{1},\Psi_{1})$ implies that the map
$\mathcal{S}_{\mathrm{I}}(\mathfrak{A})\ni\omega\mapsto s(\omega|\nu_{1})$ is
lower-semicontinuous, and so

$0\leq
s(\nu_{+}|\nu_{1})\leq\liminf_{T\to\infty}s(\nu_{0}\circ\alpha_{T\Psi}^{0\to
1}|\nu_{1})=0.$

Since $\nu_{+}\in\mathcal{S}_{\mathrm{I}}(\mathfrak{A})$, we deduce from

$0=s(\nu_{+}|\nu_{1})=-s(\nu_{+})+\nu_{+}(E_{\Psi_{1}})+P(\Psi_{1})$

that $\nu_{+}\in\mathcal{S}_{\mathrm{eq}}(\Psi_{1})=\\{\nu_{1}\\}$. It follows
that $\nu_{0}\circ\alpha_{T\Psi}^{0\to 1}\rightharpoonup\nu_{1}$ as
$T\to\infty$.

To prove that $s(\nu_{0})=s(\nu_{1})$, note that the formula (9) and
Proposition 4.2 yield

$s(\nu_{0}\circ\alpha_{T\Psi}^{0\to
1}|\nu_{1})=-s(\nu_{0})+\nu_{0}\circ\alpha_{T\Psi}^{0\to
1}(E_{\Psi_{1}})+P(\Psi_{1}).$

Hence,

$0=\lim_{T\rightarrow\infty}s(\nu_{0}\circ\alpha_{T\Psi}^{0\to
1}|\nu_{1})=-s(\nu_{0})+\nu_{1}(E_{\Psi_{1}})+P(\Psi_{1})=s(\nu_{1})-s(\nu_{0}),$

where we used that $\nu_{1}\in\mathcal{S}_{\mathrm{eq}}(\Psi_{1})$. ∎

An immediate consequence of the last result is:

######  Theorem 4.5.

Suppose that the path $\llbracket\nu,\Psi\rrbracket$ is uniquely regular. Then
the following statements are equivalent:

  1. (i)

$s(\nu_{0})=s(\nu_{\tau})$ for all $\tau\in[0,1]$ and the path adiabatic
theorem holds for $\llbracket\nu,\Psi\rrbracket$.

  2. (ii)

$s(\nu_{0})=s(\nu_{\tau})$ and
$\displaystyle\lim_{T\to\infty}\nu_{0}\circ\alpha_{T\Psi}^{0\to\tau}(E_{\Psi_{\tau}})=\nu_{\tau}(E_{\Psi_{\tau}})$
for all $\tau\in[0,1]$.

  3. (iii)

The entropic path adiabatic theorem holds for $\llbracket\nu,\Psi\rrbracket$.

Our next result is:

######  Theorem 4.6.

Suppose that the path $\llbracket\nu,\Psi\rrbracket$ is uniquely regular. If
$\Psi\in C^{1}([0,1],\mathcal{B}^{r})$, then the following statements are
equivalent:

  1. (i)

For Lebesgue a.e. $\tau\in[0,1]$,

$\lim_{T\to\infty}\nu_{0}\circ\alpha_{T\Psi}^{0\to\tau}(\partial_{\tau}E_{\Psi_{\tau}})=\nu_{\tau}(\partial_{\tau}E_{\Psi_{\tau}}).$

  2. (ii)

The entropic path adiabatic theorem holds for $\llbracket\nu,\Psi\rrbracket$.

###### Proof.

(ii)$\Rightarrow$(i) follows from Theorem 4.5.

(i)$\Rightarrow$(ii). The map $[0,1]\ni\tau\mapsto E_{\Psi_{\tau}}$ is
continuously differentiable by assumption. Moreover, it follows from Part (3)
of Proposition 4.1 that for $A\in{\mathfrak{A}_{\mathrm{loc}}}$,

$\alpha_{T\Psi}^{\sigma\to\tau}(A)=A+T\int_{\sigma}^{\tau}\alpha_{T\Psi}^{\sigma\to\upsilon}\circ\delta_{\Psi_{\upsilon}}(A)\mathrm{d}\upsilon.$

By the proof of Part (2) of [JPT23, Theorem 2.8], for any
$\tau,\upsilon\in[0,1]$ one has
$E_{\Psi_{\tau}}\in\mathop{\mathrm{Dom}}\nolimits(\delta_{\Psi_{\upsilon}})$
and

$\omega\circ\delta_{\Psi_{\tau}}(E_{\Psi_{\tau}})=0\ \text{for all
}\omega\in\mathcal{S}_{\mathrm{I}}(\mathfrak{A}).$ (10)

Moreover, there exist a sequence $(E_{\Psi_{\tau},n})_{n\in\mathbb{N}}$ in
${\mathfrak{A}_{\mathrm{loc}}}$ such that555The norm $\|\Psi\|_{r}$ of a time-
dependent interaction $\Psi$ is defined in (21).
$\|\delta_{\Psi_{\upsilon}}(E_{\Psi_{\tau,n}})\|\leq 2\|\Psi\|_{r}^{2}$, and

$\lim_{n\to\infty}E_{\Psi_{\tau},n}=E_{\Psi_{\tau}},\qquad\lim_{n\to\infty}\delta_{\Psi_{\upsilon}}(E_{\Psi_{\tau},n})=\delta_{\Psi_{\upsilon}}(E_{\Psi_{\tau}}).$

Hence, the dominated convergence theorem yields

$\alpha_{T\Psi}^{\sigma\to\tau}(E_{\Psi_{\tau}})=E_{\Psi_{\tau}}+T\int_{\sigma}^{\tau}\alpha_{T\Psi}^{\sigma\to\upsilon}\circ\delta_{\Psi_{\upsilon}}(E_{\Psi_{\tau}})\mathrm{d}\upsilon,$

from which we deduce

$\partial_{\tau}\nu_{0}\circ\alpha_{T\Psi}^{0\to\tau}(E_{\Psi_{\tau}})=T\nu_{0}\circ\alpha_{T\Psi}^{0\to\tau}\circ\delta_{\Psi_{\tau}}(E_{\Psi_{\tau}})+\nu_{0}\circ\alpha_{T\Psi}^{0\to\tau}(\partial_{\tau}E_{\Psi_{\tau}}).$

Since
$\nu_{0}\circ\alpha_{T\Psi}^{0\to\tau}\in\mathcal{S}_{\mathrm{I}}(\mathfrak{A})$
by Part (2) of Proposition 4.1, it follows from (10) that

$\nu_{0}\circ\alpha_{T\Psi}^{0\to\tau}(E_{\Psi_{\tau}})-\nu_{0}(E_{\Psi_{0}})=\int_{0}^{\tau}\nu_{0}\circ\alpha_{T\Psi}^{0\to\sigma}(\partial_{\sigma}E_{\Psi_{\sigma}})\mathrm{d}\sigma.$

Taking the limit $T\to\infty$, it follows from our hypotheses and the
dominated convergence theorem that

