# What is a Fluctuation Theorem?

Noé Cuneo, Vojkan Jakšić, Claude-Alain Pillet, Armen Shirikyan

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Noé Cuneo, Vojkan Jakšić, Claude-Alain Pillet, Armen Shirikyan. What is a Fluctuation Theorem?. 2026, 10.1007/978-3-032-02095-6. hal-05505450

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Noé Cuneo
Vojkan Jakšić
Claude-Alain Pillet
Armen Shirikyan

# What is a Fluctuation Theorem?
<figure><img src="image_2.png" /></figure>
Dedicated to David Ruelle, on the occasion
of his 90th birthday
<figure><img src="image_3.png" /></figure>
Preface

Fluctuation Relations (FRs for short) describe statistical features of the fluctua-
tions of some physical quantities in systems subject to a discrete symmetry. They
provide quantitative information on the spontaneous breakdown of this symme-
try. FRs were first discovered in numerical experiments on shear flows, in the
early 1990s, by D.J. Evans, E.G.D. Cohen and G.P. Morris.<fnref n="1" /> There, the quantity of
interest was entropy production and the symmetry was time-reversal. Its break-
down led to the emergence of the arrow of time, and the FRs gave quantitative
estimates on the probability of violation of the second law of thermodynamics.
D.J. Evans and D.J. Searles<fnref n="2" /> subsequently showed that, on finite time intervals,
the rate of entropy production of time-reversal invariant systems relaxing to a
nonequilibrium steady state satisfies a FR (such FRs are now called transient).
The last and crucial pioneering step was achieved by G. Gallavotti and E.G.D. Co-
hen<fnref n="3" /> who proved that, under an appropriate chaotic hypothesis, FRs also hold in
the steady states of time-reversal invariant systems. These early discoveries and
developments generated an enormous body of numerical, theoretical and exper-
imental works in the physics literature which have fundamentally modified our
understanding of non-equilibrium statistical physics, with applications extend-
ing to chemistry and biology.

This monograph, addressed to mathematical physicists, is an introduction to
FRs and the associated Fluctuation Theorems in the context of statistical me-
chanics, stochastic dynamics and dynamical systems.

Paris, Milan, Toulon, Cergy

April 2025

Noé Cuneo

Vojkan Jakšić

Claude-Alain Pillet

Armen Shirikyan

<footnote n="1">¹ Phys. Rev. Lett. 71, 2401–4 (1993).</footnote>

<footnote n="2">² Phys. Rev. E 50, 1645–1648 (1994).</footnote>

<footnote n="3">³ Phys. Rev. Lett. 74, 2694–7 (1995).</footnote>
Noé Cuneo

Université Paris Cité and Sorbonne Université, CNRS,
Laboratoire de Probabilités, Statistique et Modélisation,
75013 Paris, France

Vojkan Jakšić

Dipartimento di Matematica
Politecnico di Milano
Piazza Leonardo da Vinci, 32
20133 Milan, Italy

Claude-Alain Pillet

Centre de Physique Théorique

Université de Toulon — CNRS — Aix-Marseille Université
83957 La Garde, France

Armen Shirikyan

Département de Mathématiques

CY Cergy Paris Université — CNRS
2, avenue Adolphe Chauvin
95302 Cergy-Pontoise, France
Acknowledgements

We are grateful to R. Chetrite and H. Comman for providing us with useful refer-
ences, and to L. Bruneau and C.-E. Pfister for helpful comments.

The work of AS, CAP, and VJ was partly funded by the CY Initiative grant *In-vestissements d'Avenir*, grant number ANR-16-IDEX-0008. We also acknowledge the support of the ANR project *DYNACQUS*, grant number ANR-24-CE40-5714. VJ acknowledges the support of NSERC, Canada and the support of the MUR grant *Dipartimento di Eccellenza 2023-2027* of Dipartimento di Matematica, Politecnico di Milano.

This document is the preprint version of Volume 54 of the *SpringerBriefs in
Mathematical Physics* collection. The final authenticated version is available on-
line at: https://doi.org/10.1007/978-3-032-02095-6
<figure><img src="image_4.png" /></figure>
# Contents

<table><tr><td>1</td><td>Introduction</td><td>1</td></tr><tr><td>1.1</td><td>Historical Overview</td><td>1</td></tr><tr><td>1.2</td><td>Appendix: Hamiltonian Fluctuation Relations</td><td>6</td></tr><tr><td></td><td>References</td><td>10</td></tr><tr><td>2</td><td>Large Deviations and Fluctuation Relations</td><td>13</td></tr><tr><td>2.1</td><td>Large Deviations</td><td>14</td></tr><tr><td>2.2</td><td>Transient Fluctuation Relation</td><td>16</td></tr><tr><td>2.3</td><td>Fluctuation Theorem</td><td>19</td></tr><tr><td>2.4</td><td>Asymptotic Fluctuation Relation</td><td>20</td></tr><tr><td>2.5</td><td>Entropic Pressure and Rate</td><td>22</td></tr><tr><td>2.6</td><td>Higher-level Fluctuation Theorems</td><td>24</td></tr><tr><td>2.7</td><td>On the Meaning of the Fluctuation Theorem</td><td>25</td></tr><tr><td></td><td>References</td><td>40</td></tr><tr><td>3</td><td>Illustrations</td><td>43</td></tr><tr><td>3.1</td><td>Finite State Markov Chains</td><td>43</td></tr><tr><td>3.1.1</td><td>Setup</td><td>43</td></tr><tr><td>3.1.2</td><td>Transient Fluctuation Relation</td><td>44</td></tr><tr><td>3.1.3</td><td>Fluctuation Theorem</td><td>45</td></tr><tr><td>3.2</td><td>Mean-field Lattice Gas</td><td>47</td></tr><tr><td>3.2.1</td><td>Setup</td><td>47</td></tr><tr><td>3.2.2</td><td>Transient Fluctuation Relation</td><td>48</td></tr><tr><td>3.2.3</td><td>Fluctuation Theorem</td><td>49</td></tr><tr><td>3.3</td><td>Ising Model</td><td>54</td></tr><tr><td>3.3.1</td><td>Setup</td><td>54</td></tr><tr><td>3.3.2</td><td>Transient Fluctuation Relation</td><td>55</td></tr><tr><td>3.3.3</td><td>Fluctuation Theorem</td><td>56</td></tr><tr><td>3.4</td><td>Langevin Dynamics</td><td>60</td></tr><tr><td>3.4.1</td><td>Setup</td><td>60</td></tr><tr><td>3.4.2</td><td>Transient Fluctuation Relation</td><td>64</td></tr></table>
<table><tbody><tr><td>3.4.3</td><td>Fluctuation Theorems</td><td>66</td></tr><tr><td>3.5</td><td>Reversible Smooth Dynamical Systems</td><td>71</td></tr><tr><td>3.5.1</td><td>Setup</td><td>71</td></tr><tr><td>3.5.2</td><td>Phase Space Contraction Rate</td><td>73</td></tr><tr><td>3.5.3</td><td>Transient Fluctuation Relations, Fluctuation Theorem and Steady States</td><td>73</td></tr><tr><td colspan="2">Appendix</td><td>75</td></tr><tr><td colspan="2">References</td><td>80</td></tr><tr><td>4</td><td>Level-2 LDP and Fluctuation Theorem for Chaotic Maps</td><td>83</td></tr><tr><td>4.1</td><td>A Class of Continuous Dynamical Systems</td><td>83</td></tr><tr><td>4.2</td><td>A Large Deviation Principle for Empirical Measures</td><td>85</td></tr><tr><td>4.3</td><td>Periodic Orbit Ensemble</td><td>87</td></tr><tr><td>4.4</td><td>Fluctuation Theorems for the Periodic Orbit Ensemble</td><td>90</td></tr><tr><td>4.5</td><td>Weak Gibbs Measures</td><td>94</td></tr><tr><td colspan="2">References</td><td>96</td></tr><tr><td>5</td><td>The Tent Map</td><td>99</td></tr><tr><td>5.1</td><td>Basic Properties</td><td>99</td></tr><tr><td>5.2</td><td>Symbolic Dynamics</td><td>101</td></tr><tr><td>5.3</td><td>Inducing</td><td>104</td></tr><tr><td>5.4</td><td>Grand Canonical Picture</td><td>108</td></tr><tr><td>5.5</td><td>Phase Transitions</td><td>115</td></tr><tr><td>5.6</td><td>A Map of the Square</td><td>116</td></tr><tr><td>5.7</td><td>Bracketing the Critical Point and the Pressure</td><td>117</td></tr><tr><td>5.8</td><td>Numerical Calculation of the Pressure</td><td>120</td></tr><tr><td colspan="2">References</td><td>121</td></tr><tr><td colspan="2">Epilogue</td><td>127</td></tr><tr><td colspan="2">References</td><td>128</td></tr><tr><td colspan="2">Index of Notations</td><td>129</td></tr></tbody></table>
# Chapter 1
## Introduction