$\lim_{T\to\infty}\nu_{0}\circ\alpha_{\Psi,T}^{0\to\tau}(E_{\Psi_{\tau}})-\nu_{0}(E_{\Psi_{0}})=\int_{0}^{\tau}\nu_{\sigma}(\partial_{\sigma}E_{\Psi_{\sigma}})\mathrm{d}\sigma.$
(11)

The uniqueness of the equilibrium state for $\Psi_{\tau}$ implies that the
pressure $\mathcal{B}^{r}\ni\Phi\mapsto P(\Phi)$ is differentiable at
$\Psi_{\tau}$, with

$P(\Psi_{\tau}+\Phi)-P(\Psi_{\tau})+\nu_{\tau}(E_{\Phi})=o(\|\Phi\|_{r})$

for $\Phi\in\mathcal{B}^{r}$ [LR68, Theorem 3]. Invoking again the
differentiability of $\tau\mapsto E_{\Psi_{\tau}}$ yields

$\partial_{\tau}P(\Psi_{\tau})=-\nu_{\tau}(\partial_{\tau}E_{\Psi_{\tau}}).$
(12)

Combining Relations (12) and (11) we derive

$\lim_{T\to\infty}\nu_{0}\circ\alpha_{\Psi,T}^{0\to\tau}(E_{\Psi_{\tau}})+P(\Psi_{\tau})=\nu_{0}(E_{\Psi_{0}})+P(\Psi_{0}).$

Now, since $(\nu_{\tau},\Psi_{\tau})$ is a regular pair, we have

$s(\nu_{0}\circ\alpha_{\Psi,T}^{0\to\tau}|\nu_{\tau})=-s(\nu_{0})+\nu_{0}\circ\alpha_{\Psi,T}^{0\to\tau}(E_{\Psi_{\tau}})+P(\Psi_{\tau}),$

and so

$\lim_{T\to\infty}s(\nu_{0}\circ\alpha_{\Psi,T}^{0\to\tau}|\nu_{\tau})=-s(\nu_{0})+\nu_{0}(E_{\Psi_{0}})+P(\Psi_{0})=0.$

∎

The following example illustrates the last result. Let
$\Phi_{0},\Phi_{1}\in\mathcal{B}^{r}$, and let $\lambda\in C^{1}([0,1])$ be
such that $\lambda(0)=0$, $\lambda(1)=1$. Set

$\Psi_{\tau}=\Phi_{0}+\lambda(\tau)(\Phi_{1}-\Phi_{0}),$

and assume that the path $\llbracket\nu,\Psi\rrbracket$ is uniquely regular.
Then the entropic path adiabatic theorem holds for
$\llbracket\nu,\Psi\rrbracket$ iff, for all $\tau\in[0,1]$ with
$\partial_{\tau}\lambda(\tau)\not=0$,

$\lim_{T\rightarrow\infty}\nu_{0}\circ\alpha_{T\Psi}^{0\to\tau}(E_{\Phi_{1}-\Phi_{0}})=\nu_{\tau}(E_{\Phi_{1}-\Phi_{0}}).$

We finish this section with the translation invariant version Theorem 3.4.
This result is direct a consequence Theorems 4.5 and 4.6 and we state it for
completeness reason.

######  Theorem 4.7.

Suppose that the path $\llbracket\nu,\Psi\rrbracket$ is uniquely regular. If
$\Psi\in C^{1}([0,1],\mathcal{B}^{r})$, then the following statements are
equivalent:

  1. (i)

For Lebesgue a.e. $\tau\in[0,1]$,

$\lim_{T\to\infty}\nu_{0}\circ\alpha_{T\Psi}^{0\to\tau}(\partial_{\tau}E_{\Psi_{\tau}})=\nu_{\tau}(\partial_{\tau}E_{\Psi_{\tau}}).$

  2. (ii)

The entropic path adiabatic theorem holds for $\llbracket\nu,\Psi\rrbracket$.

  3. (iii)

The path adiabatic theorem holds for $\llbracket\nu,\Psi\rrbracket$.

Moreover, any of the above statements implies

  1. (iv)

For all $\tau\in[0,1]$,

$s(\nu_{\tau})=s(\nu_{0}).$

###  4.3 Adiabaticity and the approach to equilibrium

Theorem 3.1 points to a close relation between the adiabatic theorem for local
perturbations of KMS states and return to equilibrium. In view of the formal
analogy between Proposition 3.4 and Theorem 4.7, one may expect a similar
relation between the adiabatic theorem for translation invariant interactions
and approach to equilibrium. Our last result shows that this is not the case.

We first recall the framework of approach to equilibrium developed in [JPT23].
Let $\omega\in\mathcal{S}_{\mathrm{I}}(\mathfrak{A})$ and
$\Phi\in\mathcal{B}^{r}$. Set

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

We denote by $\mathcal{S}_{+}(\omega,\Phi)$ the set of
$\mathrm{weak}^{\ast}$-limit points of the net $(\bar{\omega}_{T})_{T>0}$ as
$T\to\infty$. Approach to equilibrium holds in this setting whenever
$\mathcal{S}_{+}(\omega,\Phi)\cap\mathcal{S}_{\mathrm{eq}}(\Phi)\neq\emptyset$.

######  Theorem 4.8.

Let $(\nu_{0},\Phi_{0})$ and $(\nu_{1},\Phi_{1})$ be two uniquely regular
pairs and suppose that $\Phi_{0}$ and $\Phi_{1}$ are not physically
equivalent. Then the following statements are mutually exclusive.

  1. (i)

$\nu_{1}\in\mathcal{S}_{+}(\nu_{0},\Phi_{1})$, i.e., approach to equilibrium
holds for $(\mathfrak{A},\alpha_{\Phi_{1}},\nu_{0})$.

  2. (ii)

There exists a uniquely regular path $\llbracket\nu,\Psi\rrbracket$, with
$\Psi\in C^{1}([0,1],\mathcal{B}^{r})$, connecting $(\nu_{0},\Phi_{0})$ to
$(\nu_{1},\Phi_{1})$ and satisfying the path adiabatic theorem.

###### Proof.

Approach to equilibrium has been characterized in terms of a strict increase
of specific entropy. According to [JPT23, Theorem 2.20], if
$\nu_{1}\in\mathcal{S}_{+}(\nu_{0},\Phi_{1})$, then $s(\nu_{0})<s(\nu_{1})$
since $\Phi_{0}$ and $\Phi_{1}$ are not physically equivalent. By Theorem 4.7,
this fact is incompatible with the existence of the path
$\llbracket\nu,\Psi\rrbracket$.

Conversely, Theorem 4.7 shows that the existence of the path
$\llbracket\nu,\Psi\rrbracket$ implies the equality $s(\nu_{0})=s(\nu_{1})$.
By [JPT23, Theorem 2.20], if $\nu_{1}\in\mathcal{S}_{+}(\nu_{0},\Phi_{1})$,
then $s(\nu_{0})=s(\nu_{1})$ implies $\nu_{0}=\nu_{1}$ and that $\Phi_{0}$ and
$\Phi_{1}$ are physically equivalent, which contradicts the assumption of the
theorem. ∎

##  5 Proofs of Propositions 3.2 and 4.1

###  5.1 Proof of Proposition 3.2

Our standing assumption is that the map $V\in C^{1}([0,1],\mathcal{O})$ takes
its value in the self-adjoint elements of $\mathcal{O}$, $\dot{V}$ denoting
its derivative. Denote by $[0,1]\ni s,t\mapsto\alpha_{T}^{s\to t}$ the
2-parameter family of $\ast$-automorphisms of $\mathcal{O}$ uniquely
determined by the Cauchy problem

$\frac{1}{T}\partial_{t}\alpha_{T}^{s\to t}(A)=\alpha_{T}^{s\to
t}\circ\delta_{V(t)}(A),\qquad\alpha_{T}^{s\to
s}(A)=A\in\mathop{\mathrm{Dom}}\nolimits(\delta),\qquad s,t\in[0,1].$

We start with:

######  Lemma 5.1.

For $s,t\in[0,1]$, one has

$\alpha_{T}^{s\to t}=\gamma_{T}^{s\to t}\circ\alpha_{V(t)}^{(t-s)T},$ (13)

where $\gamma_{T}^{s\to t}$ is the inner $\ast$-automorphism of $\mathcal{O}$
given by $\gamma_{T}^{s\to t}(A)=\Gamma_{T}^{s\to t}A\Gamma_{T}^{s\to t\ast}$,
and the 2-parameter family of unitaries $\Gamma_{T}$ is the solution of the
Cauchy problem

$\frac{\mathrm{i}}{T}\partial_{t}\Gamma_{T}^{s\to t}=\Gamma_{T}^{s\to
t}\int_{s}^{t}\alpha_{V(t)}^{(r-s)T}(\dot{V}(t))\mathrm{d}r,\qquad\Gamma_{T}^{s\to
s}=\mathbbm{1}\qquad s,t\in[0,1].$ (14)

Moreover, one has $\Gamma_{T}^{s\to
t}\in\mathop{\mathrm{Dom}}\nolimits(\delta)$ for all $s,t\in[0,1]$.

###### Proof.

Fix $T>0$, $s\in[0,1]$, and set

$\widetilde{H}_{t}=T\int_{s}^{t}\alpha_{V(t)}^{(r-s)T}(\dot{V}(t))\mathrm{d}r,$

so that

$\partial_{t}\gamma_{T}^{s\to t}(A)=\gamma_{T}^{s\to
t}(-\mathrm{i}[\widetilde{H}_{t},A]).$ (15)

For $A\in\mathop{\mathrm{Dom}}\nolimits(\delta)$, one has
$\partial_{u}\delta_{V(u)}(A)=\mathrm{i}[\dot{V}(u),A]$, and Duhamel’s formula
yields

$\partial_{u}\alpha_{V(u)}^{(t-s)T}(A)=T\int_{s}^{t}\alpha_{V(u)}^{(r-s)T}(\mathrm{i}[\dot{V}(u),\alpha_{V(u)}^{(t-r)T}(A)])\mathrm{d}r,$

so that

$\partial_{u}\alpha_{V(u)}^{(t-s)T}(A)\bigg{|}_{u=t}=\mathrm{i}[\widetilde{H}_{t},\alpha_{V(t)}^{(t-s)T}(A)],$

and hence

$\partial_{t}\alpha_{V(t)}^{(t-s)T}(A)=T\alpha_{V(t)}^{(t-s)T}\circ\delta_{V(t)}(A)+\mathrm{i}[\widetilde{H}_{t},\alpha_{V(t)}^{(t-s)T}(A)].$
(16)

Combining Relations (15) and (16), we derive

$\displaystyle\partial_{t}\gamma_{T}^{s\to t}\circ\alpha_{V(t)}^{(t-s)T}(A)$
$\displaystyle=\gamma_{T}^{s\to
t}(-\mathrm{i}[\widetilde{H}_{t},\alpha_{V(t)}^{(t-s)T}(A)])+\gamma_{T}^{s\to
t}(T\alpha_{V(t)}^{(t-s)T}\circ\delta_{V(t)}(A)+\mathrm{i}[\widetilde{H}_{t},\alpha_{V(t)}^{(t-s)T}(A)])$
$\displaystyle=T\gamma_{T}^{s\to
t}\circ\alpha_{V(t)}^{(t-s)T}\circ\delta_{V(t)}(A).$

Since obviously $\gamma_{T}^{s\to s}\circ\alpha_{V(t)}^{(s-s)T}(A)=A$,
Relation (13) holds.

To prove the last statement, for fixed $s,t\in[0,1]$ let the sequence
$(Q_{n})_{n\in\mathbb{N}}\subset\mathop{\mathrm{Dom}}\nolimits(\delta_{V(t)})$
be such that $\lim_{n\to\infty}Q_{n}=\dot{V}(t)$. Setting

$H_{n}=T\int_{s}^{t}\alpha_{V(t)}^{(r-s)T}(Q_{n})\mathrm{d}r,$

and observing that

$T\int_{s}^{t}\alpha_{V(t)}^{(r-s)T}(\delta_{V(t)}(Q_{n}))\mathrm{d}r=\int_{s}^{t}\partial_{r}\alpha_{V(t)}^{(r-s)T}(Q_{n})\mathrm{d}r=\alpha_{V(t)}^{(t-s)T}(Q_{n})-Q_{n},$

we derive

$\delta_{V(t)}(H_{n})=\alpha_{V(t)}^{(t-s)T}(Q_{n})-Q_{n}.$

From the obvious facts that

$\lim_{n\to\infty}H_{n}=\widetilde{H}_{t},\qquad\lim_{n\to\infty}\delta_{V(t)}(H_{n})=\alpha_{V(t)}^{(t-s)T}(\dot{V}(t))-\dot{V}(t),$

the closedness of $\delta_{V(t)}$ allows us to conclude that
$\widetilde{H}_{t}\in\mathop{\mathrm{Dom}}\nolimits(\delta_{V(t)})=\mathop{\mathrm{Dom}}\nolimits(\delta)$
and that

$\delta_{V(t)}(\widetilde{H}_{t})=\alpha_{V(t)}^{(t-s)T}(\dot{V}(t))-\dot{V}(t).$
(17)

Writing the solution to the Cauchy problem (14) as the uniformly convergent
Dyson series

$\Gamma_{T}^{s\to t}=\sum_{n\geq 0}(-\mathrm{i})^{n}\int\limits_{s\leq
t_{1}\leq\cdots\leq t_{n}\leq
t}\widetilde{H}_{t_{1}}\cdots\widetilde{H}_{t_{n}}\mathrm{d}t_{1}\cdots\mathrm{d}t_{n},$

one deduces $\Gamma_{T}^{s\to t}\in\mathop{\mathrm{Dom}}\nolimits(\delta)$ for
all $s,t\in[0,1]$ and that

$\delta(\Gamma_{T}^{s\to t})=\sum_{n\geq 0}(-\mathrm{i})^{n}\sum_{k=1}^{n}\
\int\limits_{s\leq t_{1}\leq\cdots\leq t_{n}\leq
t}\widetilde{H}_{t_{1}}\cdots\delta(\widetilde{H}_{t_{k}})\cdots\widetilde{H}_{t_{n}}\mathrm{d}t_{1}\cdots\mathrm{d}t_{n},$

where the series on right hand side converges uniformly. ∎

We now proceed with the proof of Proposition 3.2.