### 1.1 Historical Overview

Despite being random, fluctuations in physical systems do carry useful information. One can hardly overestimate the impact of the earliest and probably best known application of this fact: the determination of Avogadro's constant $N_A$ by Perrin [Per08; Per09] on the basis of a statistical analysis of Brownian motion.<fnref n="1" /> The theoretical foundation of Perrin's experiment was the Einstein-Smoluchowski relation [Ein05; Smo06], linking the mobility $\mu$ of a tracer particle immersed in a fluid to the diffusion constant $D$,

$$D = \mu k_B T, \qquad k_B = \frac{R}{N_A}, \qquad (1.1)$$

where $T$ is the equilibrium temperature of the fluid, $k_B$ is the Boltzmann constant, and $R$ is the ideal gas constant. In the modern language of statistical mechanics, this is a fluctuation-dissipation relation which, as the name indicates, relates the fluctuations of some observable, typically encoded in a correlation function, to some quantity linked to the dissipative properties of the system. In the case of Brownian motion, the left-hand side of (1.1) is related to the fluctuations of the velocity $\dot{X}$ of the tracer by<fnref n="2" />

$$3D = \int_{0}^{\infty} \langle \dot{X}_{0} \cdot \dot{X}_{t} \rangle dt.$$

The fluid's viscosity is responsible for the dissipative drag force, $F = -\mu^{-1}\dot{X}_t$, opposed to the motion of the tracer. For a spherical tracer of radius $r$, the mobility appearing on the right-hand side of (1.1) is related to the dynamic viscosity $\eta$ of

<footnote n="1">¹ The interested reader should consult [Gal99, Chapter 8] for a comprehensive account on the theory of Brownian motion and its history.</footnote>

<footnote n="2">² Here, $\langle \cdot \rangle$ denotes the ensemble average. The factor of 3 is the dimension of the configuration space $\mathbb{R}^3$, and "·" denotes the Euclidean inner product.</footnote>
the fluid by Stokes' formula

$$ \mu = \frac{1}{6\pi\eta r}. $$

We also mention that the velocity power spectrum $S$,

$$ S(\omega) \equiv \lim_{T \to \infty} \frac{1}{T} \langle |\hat{V}_T(\omega)|^2 \rangle, \quad \hat{V}_T(\omega) = \int_0^T \dot{X}_t e^{-i\omega t} dt, $$

is linked to the correlation function by the Wiener-Khinchin relation [Khi34],

$$ \langle \dot{X}_0 \cdot \dot{X}_t \rangle = \int_{-\infty}^{\infty} S(\omega) e^{i\omega t} \frac{d\omega}{2\pi}. $$

Since the instantaneous power dissipated by the Stokes force is given by the quadratic expression $P_t = \mu^{-1}|\dot{X}_t|^2$, its steady averaged value is related to the power spectrum as

$$ \langle P_0 \rangle = \frac{1}{\mu} \int_{-\infty}^{\infty} S(\omega) \frac{d\omega}{2\pi}. $$

The Johnson-Nyquist noise [Joh28; Nyq28] — the electrical counterpart of Brownian motion — was discovered a century after Brown's observations [Bro28]. The electromotive force $U$ induced by the thermally generated motion of charge carriers in a resistor of resistance $R$ kept at a constant temperature $T$ has a power spectrum given by the celebrated Nyquist formula [Kam81, Section XI.2]

$$ S(\omega) = 2k_B T R. $$

Invoking the Wiener-Khinchin relation yields the fluctuation-dissipation relation

$$ \int_{0}^{\infty} \langle U_{0}U_{t} \rangle dt = 2Rk_{B}T. $$

In nonequilibrium thermodynamics, linear response theory describes the reaction of a system driven out of equilibrium by weak external forces [GM62, Chapters III-IV]. Distinctive features of this reaction are the induced currents (particles species, heat, charges) $J_i$ and the associated entropy production rate

$$ \sigma = \sum_{i} J_{i} X_{i}, $$

where $X_i$ is the thermodynamic force, or affinity, conjugated to the current $J_i$. In the linear response regime, the currents are linear functions of the forces

$$ J_j = \sum_i L_{ji} X_i. $$

The $L_{ji}$ are called transport coefficients, and the second law of thermodynamics requires
$$ \sigma = \sum_{i,j} L_{ij} X_i X_j \geq 0. $$

In other words, the symmetric part of the matrix L is non-negative: $L + L^T \geq 0$.

A few years after the aforementioned works of Johnson and Nyquist, Onsager developed an approach to the transport coefficients based on the general theory of fluctuations laid down by Einstein [Ein10] on the basis of Boltzmann's entropy formula. The main result of these investigations [Ons31a; Ons31b] is a general formulation of the so-called reciprocity relations, the first instances of which were discovered by Thomson (Lord Kelvin), Helmholtz and others: if the system satisfies microscopic reversibility, then the matrix L is symmetric, $L = L^T \geq 0$.

Two decades later, invoking quantum mechanics, Callen and Welton derived a general form of the Nyquist formula [CW51], including a low temperature quantum correction, while Green and Kubo [Gre52; Kub57] derived a general relation between transport coefficients and current-current correlation functions, since then called the Green-Kubo formula,

$$ L_{ij} = \frac{1}{2} \int_{-\infty}^{\infty} \langle \Phi_i(0) \Phi_j(t) \rangle_{\text{eq}} dt, \qquad (1.2) $$

where $\Phi_j(t)$ denotes the instantaneous current observable at time $t$ and $\langle \cdot \cdot \rangle_{\text{eq}}$ the thermal equilibrium state.

To the best of our knowledge, Bochkov and Kuzovlev were the first to investigate the formal consequences of microscopic reversibility on the statistical properties of fluctuations away from thermal equilibrium [BK77]. They consider a classical system governed by the Hamiltonian $H_0$, assumed to be invariant under a time-reversal map, i.e., $H_0 = H_0 \circ \theta$ for some anti-symplectic involution $\theta$ of the phase space. The system can be acted upon through external “forces” $X_i$, with the associated Hamiltonian

$$ H(X) = H_0 - \sum_i X_i Q_i, $$

where $Q_i$'s are phase functions that are either even or odd under time-reversal,

$$ Q_i \circ \theta = \theta_i Q_i, \quad \theta_i = \pm 1. $$

Given a steady forcing $X$, the state of thermal equilibrium at temperature $T$ is described by the Gibbs probability measure

$$ d\mu_X = e^{-\beta(H(X)-F(X))} d\gamma, \quad F(X) = -\frac{1}{\beta} \log \int e^{-\beta H(X)} d\gamma, $$

where $\beta = 1/k_B T$, $\gamma$ denotes Liouville's measure on phase space, and $F(X)$ the Helmholtz free energy.

The system, initially in the state $\mu_0$, is driven out of thermal equilibrium by a time-dependent forcing $X: [0, \tau] \ni t \mapsto X(t)$. We denote by $[0, \tau] \ni t \mapsto \Phi^t(\cdot | X)$ the flow generated by the time-dependent Hamiltonian $[0, \tau] \ni t \mapsto H(X(t))$, and
by

$$ \Phi_X : x \mapsto (\Phi^t(x|X))_{t \in [0, \tau]} $$

the induced map between initial conditions and solutions of the Hamilton equation of motion.

Let $\mathbb{P}_X = \mu_0 \circ \Phi_X^{-1}$ be the probability measure induced by $\mu_0$ on the set of phase space paths $(x_t)_{t \in [0, \tau]}$, and denote by $\Theta : (x_t)_{t \in [0, \tau]} \to (\theta x_{\tau-t})_{t \in [0, \tau]}$ the time-reversal map. Bochkov and Kuzovlev derived the relation<fnref n="3" />

$$ \frac{d\mathbb{P}_X}{d\mathbb{P}_X \circ \Theta} = e^{S_\tau}, \qquad (1.3) $$

where

$$ S_{\tau} = \sum_{i} \int_{0}^{\tau} J_{i}(t) X_{i}(t) dt = \beta W_{\tau}^{\text{ex}}, $$

$J_i(t) = \beta \partial_t Q_i \circ \Phi^t$ is the current conjugated to $X_i$, and $W_\tau^{\text{ex}} = H_0 \circ \Phi^\tau - H_0$ is the increase of the internal energy $H_0$ during the period $[0, \tau]$. For later reference, we mention that $W_\tau^{\text{ex}}$ is sometimes called the exclusive work performed on the system by the external forces. The term “exclusive” is motivated by the fact that the free Hamiltonian $H_0$, and not the full one $H(X(t))$ is used in the energy book-keeping. The inclusive work

$$ W_{\tau}^{\text{in}} = H(X(\tau)) \circ \Phi^{\tau} - H(X(0)) = - \sum_{i} \int_{0}^{\tau} Q_{i}(t) \dot{X}_{i}(t) dt $$

will enter the scene below. Rewriting (1.3) as $e^{-S_\tau} d\mathbb{P}_X = d\mathbb{P}_X \circ \Theta$ leads to

$$ \int e^{-\beta W_{\tau}^{\text{ex}}} d\mathbb{P}_X = 1. \qquad (1.4) $$

For an obvious reason, such an identity is often referred to as an *integral fluctuation relation*, to distinguish it from a more *detailed fluctuation relation* of the form (1.3).

From (1.3), Bochkov and Kuzovlev further deduced the existence of an infinite hierarchy of fluctuation-dissipation relations, the Green-Kubo formula (1.2) being just a limiting form of one of them. A simple extension of (1.3) to more general initial states [BК79; BK81] allowed them to deal also with thermally forced systems and to derive fluctuation relations for nonequilibrium steady states of open systems.

A decade later, while investigating microscopic violations of the 2nd Law in numerical simulations of sheared viscous fluids, Evans, Cohen and Morriss [ECM93] discovered a fluctuation relation formally similar to (1.3). Their model consists of a many particles system driven by non-conservative forces and coupled to a Gaussian thermostat to prevent heating up and ensure relaxation toward a non-

<footnote n="3">³ Strictly speaking, their main result, [BК77, Equ. (7)], is slightly less general. See the appendix to this introduction for a derivation of (1.3) and comparison with other fluctuation relations.</footnote>
equilibrium steady state. As pointed out by Gallavotti and Cohen [CG99], a precise statement of this new relation writes

$$ \lim_{\tau \to \infty} \frac{1}{\tau} \log \frac{dP_{\tau}}{dP_{\tau} \circ \theta}(s) = s, \qquad (1.5) $$

where $P_{\tau}$ denotes the law of the entropy production rate<fnref n="4" /> over the time period $[0, \tau]$ induced by the steady state of the system, and the map $\theta$ time-reversal at the entropic level, i.e., $\theta(s) = -s$. Evans and Searles [ES94] derived a relation for transient states of thermostated systems that is formally similar to (1.3). In their version of this relation, the measure $\mathbb{P}_{\tau}$ is induced by a time-reversal invariant initial state (e.g., Liouville's measure or the Gibbs state $\mu_0$) and pertains to a trajectory segment of duration $\tau$. The time-reversal $\Theta$ maps a trajectory with entropy production $S_{\tau}$ to a trajectory with the reversed entropy production $-S_{\tau}$, and in this case

$$ \frac{1}{\tau} \log \frac{dP_{\tau}}{dP_{\tau} \circ \theta}(s) = s $$

holds for all $\tau$. Relations of this kind are often referred to as *Evans–Searles, transient* or *finite-time fluctuation relations*, due to the fact that they hold on any time intervals $[0, \tau]$ of finite durations, with a time-reversal invariant initial state, as opposed to the *Gallavotti–Cohen* or *steady state fluctuation relations* (1.5), which only hold in the limit $\tau \to \infty$, with a nonequilibrium steady state typically breaking time-reversal symmetry. While transient fluctuation relations usually follow from simple calculations, their steady state counterparts are much deeper and rely on a precise analysis of the relaxation process leading to the steady state or on detailed properties of the latter. Gallavotti and Cohen [GC95a; GC95b] proved that the steady state fluctuation relation holds for systems with sufficiently chaotic dynamics, more precisely for the SRB measure of Anosov dynamical systems. Gallavotti [Gal95] further showed how, in such cases, large deviation estimates can be related to the fluctuation relation, giving rise to what we shall call a *fluctuation theorem*.