The previous lemma gives

$S(\omega_{V(s)}\circ\alpha_{T}^{s\to
t}|\omega_{V(t)})=S(\omega_{V(s)}\circ\gamma_{T}^{s\to
t}\circ\alpha_{V(t)}^{(t-s)T}|\omega_{V(t)})=S(\omega_{V(s)}\circ\gamma_{T}^{s\to
t}|\omega_{V(t)}),$

and invoking [JP03, Theorem 1.1]666We note that the definition of relative
entropy used in [JP03, DJP03] differs in its sign with the one used here., we
can write

$S(\omega_{V(s)}\circ\alpha_{T}^{s\to
t}|\omega_{V(t)})=S(\omega_{V(s)}|\omega_{V(t)})-\mathrm{i}\beta\omega_{V(s)}(\Gamma_{T}^{s\to
t}\delta_{V(t)}(\Gamma_{T}^{s\to t\ast})).$ (18)

To deal with the first term on the right-hand side, we invoke Araki’s
perturbation theory (see, e.g., [DJP03, Theorem 5.1]). Setting
$\nu=\omega_{V(s)}$ and $W_{t}=V(t)-V(s)$, we have

$S(\omega_{V(s)}|\omega_{V(t)})=S(\nu|\nu_{W_{t}})=\beta\nu(W_{t})+\log\langle\Omega_{\nu},\mathrm{e}^{-\beta(L+\pi_{\nu}(W_{t}))}\Omega_{\nu}\rangle,$

where $(\mathcal{H}_{\nu},\pi_{\nu},\Omega_{\nu})$ denotes the GNS
representation of $\mathcal{O}$ induced by $\nu$, and $L$ is the standard
Liouvillean generating the unitary implementation of the $C^{\ast}$-dynamics
$\alpha_{V(s)}$ on $\mathcal{H}_{\nu}$. Taking the derivative w.r.t. $t$ of
the logarithmic term and using the facts that $W_{s}=0$ and $L\Omega_{\nu}=0$
gives

$\log\langle\Omega_{\nu},\mathrm{e}^{-\beta(L+\pi_{\nu}(W_{t}))}\Omega_{\nu}\rangle=\int_{s}^{t}\partial_{u}\log\langle\Omega_{\nu},\mathrm{e}^{-\beta(L+\pi_{\nu}(W_{u}))}\Omega_{\nu}\rangle\mathrm{d}u.$

Duhamel’s formula further yields

$\displaystyle\partial_{u}\log\langle\Omega_{\nu},\mathrm{e}^{-\beta(L+\pi_{\nu}(W_{u}))}\Omega_{\nu}\rangle$
$\displaystyle=-\int_{0}^{\beta}\frac{\langle\Omega_{\nu},\mathrm{e}^{-\gamma(L+\pi_{\nu}(W_{u}))}\pi_{\nu}(\dot{V}(u))\mathrm{e}^{-(\beta-\gamma)(L+\pi_{\nu}(W_{u}))}\Omega_{\nu}\rangle}{\|\Psi_{u}\|^{2}}\mathrm{d}\gamma$
$\displaystyle=-\int_{-\beta/2}^{\beta/2}\frac{\langle\Psi_{u},\mathrm{e}^{-\gamma(L+\pi_{\nu}(W_{u}))}\pi_{\nu}(\dot{V}(u))\mathrm{e}^{\gamma(L+\pi_{\nu}(W_{u}))}\Psi_{u}\rangle}{\|\Psi_{u}\|^{2}}\mathrm{d}\gamma,$

where $\Psi_{u}=\mathrm{e}^{-\beta(L+\pi_{\nu}(W_{u}))/2}\Omega_{\nu}$ is, up
to normalization, the vector representative of $\omega_{V(u)}$ in
$\mathcal{H}_{\nu}$. It follows that

$\partial_{u}\log\langle\Omega_{\nu},\mathrm{e}^{-\beta(L+\pi_{\nu}(W_{u}))}\Omega_{\nu}\rangle=-\int_{-\beta/2}^{\beta/2}\omega_{V(u)}\left(\alpha_{V(u)}^{\mathrm{i}\gamma}(\dot{V}(u))\right)\mathrm{d}\gamma=-\beta\omega_{V(u)}(\dot{V}(u)),$

and hence

$S(\omega_{V(s)}|\omega_{V(t)})=\beta\omega_{V(s)}(V(t)-V(s))-\beta\int_{s}^{t}\omega_{V(u)}(\dot{V}(u))\mathrm{d}u=-\beta\int_{s}^{t}(\omega_{V(u)}-\omega_{V(s)})(\dot{V}(u))\mathrm{d}u.$

To deal with the second term on the right-hand side of (18), we note that

$\mathrm{i}\Gamma_{T}^{s\to t}\delta_{V(t)}(\Gamma_{T}^{s\to
t\ast})=\int_{s}^{t}\mathrm{i}\partial_{u}\Gamma_{T}^{s\to
u}\delta_{V(u)}(\Gamma_{T}^{s\to u\ast})\mathrm{d}u,$

and so, using Relation (17), we have that

$\displaystyle\mathrm{i}\partial_{u}\Gamma_{T}^{s\to
u}\delta_{V(u)}(\Gamma_{T}^{s\to u\ast})$ $\displaystyle=\Gamma_{T}^{s\to
u}\widetilde{H}_{u}\delta_{V(u)}(\Gamma_{T}^{s\to u\ast})-\Gamma_{T}^{s\to
u}\delta_{V(u)}(\widetilde{H}_{u}\Gamma_{T}^{s\to u\ast})-\Gamma_{T}^{s\to
u}[\dot{V}(u),\Gamma_{T}^{s\to u\ast}]$ $\displaystyle=-\Gamma_{T}^{s\to
u}(\delta_{V(u)}(\widetilde{H}_{u})+\dot{V}(u))\Gamma_{T}^{s\to
u\ast}+\dot{V}(u)$ $\displaystyle=-\Gamma_{T}^{s\to
u}\alpha_{V(u)}^{(u-s)T}(\dot{V}(u))\Gamma_{T}^{s\to u\ast}+\dot{V}(u)$
$\displaystyle=-\alpha_{T}^{s\to u}(\dot{V}(u))+\dot{V}(u).$

It follows that

$-\mathrm{i}\beta\omega_{V(s)}(\Gamma_{T}^{s\to
t}\delta_{V(t)}(\Gamma_{T}^{s\to
t\ast}))=\beta\int_{s}^{t}\left(\omega_{V(s)}\circ\alpha_{T}^{s\to
u}(\dot{V}(u))-\omega_{V(s)}(\dot{V}(u))\right)\mathrm{d}u,$

and finally, Relation (18) gives

$S(\omega_{V(s)}\circ\alpha_{T}^{s\to
t}|\omega_{V(t)})=\beta\int_{s}^{t}\left(\omega_{V(s)}\circ\alpha_{T}^{s\to
u}(\dot{V}(u))-\omega_{V(u)}(\dot{V}(u))\right)\mathrm{d}u.$