Unaware of the works of Bochkov and Kuzovlev, but working in the same Hamiltonian setup, Jarzynski [Jar97a; Jar04; Jar07] derived the integral fluctuation relation resulting from the replacement of the exclusive work $W_{\tau}^{\text{ex}}$ by the inclusive one $W_{\tau}^{\text{in}}$ in (1.4), namely

$$ \int e^{-\beta W_{\tau}^{\text{in}}} d\mathbb{P}_{X} = e^{-\beta(F(X(\tau)) - F(X(0)))}, \qquad (1.6) $$

where $\mathbb{P}_X = \mu_{X(0)} \circ \Phi_X^{-1}$. What makes Jarzynski's fluctuation relation particularly interesting is that its two sides have a radically distinct nature. On the left-hand side, one has the ensemble average of a quantity depending on the work done by a non-reversible process transforming the equilibrium state $\mu_{X(0)}$ into the non-equilibrium state $\mu_{X(0)} \circ (\Phi^{\tau})^{-1}$. On the right-hand side, the free energy difference

<footnote n="4">⁴ In the model considered, entropy production rate is simply the phase-space contraction rate.</footnote>
$\Delta F = F(X(\tau)) - F(X(0))$ is the reversible work, i.e., the work that has to be done on the system to bring it from the initial equilibrium state $\mu_{X(0)}$ to the final one $\mu_{X(\tau)}$ by an arbitrary reversible process. This connection between a nonequilibrium process and equilibrium quantities provides an opportunity to measure the latter, see [Jar08] and references therein for examples of such applications.

Jarzynski, Crooks and Kurchan [Jar97b; Cro98; Kur98] extended the derivation of the above transient and integral fluctuation relations from the deterministic Hamiltonian setting to stochastic dynamics, while Lebowitz and Spohn developed the Gallavotti-Cohen steady-state fluctuation theorem [LS99] in the Markovian setting. Maes [Mae99] interpreted fluctuation theorem within Gibbs formalism. A detailed version of Jarzynski's integral relation (1.6) was obtained by Crooks [Cro99] in the stochastic case, and by Jarzynski [Jar00] in the deterministic case. The results of Gallavotti-Cohen [GC95a; GC95b] were extended to general chaotic dynamical systems on compact metric spaces in [MV03].

The difficulties involved with experimental studies of the fluctuation relations explain, at least partly, why the works of Bochkov and Kuzovlev remained largely unnoticed [Pit11]. In this respect, two decades later, new techniques became available, and the subject generated an enormous body of theoretical, numerical and experimental works which have fundamentally altered our understanding of non-equilibrium physics, with applications extending to chemistry and biology. For reviews of these various developments we refer the reader to [Rue99; ES02; JQQ04; Gas05; RM07; Mar+08; JPR11]

## 1.2 Appendix: Hamiltonian Fluctuation Relations

We finish this introduction with a formal derivation of various forms of detailed fluctuation relations for Hamiltonian dynamics. The setup is similar to that of [BK77] and [Jar00], and any necessary regularity is assumed without notice.

We consider a system with phase space $\Gamma$, equipped with its symplectic form, Liouville measure $\gamma$, and anti-symplectic time-reversal map $\theta$. We assume that the system Hamiltonian $\Gamma \ni x \mapsto H(x|X)$ depends on some control parameters $X = (X_1, \dots, X_n) \in \mathbb{R}^n$, describing adjustable conditions of the model, e.g., some applied voltage bias, the coupling strengths between some of its subsystems, etc. A protocol is a scenario specifying how these controls vary in the course of time.

For a fixed time $\tau$ and given the protocol

$$X: [0, \tau] \to \mathbb{R}^n, \qquad (1.7)$$

we denote by $[0, \tau] \ni t \mapsto \Phi^t(\cdot|X)$ the phase space flow associated with the time-dependent Hamiltonian $[0, \tau] \ni t \mapsto H(\cdot|X(t))$. We also assume the time-reversal covariance of the Hamiltonian,

$$H(\theta(x)|X) = H(x|\theta(X)), \qquad (1.8)$$
where $\theta$ is a linear involution of $\mathbb{R}^n$, and define the time-reversal of the protocol (1.7) by

$$ \hat{X} : [0, \tau] \to \theta(X(\tau - t)). $$

For a given protocol $X$, $\Phi_X : x \mapsto (\Phi^t(x|X))_{t \in [0, \tau]}$ maps an initial condition to its phase space paths, and $\Theta : (x_t)_{t \in [0, \tau]} \mapsto (\theta(x_{\tau-t}))_{t \in [0, \tau]}$ maps such a path to its time-reversal.

The following is the mother of all fluctuation relations concerning deterministic Hamiltonian systems.

**Proposition 1.1** Let

$$ v_X(\mathrm{d}x) := e^{-S(x|X(0))} \gamma(\mathrm{d}x) $$

be a probability measure on $\Gamma$ and $\mathbb{P}_X = v_X \circ \Phi_X^{-1}$ the induced path-space measure. Then, the time-reversed measure $\mathbb{P}_{\hat{X}} \circ \Theta$ is absolutely continuous w.r.t. $\mathbb{P}_X$, with the Radon-Nikodym derivative

$$ \mathbf{x} = (x_t)_{t \in [0, \tau]} \mapsto \frac{\mathrm{d}\mathbb{P}_{\hat{X}} \circ \Theta}{\mathrm{d}\mathbb{P}_X}(\mathbf{x}) = e^{-S(\theta(x_\tau)|\theta(X(\tau))) + S(x_0|X(0))}, \quad (1.9) $$

i.e., for any path function $F$,

$$ \int F(\Theta(\mathbf{x})) \mathbb{P}_{\hat{X}}(\mathrm{d}\mathbf{x}) = \int e^{-S(\theta(x_\tau)|\theta(X(\tau))) + S(x_0|X(0))} F(\mathbf{x}) \mathbb{P}_X(\mathrm{d}\mathbf{x}). $$

**Proof.** Defining the time-reversed Hamiltonian flow by

$$ \hat{\Phi}^t(x|X) = \theta \circ \Phi^{\tau-t}(\theta(x)|\hat{X}), \quad (1.10) $$

one easily checks that

$$ \Phi^t(x|X) = \hat{\Phi}^t(\Phi^\tau(x|X)|X) \quad (1.11) $$

holds for all $x \in \Gamma$ and $t \in [0, \tau]$ (both sides of this identity satisfy Hamilton's equation of motion with the same final condition). Denoting $\mathbb{E}_X$ the expectation functional associated to $\mathbb{P}_X$, for $F(\mathbf{x}) = f(x_{t_1}, \dots, x_{t_n})$ one has

$$ \mathbb{E}_{\hat{X}}[F \circ \Theta] = \int_{\Gamma} f(\theta \circ \Phi^{\tau-t_1}(x|\hat{X}), \dots, \theta \circ \Phi^{\tau-t_n}(x|\hat{X})) e^{-S(x|\hat{X}(0))} \gamma(\mathrm{d}x), $$

and invoking (1.10),

$$ \mathbb{E}_{\hat{X}}[F \circ \Theta] = \int_{\Gamma} f(\hat{\Phi}^{t_1}(\theta(x)|X), \dots, \hat{\Phi}^{t_n}(\theta(x)|X)) e^{-S(x|\hat{X}(0))} \gamma(\mathrm{d}x). $$

Using the fact that Liouville's measure is $\theta$-invariant, we can write

$$ \mathbb{E}_{\hat{X}}[F \circ \Theta] = \int_{\Gamma} f(\hat{\Phi}^{t_1}(x|X), \dots, \hat{\Phi}^{t_n}(x|X)) e^{-S(\theta(x)|\hat{X}(0))} \gamma(\mathrm{d}x), $$

and its invariance under the Hamiltonian flow further yields
$$ \mathbb{E}_{\hat{X}}[F \circ \Theta] = \int_{\Gamma} f(\hat{\Phi}^{t_1}(\Phi^{\tau}(x|X)|X), \dots, \hat{\Phi}^{t_n}(\Phi^{\tau}(x|X)|X)) e^{-S(\theta(\Phi^{\tau}(x|X))|\hat{X}(0))} \gamma(dx). $$

In view of (1.11), we finally get

$$ \begin{aligned} \mathbb{E}_{\hat{X}}[F \circ \Theta] &= \int_{\Gamma} f(\Phi^{t_1}(x|X), \dots, \Phi^{t_n}(x|X)) e^{S(x|X(0))-S(\theta(\Phi^{\tau}(x|X))|\hat{X}(0))} \nu_X(dx) \\ &= \mathbb{E}_X \left[ F e^{S(x_0|X(0))-S(\theta(x_\tau)|\theta(X(\tau))} \right]. \end{aligned} $$

<figure><img src="image_5.png" /></figure>

Setting

$$ F(X) := -\frac{1}{\beta} \log \int_{\Gamma} e^{-\beta H(x|X)} \gamma(dx), $$

we now discuss a few consequences of Proposition 1.1.

**The Bochkov–Kuzovlev FR.** It follows from (1.8) that the Hamiltonian $H_0 = H(\cdot|0)$ is time-reversal invariant. With $S(\cdot|X) = \beta(H_0 - F(0))$, the initial measure $\nu_X = \nu_{\hat{X}}$ is the associated equilibrium state at inverse temperature $\beta$ and (1.9) gives the detailed Bochkov–Kuzovlev fluctuation relation (1.3),

$$ \frac{d\mathbb{P}_{\hat{X}} \circ \Theta}{d\mathbb{P}_X}(\mathbf{x}) = e^{-\beta(H_0(x_\tau) - H_0(x_0))} = e^{-\beta W_\tau^{\text{ex}}}. $$

**The Jarzynski–Crooks FR.** With $S(\cdot|X) = \beta(H(\cdot|X) - F(X))$, the initial measures

$$ \nu_X(dx) = e^{-\beta(H(x|X(0))-F(X(0)))} \gamma(dx), $$

and

$$ \nu_{\hat{X}} \circ \theta(dx) = e^{-\beta(H(x|X(\tau))-F(X(\tau)))} \gamma(dx), $$

are, respectively, the equilibrium measure at inverse temperature $\beta$ for the initial and final Hamiltonians $H(\cdot|X(0))$ and $H(\cdot|X(\tau))$. In this case, and in view of (1.8), the relation (1.9) reads

$$ \frac{d\mathbb{P}_{\hat{X}} \circ \Theta}{d\mathbb{P}_X}(\mathbf{x}) = e^{\beta(F(X(\tau))-F(X(0))}} e^{-\beta(H(x_\tau|X(\tau))-H(x_0|X(0)))} = e^{\beta\Delta F} e^{-\beta W_\tau^{\text{in}}}, $$

which is the detailed form of the Jarzynski–Crooks fluctuation relation, from which (1.6) immediately follows.

**FR in open systems.** We consider an open system $\mathcal{Z}$ coupled to thermal reservoirs $\mathcal{R}_1, \dots, \mathcal{R}_m$. The phase space of the joint system $\mathcal{Z} + \mathcal{R}_1 + \dots + \mathcal{R}_m$ is

$$ \Gamma = \Gamma_{\mathcal{Z}} \times \Gamma_{\mathcal{R}}, \quad \Gamma_{\mathcal{R}} = \bigtimes_{k=1}^{m} \Gamma_{\mathcal{R}_k}. $$

Accordingly, we denote phase space points by $x = (z, r)$, $r = (r_1, \dots, r_m)$ and Liouville's measure by $dx = dz \otimes dr$, $dr = \otimes_{k=1}^m dr_k$. The controls are $X = (Z, R)$ and the
Hamiltonian of the coupled system is

$$H(x|X) = H_Z(z|Z) + \sum_{k=1}^{m} H_k(r_k) + \sum_{k=1}^{m} R_k Q_k(z, r_k).$$

Assuming $H_Z(\theta(z)|Z) = H_Z(z|\theta(Z))$, $H_k \circ \theta = H_k$ and $Q_k \circ \theta = \theta_k Q_k$ with $\theta_k = \pm 1$ for each $k$, implies (1.8) with $\theta(R)_k = \theta_k R_k$.

Given $\beta = (\beta_1, \dots, \beta_m)$, consider the initial measure $v_X = \mu_Z \otimes \mu_\beta$, where the product of Gibbs states

$$\mu_{\beta}(dr) = \bigotimes_{k=1}^{m} e^{-\beta_k(H_k(r_k) - F_k)} dr_k, \quad F_k = -\frac{1}{\beta_k} \log \int_{\Gamma_{\mathcal{R}_k}} e^{-\beta_k H_k(r_k)} dr_k,$$

describes each reservoir $\mathcal{R}_k$ in thermal equilibrium at inverse temperature $\beta_k$ and

$$\mu_Z(dz) = e^{-s(z|Z)} dz,$$

so that

$$S(x|X) = s(z|Z) + \sum_{k=1}^{n} \beta_k (H_k(r_k) - F_k).$$

Then the relation (1.9) reads

$$\frac{d\mathbb{P}_{\hat{X}} \circ \Theta}{d\mathbb{P}_X}(\mathbf{x}) = e^{-(s(\theta(z_{\tau})|\theta(Z(\tau))) - s(z_0|Z(0)))} e^{-\sum_{k=1}^{n} \beta_k \Delta E_k},$$

where

$$\Delta E_k = H_k(r_{k,\tau}) - H_k(r_{k,0})$$

is the energy dissipated in the reservoir $\mathcal{R}_k$. Note that, with $\{\cdot, \cdot\}$ denoting the Poisson bracket,

$$\beta_k \Delta E_k = \int_0^\tau R_k(t) J_k(t) dt, \quad J_k(t) = \beta_k \{Q_k, H_k\}(x_t),$$

so that $J_k$ can be identified as the current conjugated to the affinity $R_k$.

Assuming the reservoirs to be large enough to remain in thermal equilibrium during the whole process, one can also identify

$$\Delta S = \sum_{k=1}^{n} \beta_k \Delta E_k = \sum_{k=1}^{n} \int_{0}^{\tau} R_k(t) J_k(t) dt$$

with the entropy dumped in the reservoirs, that is the total entropy generated by the process.

In the special case

$$s(z|Z) = \beta(H_Z(z|Z) - F(Z)), \quad F(Z) = -\frac{1}{\beta} \log \int_{\Gamma_Z} e^{-\beta H_Z(z|Z)} dz,$$
one has

$$ \frac{d\mathbb{P}_{\bar{X}} \circ \Theta}{d\mathbb{P}_X}(\mathbf{x}) = e^{\beta \Delta F} e^{-\beta \Delta E - \Delta S} $$

where

$$ \Delta F = F(Z(\tau)) - F(Z(0)), \quad \Delta E = H_Z(z_\tau|Z(\tau)) - H_Z(0|Z(0)), $$

are, respectively, the free energy difference between the two equilibrium states $\mu_{Z(0)}$ and $\mu_{Z(\tau)}$ of the system $\mathcal{Z}$, and the change in the internal energy of this system during the process. This leads to the fluctuation relation

$$ \mathbb{E}_X[e^{-\beta \Delta E - \Delta S}] = e^{-\beta \Delta F}. $$

References

[BK77] G. N. BOCHKOV AND Y. E. KUZOVLEV: General theory of thermal fluctuations in nonlinear systems. Sov. Phys. JETP 45 (1977), 125–130.