###  5.2 Proof of Proposition 4.1

We start with some preliminary observations. By [BR81, Theorem 6.2.4], the
time-dependent derivation $[0,1]\ni\tau\mapsto\delta_{\Phi_{\tau}}$ defined on
${\mathfrak{A}_{\mathrm{loc}}}$ by

$\delta_{\Phi_{\tau}}(A)=\sum_{X\cap\mathop{\mathrm{supp}}\nolimits(A)\neq\emptyset}\mathrm{i}[\Phi_{\tau}(X),A],$

is closable and its closure, which we shall denote by the same symbol,
generates the frozen $C^{\ast}$-dynamics

$\mathbb{R}\ni
t\mapsto\alpha_{\Phi_{\tau}}^{t}=\mathrm{e}^{t\delta_{\Phi_{\tau}}}.$

The translation invariance of the interaction $\Phi_{\tau}$ implies that
$\varphi^{x}(\mathop{\mathrm{Dom}}\nolimits(\delta_{\Phi_{\tau}}))=\mathop{\mathrm{Dom}}\nolimits(\delta_{\Phi_{\tau}})$
and that

$\delta_{\Phi_{\tau}}\circ\varphi^{x}=\varphi^{x}\circ\delta_{\Phi_{\tau}},$
(19)

for all $x\in\mathbb{Z}^{d}$.

Note that, for $A\in{\mathfrak{A}_{\mathrm{loc}}}$, one has

$\delta_{\Phi_{\tau}}(A)=\lim_{\Lambda\uparrow\mathbb{Z}^{d}}\delta_{\Phi_{\tau},\Lambda}(A),$

where the convergence is in norm and

$\delta_{\Phi_{\tau},\Lambda}(A)=\sum_{X\subset\Lambda}\mathrm{i}[\Phi_{\tau}(X),A]=\mathrm{i}[H_{\Lambda}(\Phi_{\tau}),A],$

is such that
$\mathop{\mathrm{supp}}\nolimits(\delta_{\Phi_{\tau},\Lambda}(A))\subset\mathop{\mathrm{supp}}\nolimits(A)\cup\Lambda$,
i.e.,
$\delta_{\Phi_{\tau},\Lambda}({\mathfrak{A}_{\mathrm{loc}}})\subset{\mathfrak{A}_{\mathrm{loc}}}$.
It follows that the non-autonomous $C^{\ast}$-dynamics generated by the time-
dependent local Hamiltonian $H_{\Lambda}(\Phi_{\tau})$ is given by the norm-
convergent Dyson expansion

$\alpha_{\Phi,\Lambda}^{\sigma\to\tau}(A)=\begin{cases}\displaystyle
A+\sum_{n\geq
1}\int_{\Delta_{n}(\sigma,\tau)}\delta_{\Phi_{\tau_{1}},\Lambda}\circ\cdots\circ\delta_{\Phi_{\tau_{n}},\Lambda}(A)\,\mathrm{d}\tau_{1}\cdots\mathrm{d}\tau_{n},&\text{
for }0\leq\sigma\leq\tau\leq 1;\\\\[12.0pt] \displaystyle A+\sum_{n\geq
1}\int_{\Delta_{n}(\sigma,\tau)}\delta_{\Phi_{\tau_{n}},\Lambda}\circ\cdots\circ\delta_{\Phi_{\tau_{1}},\Lambda}(A)\,\mathrm{d}\tau_{1}\cdots\mathrm{d}\tau_{n},&\text{
for }0\leq\tau\leq\sigma\leq 1;\end{cases}$ (20)

for $A\in{\mathfrak{A}_{\mathrm{loc}}}$, where
$\Delta_{n}(\sigma,\tau)=\\{(\tau_{1},\ldots,\tau_{n})\mid
0\leq\min(\sigma,\tau)\leq\tau_{1}\leq\cdots\leq\tau_{n}\leq\max(\sigma,\tau)\\}$.

In the following, the norm of a time-dependent continuous interaction
$[0,1]\ni\tau\mapsto\Phi(\tau)\in\mathcal{B}^{r}$ is taken to be

$\|\Phi\|_{r}=\sup_{\tau\in[0,1]}\|\Phi_{\tau}\|_{r}.$ (21)

(1) A trivial extension of [Isr79, Lemma III.3.5] gives that, for
$A\in{\mathfrak{A}_{\mathrm{loc}}}$ and $\tau_{1},\ldots,\tau_{n}\in[0,1]$,

$\sum_{X_{1},\ldots,X_{n}\in\mathcal{F}}\left\|\mathrm{i}[\Phi_{\tau_{1}}(X_{1}),\mathrm{i}[\Phi_{\tau_{2}}(X_{2}),\cdots,\mathrm{i}[\Phi_{\tau_{n}}(X_{n}),A]\cdots]]\right\|\leq\|A\|\mathrm{e}^{r|\mathop{\mathrm{supp}}\nolimits(A)|}\left(\frac{2\|\Phi\|_{r}}{r}\right)^{n}n!.$

Hence,

$\sup_{\tau_{1},\ldots,\tau_{n}\in[0,1]}\|\delta_{\Phi_{\tau_{1}}}\circ\cdots\circ\delta_{\Phi_{\tau_{n}}}(A)\|\leq\|A\|\mathrm{e}^{r|\mathop{\mathrm{supp}}\nolimits(A)|}\left(\frac{2\|\Phi\|_{r}}{r}\right)^{n}n!,$
(22)

and the same estimate holds with $\delta_{\Phi_{\tau_{k}}}$ replaced by
$\delta_{\Phi_{\tau_{k}},\Lambda}$. Thus, taking the thermodynamic limit in
(20) yields that, for $|\tau-\sigma|<\epsilon=r/2\|\Phi\|_{r}$,

$\lim_{\Lambda\uparrow\mathbb{Z}^{d}}\alpha_{\Phi,\Lambda}^{\sigma\to\tau}(A)=\begin{cases}\displaystyle
A+\sum_{n\geq
1}\int_{\Delta_{n}(\sigma,\tau)}\delta_{\Phi_{\tau_{1}}}\circ\cdots\circ\delta_{\Phi_{\tau_{n}}}(A)\,\mathrm{d}\tau_{1}\cdots\mathrm{d}\tau_{n},&\text{
for }0\leq\sigma\leq\tau\leq 1;\\\\[12.0pt] \displaystyle A+\sum_{n\geq
1}\int_{\Delta_{n}(\sigma,\tau)}\delta_{\Phi_{\tau_{n}}}\circ\cdots\circ\delta_{\Phi_{\tau_{1}}}(A)\,\mathrm{d}\tau_{1}\cdots\mathrm{d}\tau_{n},&\text{
for }0\leq\tau\leq\sigma\leq 1;\end{cases}$ (23)

the limit being uniform for $\tau-\sigma$ in compact subsets of
${]}-\epsilon,\epsilon{[}$. Since the maps
$\alpha_{\Phi,\Lambda}^{\sigma\to\tau}:{\mathfrak{A}_{\mathrm{loc}}}\to\mathfrak{A}$
are isometric, their limit $\alpha_{\Phi}^{\sigma\to\tau}$ is norm continuous,
and hence uniquely extends by continuity to an isometry on $\mathfrak{A}$. It
follows that (7) holds for all $A\in\mathfrak{A}$ and $\sigma,\tau\in[0,1]$
satisfying $|\tau-\sigma|<\epsilon$. Moreover, as norm-limits of
$\ast$-morphisms, the maps $\alpha_{\Phi}^{\sigma\to\tau}$ are themselves
$\ast$-morphisms. For $\sigma,\tau,\upsilon\in[0,1]$ such that
$\max(|\tau-\sigma|,|\sigma-\upsilon|)<\epsilon$, it follows from the
continuity of $\alpha_{\Phi}^{\sigma\to\tau}$ that