[BK79] ———, Fluctuation-dissipation relations for nonequilibrium processes in open systems. Sov. Phys. JEPT 49 (1979), 543–551.

[BK81] ———, Nonlinear fluctuation-dissipation relations and stochastic models in nonequilibrium thermodynamics I. Generalized fluctuation-dissipation theorem, Physica A: Statistical Mechanics and its Applications 106 (1981), 443–479. DOI: 10.1016/0378-4371(81)90122-9.

[Bro28] R. BROWN: A brief account of microscopical observations made in the months of June, July and August 1827, on the particles contained in the pollen of plants; and on the general existence of active molecules in organic and inorganic bodies. The Philosophical Magazine 4 (1828), 161–173. DOI: 10.1080/14786442808674769.

[CG99] E. G. D. COHEN AND G. GALLAVOTTI: Note on two theorems in nonequilibrium statistical mechanics. J. Stat. Phys. 96 (1999), 1343–1349. DOI: 10.1023/A:1004604804070.

[Cro98] G. E. CROOKS: Nonequilibrium measurements of free energy differences for microscopically reversible Markovian systems. J. Stat. Phys. 90 (1998), 1481–1487. DOI: 10.1023/A:1023208217925.

[Cro99] ———, Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences. Phys. Rev. E 60 (1999), 2721–2726. DOI: 10.1103/PhysRevE.60.2721.

[CW51] H. B. CALLEN AND T. A. WELTON: Irreversibility and generalized noise. Phys Rev 83 (1951), 34–40. DOI: 10.1103/PhysRev.83.34.

[ECM93] D. J. EVANS, E. G. D. COHEN, AND G. P. MORRISS: Probability of second law violations in shearing steady states. Phys. Rev. Lett. 71 (1993), 2401–2404. DOI: 10.1103/PhysRevLett.71.2401.

[Ein05] A. EINSTEIN: Uber die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen. Annalen der Physik 17 (1905), 549–560. DOI: 10.1002/andp.19053220806.

[Ein10] ———, Theorie der Opaleszenz von homogenen Flüssigkeiten und Flüssigkeitsgemischen in der Nähe des kritischen Zustandes. Annalen der Physik 33 (1910), 1275–1298. DOI: 10.1002/andp.19103381612.

[ES02] D. J. EVANS AND D. J. SEARLES: The fluctuation theorem. Adv. Phys. 51 (2002), 1529–1585. DOI: 10.1080/00018730210155133.
[ES94] ______, Equilibrium microstates which generate second law violating steady states. Phys. Rev. E 50 (1994), 1645-1648. DOI: 10.1103/PhysRevE.50.1645.

[Gal95] G. GALLAVOTTI: Reversible Anosov diffeomorphisms and large deviations. Math. Phys. Electron. J. 1 (1995), Paper 1, 1-12.

[Gal99] G. GALLAVOTTI: *Statistical Mechanics*. Springer Berlin Heidelberg, 1999. DOI: 10.1007/978-3-662-03952-6.

[Gas05] P. GASPARD: *Dynamical systems theory of irreversibility*. Chaotic dynamics and transport in classical and quantum systems. Ed. by P. Collet, M. Courbage, S. Métens, A. Neishtadt, and G. Zaslavsky. Kluwer Acad. Publ., Dordrecht, 2005, 107-157. DOI: 10.1007/1-4020-2947-0.

[GC95a] G. GALLAVOTTI AND E. G. D. COHEN: Dynamical ensembles in nonequilibrium statistical mechanics. Phys. Rev. Lett. 74 (1995), 2694-2697. DOI: 10.1103/PhysRevLett.74.2694.

[GC95b] ______, Dynamical ensembles in stationary states. J. Stat. Phys. 80 (1995), 931-970. DOI: 10.1007/BF02179860.

[GM62] S. R. DE GROOT AND P. MAZUR: *Non-equilibrium thermodynamics*. Series in Physics. North-Holland Publishing Co., Amsterdam; Interscience Publishers Inc., New York, 1962, x+510.

[Gre52] M. S. GREEN: Markoff random processes and the statistical mechanics of time-dependent phenomena. J. Chem. Phys. 20 (1952), 1281-1295. DOI: 10.1063/1.1700722.

[Jar00] C. JARZYNSKI: Hamiltonian derivation of a detailed fluctuation theorem. J. Stat. Phys. 98 (2000), 77-102. DOI: 10.1023/A:1018670721277.

[Jar04] ______, Nonequilibrium work theorem for a system strongly coupled to a thermal environment. Journal of Statistical Mechanics: Theory and Experiment 2004 (2004), P09005. DOI: 10.1088/1742-5468/2004/09/P09005.

[Jar07] C. JARZYNSKI: Comparison of far-from-equilibrium work relations. Comptes Rendus Physique 8 (2007), 495-506. DOI: 10.1016/j.crhy.2007.04.010.

[Jar08] C. JARZYNSKI: Nonequilibrium work relations: foundations and applications. Eur. Phys. J. B 64 (2008), 331-340. DOI: 10.1140/epjb/e2008-00254-2.

[Jar97a] ______, Nonequilibrium equality for free energy differences. Phys. Rev. Lett. 78 (1997), 2690-2693. DOI: 10.1103/PhysRevLett.78.2690.

[Jar97b] ______, Equilibrium free-energy differences from nonequilibrium measurements: A master-equation approach. Physical Review E 56 (1997), 5018-5035. DOI: 10.1103/PhysRevE.56.5018.

[Joh28] J. B. JOHNSON: Thermal agitation of electricity in conductors. Phys. Rev. 32 (1928), 97-109. DOI: 10.1103/PhysRev.32.97.

[JPR11] V. JAKSKIĆ, C.-A. PILLET, AND L. REY-BELLET: Entropic fluctuations in statistical mechanics: I. Classical dynamical systems. Nonlinearity 24 (2011), 699-763. DOI: 10.1088/0951-7715/24/3/003.

[JQQ04] D.-Q. JIANG, M. QIAN, AND M.-P. QIAN: *Mathematical theory of nonequilibrium steady states*. Vol. 1833. Lecture Notes in Mathematics. On the frontier of probability and dynamical systems. Springer, Berlin, 2004, x+280. DOI: 10.1007/b94615.

[Kam81] N. G. VAN KAMPEN: *Stochastic processes in physics and chemistry*. Vol. 888. Lecture Notes in Mathematics. North-Holland Publishing Co., Amsterdam-New York, 1981, xiv+419.

[Khi34] A. KHINTCHINE: Korrelationstheorie der stationären stochastischen Prozesse. Mathematische Annalen 109 (1934), 604-615. DOI: 10.1007/BF01449156.