$\lim_{\Lambda\uparrow\mathbb{Z}^{d}}\alpha_{\Phi}^{\sigma\to\tau}\circ\alpha_{\Phi,\Lambda}^{\tau\to\upsilon}(A)=\alpha_{\Phi}^{\sigma\to\tau}\circ\alpha_{\Phi}^{\tau\to\upsilon}(A),$

for $A\in{\mathfrak{A}_{\mathrm{loc}}}$. Writing

$\alpha_{\Phi}^{\sigma\to\tau}\circ\alpha_{\Phi,\Lambda}^{\tau\to\upsilon}(A)=\alpha_{\Phi,\Lambda}^{\sigma\to\upsilon}(A)+(\alpha_{\Phi}^{\sigma\to\tau}-\alpha_{\Phi,\Lambda}^{\sigma\to\tau})\circ\alpha_{\Phi}^{\tau\to\upsilon}(A)+(\alpha_{\Phi}^{\sigma\to\tau}-\alpha_{\Phi,\Lambda}^{\sigma\to\tau})\circ(\alpha_{\Phi,\Lambda}^{\tau\to\upsilon}-\alpha_{\Phi}^{\tau\to\upsilon})(A),$

and using that

$\displaystyle\lim_{\Lambda\uparrow\mathbb{Z}^{d}}(\alpha_{\Phi}^{\sigma\to\tau}-\alpha_{\Phi,\Lambda}^{\sigma\to\tau})\circ\alpha_{\Phi}^{\tau\to\upsilon}(A)=0,$
$\displaystyle\lim_{\Lambda\uparrow\mathbb{Z}^{d}}\|(\alpha_{\Phi}^{\sigma\to\tau}-\alpha_{\Phi,\Lambda}^{\sigma\to\tau})\circ(\alpha_{\Phi,\Lambda}^{\tau\to\upsilon}-\alpha_{\Phi}^{\tau\to\upsilon})(A)\|\leq
2\lim_{\Lambda\uparrow\mathbb{Z}^{d}}\|\alpha_{\Phi,\Lambda}^{\tau\to\upsilon}(A)-\alpha_{\Phi}^{\tau\to\upsilon}(A)\|=0,$

we derive

$\alpha_{\Phi}^{\sigma\to\tau}\circ\alpha_{\Phi}^{\tau\to\upsilon}(A)=\alpha_{\Phi}^{\sigma\to\upsilon}(A),$
(24)

provided that $|\upsilon-\sigma|<\epsilon$. This identity extends by
continuity to all $A\in\mathfrak{A}$, and allows to extend the family of
$\ast$-morphisms $(\alpha_{\Phi}^{\sigma\to\tau})_{|\tau-\sigma|<\epsilon}$ to
arbitrary $\sigma,\tau\in[0,1]$ by setting

$\alpha_{\Phi}^{\sigma\to\tau}=\alpha_{\Phi}^{\sigma\to\tau_{1}}\circ\alpha_{\Phi}^{\tau_{1}\to\tau_{2}}\circ\cdots\circ\alpha_{\Phi}^{\tau_{n-1}\to\tau_{n}}\circ\alpha_{\Phi}^{\tau_{n}\to\tau},$
(25)

where $\tau_{0}=\sigma,\tau_{1},\ldots,\tau_{n+1}=\tau$ are such that
$\max_{k}|\tau_{k+1}-\tau_{k}|<\epsilon$. One easily checks that the left-hand
side of this expression does not depend on the choice of the subdivisions
$\tau_{1},\ldots,\tau_{n}$ on the right-hand side. By construction, this
extension satisfies (24) for all $\sigma,\tau,\upsilon\in[0,1]$. From the
telescopic expansion based on (25),

$\alpha_{\Phi}^{\sigma\to\tau}(A)-\alpha_{\Phi,\Lambda}^{\sigma\to\tau}(A)=\sum_{k=0}^{n}\alpha_{\Phi,\Lambda}^{\sigma\to\tau_{1}}\circ\cdots\circ\alpha_{\Phi,\Lambda}^{\tau_{k-1}\to\tau_{k}}\circ(\alpha_{\Phi}^{\tau_{k}\to\tau_{k+1}}-\alpha_{\Phi,\Lambda}^{\tau_{k}\to\tau_{k+1}})\circ\alpha_{\Phi}^{\tau_{k+1}\to\tau_{k+2}}\circ\cdots\circ\alpha_{\Phi}^{\tau_{n}\to\tau}(A),$

we derive

$\|\alpha_{\Phi}^{\sigma\to\tau}(A)-\alpha_{\Phi,\Lambda}^{\sigma\to\tau}(A)\|\leq\sum_{k=0}^{n}\|(\alpha_{\Phi}^{\tau_{k}\to\tau_{k+1}}-\alpha_{\Phi,\Lambda}^{\tau_{k}\to\tau_{k+1}})(A_{k})\|,$

with
$A_{k}=\alpha_{\Phi}^{\tau_{k+1}\to\tau_{k+2}}\circ\cdots\circ\alpha_{\Phi}^{\tau_{n}\to\tau}(A)$.
It follows that (7) holds for all $A\in\mathfrak{A}$ and all
$\sigma,\tau\in[0,1]$. Note that (24) implies that
$\alpha_{\Phi}^{\sigma\to\tau}\circ\alpha_{\Phi}^{\tau\to\sigma}=\alpha_{\Phi}^{\sigma\to\sigma}=\mathrm{Id}$,
which shows that the maps $\alpha_{\Phi}^{\sigma\to\tau}$ are
$\ast$-automorphisms of $\mathfrak{A}$.

(2) The translation invariance
$\alpha_{\Phi}^{\sigma\to\tau}\circ\varphi^{x}=\varphi^{x}\circ\alpha_{\Phi}^{\sigma\to\tau}$
follows immediately from Relation (19).

(3) For $A\in{\mathfrak{A}_{\mathrm{loc}}}$ and $|\tau-\sigma|<\epsilon/2$,
combining the Dyson expansion (23) with the estimate (22), we get

$\|\alpha_{\Phi}^{\sigma\to\tau}(A)-A\|\leq
2\epsilon^{-1}\|A\|\mathrm{e}^{d|\mathop{\mathrm{supp}}\nolimits(A)|}|\tau-\sigma|.$

Since $\alpha_{\Phi}^{\sigma\to\tau}$ is an isometry and
${\mathfrak{A}_{\mathrm{loc}}}$ is dense in $\mathfrak{A}$, we conclude that
the map $(\sigma,\tau)\mapsto\alpha_{\Phi}^{\sigma\to\tau}$ is strongly
continuous on the diagonal $(\tau,\tau)\in[0,1]\times[0,1]$. For
$(\sigma_{0},\tau_{0})\in[0,1]\times[0,1]$, Property (24) gives

$\alpha_{\Phi}^{\sigma\to\tau}(A)-\alpha_{\Phi}^{\sigma_{0}\to\tau_{0}}(A)=\alpha_{\Phi}^{\sigma\to\tau_{0}}(\alpha_{\Phi}^{\tau_{0}\to\tau}(A)-A)+(\alpha_{\Phi}^{\sigma\to\sigma_{0}}(B)-B),$

with $B=\sigma_{\Phi}^{\sigma_{0}\to\tau_{0}}(A)$, which shows strong
continuity at $(\sigma_{0},\tau_{0})$.

From (24) and the Dyson expansion (23), we further derive that

$\partial_{\tau}\alpha_{\Phi}^{\sigma\to\tau}(A)=\lim_{\upsilon\to
0}\frac{1}{\upsilon}\left(\alpha_{\Phi}^{\sigma\to\tau+\upsilon}(A)-\alpha_{\Phi}^{\sigma\to\tau}(A)\right)=\lim_{\upsilon\to
0}\frac{1}{\upsilon}\alpha_{\Phi}^{\sigma\to\tau}(\alpha_{\Phi}^{\tau\to\tau+\upsilon}(A)-A)=\alpha_{\Phi}^{\sigma\to\tau}\circ\delta_{\Phi_{\tau}}(A),$

holds in norm for arbitrary $\sigma,\tau\in[0,1]$ and
$A\in{\mathfrak{A}_{\mathrm{loc}}}$.

(4) We will consider the case $\sigma<\tau$, the opposite case is similar. We
also observe that Part (1) extends to time-dependent piecewise continuous
interactions $\Phi$. Indeed, the continuity was only tacitly used to justify
the Riemann integration in the Dyson expansions (20) and (23).

Let $\Phi,\Psi:[0,1]\to\mathcal{B}^{r}$ be piecewise continuous and set
$\bar{\epsilon}=r/2\max(\|\Phi\|_{r},\|\Psi\|_{r})$. The telescopic expansion

$\delta_{\Phi_{\tau_{1}}}\circ\cdots\circ\delta_{\Phi_{\tau_{n}}}-\delta_{\Psi_{\tau_{1}}}\circ\cdots\circ\delta_{\Psi_{\tau_{n}}}=\sum_{k=1}^{n}\delta_{\Phi_{\tau_{1}}}\circ\cdots\circ\delta_{\Phi_{\tau_{k-1}}}\circ\delta_{(\Phi-\Psi)_{\tau_{k}}}\circ\delta_{\Psi_{\tau_{k+1}}}\circ\cdots\circ\delta_{\Psi_{\tau_{n}}},$

combined with the estimate (22), gives

$\sup_{\tau_{1},\ldots,\tau_{n}\in[0,1]}\|\delta_{\Phi_{\tau_{1}}}\circ\cdots\circ\delta_{\Phi_{\tau_{n}}}(A)-\delta_{\Psi_{\tau_{1}}}\circ\cdots\circ\delta_{\Psi_{\tau_{n}}}(A)\|\leq
2n\|A\|\mathrm{e}^{r|\mathop{\mathrm{supp}}\nolimits(A)|}\bar{\epsilon}^{-n+1}\frac{\|\Phi-\Psi\|_{r}}{r}n!,$

for $A\in{\mathfrak{A}_{\mathrm{loc}}}$. Thus, it follows from the Dyson
expansion (23) that

$\sup_{|\tau-\sigma|\leq\bar{\epsilon}/2}\|\alpha_{\Phi}^{\sigma\to\tau}(A)-\alpha_{\Psi}^{\sigma\to\tau}(A)\|\leq\frac{4\bar{\epsilon}}{r}\|A\|\mathrm{e}^{r|\mathop{\mathrm{supp}}\nolimits(A)|}\|\Phi-\Psi\|_{r}.$

By density/continuity, we derive that

$\lim_{\Psi\to\Phi}\|\alpha_{\Phi}^{\sigma\to\tau}(A)-\alpha_{\Psi}^{\sigma\to\tau}(A)\|=0$
(26)

for all $A\in\mathfrak{A}$ and $|\tau-\sigma|\leq\bar{\epsilon}/2$. Finally, a
telescopic expansion analogous to (25) and with
$\max_{k}(|\tau_{k+1}-\tau_{k}|)<\bar{\epsilon}/2$ gives

$\|\alpha_{\Phi}^{\sigma\to\tau}(A)-\alpha_{\Psi}^{\sigma\to\tau}(A)\|\leq\sum_{k=0}^{n}\|(\alpha_{\Phi}^{\tau_{k}\to\tau_{k+1}}-\alpha_{\Psi}^{\tau_{k}\to\tau_{k+1}})(A_{k})\|,$

with
$A_{k}=\alpha_{\Phi}^{\tau_{k+1}\to\tau_{k+2}}\circ\cdots\circ\alpha_{\Phi}^{\tau_{n}\to\tau}(A)$,
from which we conclude that (26) holds for all $A\in\mathfrak{A}$ and all
$\sigma,\tau\in[0,1]$.

To complete the proof, given the continuous time-dependent interaction $\Phi$,
set

$\Phi_{t}^{(N)}=1_{[0,\upsilon_{0}{[}}(t)\Phi(t)+\sum_{k=0}^{N-1}1_{[\upsilon_{k},\upsilon_{k+1}{[}}(t)\Phi_{\upsilon_{k}}+1_{[\upsilon_{N},1]}(t)\Phi_{t},$

where $1_{I}$ denotes the indicator function of the set $I$. Since

$\sup_{t\in[0,1]}\|\Phi_{t}-\Phi^{(N)}_{t}\|_{r}\leq\sup_{|t-s|<\xi_{N}}\|\Phi_{t}-\Phi_{s}\|_{r},$

the uniform continuity of the map
$[0,1]\ni\tau\mapsto\Phi_{\tau}\in\mathcal{B}^{r}$ implies

$\lim_{N\to\infty}\|\Phi-\Phi^{(N)}\|_{r}=0.$

Thus, the claim follows from (26) and the fact that

$\alpha_{\Phi^{(N)}}^{\sigma\to\tau}(A)=\alpha_{\Phi_{\upsilon_{0}}}^{\xi_{N}}\circ\cdots\circ\alpha_{\Phi_{\upsilon_{N-1}}}^{\xi_{N}}(A).$

##  Appendix A Glossary of terms

More details can be found in [JPT23].

  * •

$\mathfrak{A}$: the $C^{\ast}$-algebra of a spin system on $\mathbb{Z}^{d}$,
or equivalently, the gauge-invariant sector of the fermionic CAR algebra on
$\ell^{2}(\mathbb{Z}^{d})$.

  * •

$\mathcal{F}$: the set of all finite subsets of $\mathbb{Z}^{d}$.