[Kub57] R. KUBO: Statistical-mechanical theory of irreversible processes. I. General theory and simple applications to magnetic and conduction problems. J. Phys. Soc. Japan 12 (1957), 570-586. DOI: 10.1143/JPSJ.12.570.

[Kur98] J. KURCHAN: Fluctuation theorem for stochastic dynamics. J. Phys. A: Math. Gen. 31 (1998), 3719-3729. DOI: 10.1088/0305-4470/31/16/003.
[LS99] J. L. LEBOWITZ AND H. SPOHN: A Gallavotti-Cohen-type symmetry in the large deviation functional for stochastic dynamics. J. Stat. Phys. **95** (1999), 333-365. DOI: **10.1023/A:1004589714161**.

[Mae99] C. MAES: The fluctuation theorem as a Gibbs property. J. Stat. Phys. **95** (1999), 367-392. DOI: **10.1023/A:1004541830999**.

[Mar+08] U. MARCONI, A. PUGLISI, L. RONDONI, AND A. VULPIANI: Fluctuation-dissipation: Response theory in statistical physics. Phys. Rep. **461** (2008), 111-195. DOI: **10.1016/j.physrep.2008.02.002**.

[MV03] C. MAES AND E. VERBITSKIY: Large deviations and a fluctuation symmetry for chaotic homeomorphisms. Commun. Math. Phys. **233** (2003), 137-151. DOI: **10.1007/s00220-002-0764-6**.

[Nyk28] H. NYQUIST: Thermal agitation of electric charge in conductors. Phys. Rev. **32** (1928), 110-113. DOI: **10.1103/PhysRev.32.110**.

[Ons31a] L. ONSAGER: Reciprocal relations in irreversible processes. I. Phys. Rev. **37** (1931), 405-426. DOI: **10.1103/PhysRev.37.405**.

[Ons31b] ________, Reciprocal relations in irreversible processes. II. Phys. Rev. **38** (1931), 2265-2279. DOI: **10.1103/physrev.38.2265**.

[Per08] J. PERRIN: L'agitation moléculaire et le mouvement brownien. Comptes rendus hebdomadaires des séances de l'académie des sciences **146** (1908), 967-970.

[Per09] ________, Mouvement brownien et réalité moléculaire. Annales de Chimie et de Physique **18** (1909), 1-114.

[Pit11] L. P. PITAEVSKII: Rigorous results of nonequilibrium statistical physics and their experimental verification. Physics-Uspekhi **54** (2011), 625-632. DOI: **10.3367/UFNE.0181.201106d.0647**.

[RM07] L. RONDONI AND C. MEJIA-MONASTERIO: Fluctuations in nonequilibrium statistical mechanics: models, mathematical theory, physical mechanisms. Nonlinearity **20** (2007), R1-R37. DOI: **10.1088/0951-7715/20/10/R01**.

[Rue99] D. RUELLE: Smooth dynamics and new theoretical ideas in nonequilibrium statistical mechanics. J. Stat. Phys. **95** (1999), 393-468. DOI: **10.1023/A:1004593915069**.

[Smo06] M. VON SMOLUCHOWSKI: Zur kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen. Annalen der Physik **21** (1906), 756-780. DOI: **10.1002/andp.19063261405**.
Chapter 2
Large Deviations and Fluctuation Relations

**Abstract** After introducing the minimal framework needed to make sense of a fluctuation theorem, we briefly review some basic facts about large deviations estimates that play a crucial role in the formulation of fluctuation theorems. The reader is referred to [DZ00; Hol00] for detailed introductions to this vast subject.

Our starting point is a family of measurable spaces (Ω_λ, F_λ) indexed by a pa-
rameter λ varying in a directed set L. Each of these spaces is equipped with a
pair (P_λ, ᾃP_λ) of probability measures. We say that the family (P_λ, ᾃP_λ)_{λ∈L} is in in-
volution whenever, for each λ ∈ L, ᾃP_λ = P_λ ○ Θ_λ where Θ_λ : Ω_λ → Ω_λ is a measure
space automorphism satisfying Θ_λ ○ Θ_λ = Id_{Ω_λ}. Such an involution implements
a Z_2-action on Ω_λ. In many cases of interest one takes<fnref n="1" /> L = Z_+ or L = R_+ and
interprets the values taken by λ as discrete or continuous time instances. In such
a situation, (Ω_λ, F_λ, P_λ) describes the space-time statistics of the physical system
under consideration over the finite time interval [0, λ] and ᾃP_λ is a reference mea-
sure related to the same system running under a different protocol.<fnref n="2" /> A possible
candidate for this reference measure is ᾃP_λ = P_λ ○ Θ_λ where Θ_λ implements the
time-reversal symmetry of the system, in which case the resulting pair of mea-
sures is in involution. However, as we shall illustrate in Chapter 3, our general
setup can also be applied to other physical problems not necessarily related to
dynamical processes. We shall denote by E_λ and ᾃE_λ the expectation functionals
associated to P_λ and ᾃP_λ.

<footnote n="1">¹ $\mathbb{Z}_+ = \{0, 1, 2, \dots\}$ and $\mathbb{R}_+ = [0, \infty[$.</footnote>

<footnote n="2">² In the physics literature, the process $(\Omega_{\lambda}, \mathcal{F}_{\lambda}, \hat{P}_{\lambda})$ is sometimes called *conjugate* to $(\Omega_{\lambda}, \mathcal{F}_{\lambda}, \mathbb{P}_{\lambda})$.</footnote>
## 2.1 Large Deviations

Assume that, for each $\lambda$ in the directed set $\mathcal{L}$, $\xi_\lambda$ is a random variable on the measurable space $(\Omega_\lambda, \mathcal{F}_\lambda)$ taking its values in a Polish (i.e., separable complete metric) space $\mathcal{X}$, and the function $\mathcal{L} \ni \lambda \mapsto r_\lambda > 0$ is such that $\lim_{\lambda \in \mathcal{L}} r_\lambda = +\infty$. One says that the family $(\xi_\lambda)_{\lambda \in \mathcal{L}}$ satisfies the **Large Deviation Principle** (LDP) w.r.t. the family of probability measures $(\mathbb{P}_\lambda)_{\lambda \in \mathcal{L}}$ with scale $(r_\lambda)_{\lambda \in \mathcal{L}}$ and rate $I$ whenever $I: \mathcal{X} \to [0, +\infty]$ has closed level sets,<fnref n="3" /> and is such that for any Borel set $X \subset \mathcal{X}$,

$$ 
\begin{aligned}
- \inf_{x \in \mathcal{X}} I(x) &\le \liminf_{\lambda \in \mathcal{L}} r_\lambda^{-1} \log \mathbb{P}_\lambda \{\xi_\lambda \in X\} \\
&\le \limsup_{\lambda \in \mathcal{L}} r_\lambda^{-1} \log \mathbb{P}_\lambda \{\xi_\lambda \in X\} \le - \inf_{x \in \bar{\mathcal{X}}} I(x),
\end{aligned} 
\quad (2.1) $$

where $\mathcal{X}/\bar{\mathcal{X}}$ denotes the interior/closure of $X$. $I$ is said to be a good rate whenever it has compact level sets. If the lower bound in (2.1) only holds for Borel sets $X$ that are subsets of $\mathcal{X} \subset \mathcal{X}$, we shall say that the **LDP holds locally** in $\mathcal{X}$. We remark that considering $X = \mathcal{X}$ in (2.1) implies that $\inf_{x \in \mathcal{X}} I(x) = 0$, and in particular that $I$ is not everywhere $+\infty$.