  * •

$\Phi$: an interaction, namely a translation invariant family
$\\{\Phi(X)\\}_{X\in\mathcal{F}}$ of self-adjoint elements of $\mathfrak{A}$
with $\mathop{\mathrm{supp}}\nolimits(\Phi)=X$.

  * •

$\mathcal{B}^{r}$: the Banach space of interactions satisfying
$\displaystyle\|\Phi\|_{r}=\sum_{X\ni
0}\mathrm{e}^{r(|X|-1)}\|\Phi(X)\|<\infty$, ($r>0$).

  * •

$\mathbb{Z}^{d}\ni x\mapsto\varphi^{x}$: the group action of $\mathbb{Z}^{d}$
on $\mathfrak{A}$.

  * •

$\mathcal{S}_{\mathrm{I}}(\mathfrak{A})$: the set of translation invariant
states on $\mathfrak{A}$.

  * •

$s(\nu)$: the specific entropy of a state
$\nu\in\mathcal{S}_{\mathrm{I}}(\mathfrak{A})$.

  * •

$s(\nu|\omega)$: the specific relative entropy of two states
$\nu,\omega\in\mathcal{S}_{\mathrm{I}}(\mathfrak{A})$.

  * •

$P(\Phi)$: the pressure of the interaction $\Phi$.

  * •

$E_{\Phi}=\sum_{X\ni 0}|X|^{-1}\Phi(X)$, so that $\nu(E_{\Phi})$ is the
expected specific energy of the interaction $\Phi$ in the state
$\nu\in\mathcal{S}_{\mathrm{I}}(\mathfrak{A})$.

  * •

$\mathcal{S}_{\mathrm{eq}}(\Phi)=\\{\nu\in\mathcal{S}_{\mathrm{I}}(\mathfrak{A})\mid
P(\Phi)=s(\nu)-\nu(E_{\Phi})\\}$: the set of equilibrium states for $\Phi$
(from the Gibbs variational principle).

  * •

$\mathrm{WG}(\Phi)$: the set of weak Gibbs states for $\Phi$, see [JPT23,
Section 2.2].

  * •

$\alpha_{\Phi}$: the Heisenberg dynamics generated by the (time-independent)
interaction $\Phi$.

  * •

Physical equivalence: two interactions $\Phi,\Psi\in\mathcal{B}^{r}$ are
physically equivalent iff $\alpha_{\Phi}=\alpha_{\Psi}$, see [JPT23, Theorem
2.7].

## References

  * [AE99] Avron, J. E. and Elgart, A.: Adiabatic theorem without a gap condition. Commun. Math. Phys. 203, 445–463 (1999), [DOI:10.1007/s002200050620].
  * [ASF07] Abou Salem, W. K. and Fröhlich, J.: Status of the fundamental laws of thermodynamics. J. Stat. Phys. 126, 1045–1068 (2007), [DOI:10.1007/s10955-006-9222-8].
  * [ASY87] Avron, J. E., Seiler, R. and Yaffe, L. G.: Adiabatic theorems and applications to the quantum Hall effect. Commun. Math. Phys. 110, 33–49 (1987), [DOI:10.1007/bf01209015].
  * [BDRF17] Bachmann, S., De Roeck, W. and Fraas, M.: Adiabatic theorem for quantum spin systems. Phys. Rev. Lett. 119, 060201 (2017), [DOI:10.1103/PhysRevLett.119.060201].
  * [BDRF18] : The adiabatic theorem and linear response theory for extended quantum systems. Commun. Math. Phys. 361, 997–1027 (2018), [DOI:10.1007/s00220-018-3117-9].
  * [BF28] Born, M. and Fock, V.: Beweis des Adiabatensatzes. Z. Phys. 51, 165–169 (1928), [DOI:10.1007/bf01343193].
  * [BFJP] Benoist, T., Fraas, M., Jakšić, V. and Pillet, C.-A.: Adiabatic theorem in quantum statistical mechanics. In preparation.
  * [BR81] Bratteli, O. and Robinson, D. W.: _Operator Algebras and Quantum-Statistical Mechanics. II_. Texts and Monographs in Physics, Springer-Verlag, New York-Berlin, 1981, [DOI:10.1007/978-3-662-03444-6].
  * [DJP03] Dereziński, J., Jakšić, V. and Pillet, C.-A.: Perturbation theory of $W^{*}$-dynamics, Liouvilleans and KMS-states. Rev. Math. Phys. 15, 447–489 (2003), [DOI:10.1142/S0129055X03001679].
  * [GLMP22] Greenblatt, R. L., Lange, M., Marcelli, G. and Porta, M.: Adiabatic evolution of low-temperature many-body systems. Preprint (2022), [DOI:10.48550/arXiv.2211.16836].
  * [Isr79] Israel, R. B.: _Convexity in the Theory of Lattice Gases_. Princeton Series in Physics, Princeton University Press, Princeton, N.J., 1979.
  * [JP] Jakšić, V. and Pillet, C.-A.: Adiabatic theorem for KMS states. Unpublished.
  * [JP91] Joye, A. and Pfister, C.-E.: Exponentially small adiabatic invariant for the Schrödinger equation. Commun. Math. Phys. 140, 15–41 (1991), [DOI:10.1007/bf02099288].
  * [JP03] Jakšić, V. and Pillet, C.-A.: A note on the entropy production formula. In _Advances in differential equations and mathematical physics (Birmingham, AL, 2002)_ , Contemp. Math., vol. 327, Amer. Math. Soc., Providence, RI, 2003, p. 175–180, [DOI:10.1090/conm/327/05813].
  * [JP14] : A note on the Landauer principle in quantum statistical mechanics. J. Math. Phys. 55, 075210, 21 (2014), [DOI:10.1063/1.4884475].
  * [JPT23] Jakšić, V., Pillet, C.-A. and Tauber, C.: Approach to equilibrium in translation invariant quantum systems: some structural results. Preprint (2023), [DOI:10.48550/arXiv.2204.00440].
  * [Kat50] Kato, T.: On the adiabatic theorem of quantum mechanics. J. Phys. Soc. Jpn. 5, 435–439 (1950), [DOI:10.1143/JPSJ.5.435].
  * [LR68] Lanford, O. E. and Robinson, D. W.: Statistical mechanics of quantum spin systems. III. Commun. Math. Phys. 9, 327–338 (1968), [DOI:10.1007/bf01654286].
  * [Nen93] Nenciu, G.: Linear adiabatic theory. Exponential estimates. Commun. Math. Phys. 152, 479–496 (1993), [DOI:10.1007/bf02096616].
  * [NSY19] Nachtergaele, B., Sims, R. and Young, A.: Quasi-locality bounds for quantum lattice systems. I. Lieb-Robinson bounds, quasi-local maps, and spectral flow automorphisms. J. Math. Phys. 60, 061101 (2019), [DOI:10.1063/1.5095769].
  * [OP93] Ohya, M. and Petz, D.: _Quantum Entropy and its Use_. Texts and Monographs in Physics, Springer-Verlag, Berlin, 1993, [DOI:10.1007/978-3-642-57997-4].
  * [Teu01] Teufel, S.: A note on the adiabatic theorem without gap condition. Lett. Math. Phys. 58, 261–2662002 (2001), [DOI:10.1023/A:1014556511004].