In most applications, the index $\lambda$ parametrizes the size of a sample and $\xi_\lambda$ describes some property of samples of size $\lambda$. Suppose that the law of large numbers holds as the size of the sample increases, i.e., that there is $\tilde{\xi} \in \mathcal{X}$ such that $\lim_{\lambda \in \mathcal{L}} \xi_\lambda = \tilde{\xi}$ holds in probability. The LDP then describes, on the exponential scale, the rate at which this limit is achieved, and the rate function $I$ has a single zero at $\tilde{\xi}$.

The **contraction principle** [DZ00, Theorem 4.2.1]<fnref n="4" /> states that if the above LDP holds with a good rate $I$, and $\phi: \mathcal{X} \to \mathcal{Y}$ is a continuous map taking its values in a Polish space $\mathcal{Y}$, then the family $(\eta_\lambda)_{\lambda \in \mathcal{L}}$ defined by $\eta_\lambda = \phi(\xi_\lambda)$ satisfies the LDP w.r.t. $(\mathbb{P}_\lambda)_{\lambda \in \mathcal{L}}$ with scale $(r_\lambda)_{\lambda \in \mathcal{L}}$ and good rate $J$, where

$$ J(y) = \inf \{ I(x) \mid x \in \phi^{-1}(\{y\}) \}. $$

The same conclusion holds [DZ00, Theorem 4.2.23] if $\phi$ is merely measurable and such that, for a sequence of continuous maps $\phi_n: \mathcal{X} \to \mathcal{Y}$, one has

$$ \limsup_{n \to \infty} \{d(\phi(x), \phi_n(x)) \mid x \in I^{-1}([0, \alpha])\} = 0 $$

for all $\alpha \in \mathbb{R}$.

**Varadhan's Lemma** [DZ00, Theorem 4.3.1] asserts that if the family $(\xi_\lambda)_{\lambda \in \mathcal{L}}$ satisfies the LDP w.r.t. $(\mathbb{P}_\lambda)_{\lambda \in \mathcal{L}}$ with scale $(r_\lambda)_{\lambda \in \mathcal{L}}$ and good rate $I$, and $\phi: \mathcal{X} \to \mathbb{R}$

<footnote n="3">³ $I$ has closed/compact level sets whenever $\{x \in \mathcal{X} \mid I(x) \le \alpha\}$ is closed/compact for all $\alpha \in \mathbb{R}$. This implies in particular that $I$ is lower semicontinuous.</footnote>

<footnote n="4">⁴ [DZ00] does not address the case where $\mathcal{L}$ is a general directed set, but the proof is almost identical.</footnote>
is a continuous function such that

$$ \limsup_{\lambda \in \mathcal{L}} \frac{1}{r_{\lambda}} \log \mathbb{E}_{\lambda} \left[ e^{\alpha r_{\lambda} \phi(\xi_{\lambda})} \right] < \infty $$

for some $\alpha > 1$, then

$$ \lim_{\lambda \in \mathcal{L}} \frac{1}{r_{\lambda}} \log \mathbb{E}_{\lambda} \left[ e^{r_{\lambda} \phi(\xi_{\lambda})} \right] = \sup \{ \phi(x) - I(x) \mid x \in \mathcal{X} \}. $$

Moreover [Hol00, Theorem III.17], defining the tilted measure$^5$ $\mathbb{P}_{\lambda}^{\phi} \ll \mathbb{P}_{\lambda}$ with density

$$ \frac{d\mathbb{P}_{\lambda}^{\phi}}{d\mathbb{P}_{\lambda}} = \frac{e^{r_{\lambda}\phi(\xi_{\lambda})}}{\mathbb{E}_{\lambda}[e^{r_{\lambda}\phi(\xi_{\lambda})}]}, $$

on $(\Omega_{\lambda}, \mathcal{F}_{\lambda})$ and setting

$$ J(x) = I(x) - \phi(x) - \inf\{I(y) - \phi(y) \mid y \in \mathcal{X}\}, \quad (2.2) $$

the family $(\xi_{\lambda})_{\lambda \in \mathcal{L}}$ satisfies the LDP w.r.t. $(\mathbb{P}_{\lambda}^{\phi})_{\lambda \in \mathcal{L}}$ with scale $(r_{\lambda})_{\lambda \in \mathcal{L}}$ and rate $J$. Given the scale $(r_{\lambda})_{\lambda \in \mathcal{L}}$, two families $(P_{\lambda})_{\lambda \in \mathcal{L}}$ and $(Q_{\lambda})_{\lambda \in \mathcal{L}}$ of probability measures on $\mathcal{X}$ are **exponentially equivalent** whenever there exists a family of probability spaces $(\Omega_{\lambda}, \mathcal{F}_{\lambda}, \mathbb{P}_{\lambda})$ and $\mathcal{X}$-valued random variables $(\xi_{\lambda})_{\lambda \in \mathcal{L}}$ and $(\eta_{\lambda})_{\lambda \in \mathcal{L}}$ whose laws under $\mathbb{P}_{\lambda}$ are given by $P_{\lambda}$ and $Q_{\lambda}$ respectively, and such that for all $\delta > 0$, the sets $\Gamma_{\lambda, \delta} = \{\omega \in \Omega_{\lambda} \mid d(\xi_{\lambda}(\omega), \eta_{\lambda}(\omega)) > \delta\} \in \mathcal{F}_{\lambda}$ satisfy

$$ \limsup_{\lambda \in \mathcal{L}} \frac{1}{r_{\lambda}} P_{\lambda}(\Gamma_{\lambda, \delta}) = -\infty. $$

Suppose that $\mathcal{X} = \mathbb{R}^d$ and that for all $\alpha \in \mathbb{R}^d$ the limit$^6$

$$ \Lambda(\alpha) := \lim_{\lambda \in \mathcal{L}} \frac{1}{r_{\lambda}} \log \mathbb{E}_{\lambda} \left[ e^{r_{\lambda} \langle \alpha, \xi_{\lambda} \rangle} \right] \in [-\infty, \infty] \quad (2.3) $$

exists and defines a lower semicontinuous function. Then, $\Lambda$ is a convex function, its essential domain $\text{Dom}(\Lambda) = \{\alpha \in \mathbb{R}^d \mid \Lambda(\alpha) < \infty\}$ is convex, and if $d = 1$ the map $\text{Dom}(\Lambda) \ni \alpha \mapsto \Lambda(\alpha)$ is continuous.$^7$ Suppose that $0$ belongs to the interior of $\text{Dom}(\Lambda)$ and that $\Lambda$ is differentiable on this interior. If either $\text{Dom}(\Lambda) = \mathbb{R}$ or

$$ \lim_{\substack{\alpha \to \partial \text{Dom}(\Lambda) \\ \alpha \in \text{Dom}(\Lambda)}} |\Lambda'(\alpha)| = +\infty, \quad (2.4) $$

<footnote>$^5$ The notation $\mathbb{P}_{\lambda}^{\phi} \ll \mathbb{P}_{\lambda}$ means that $\mathbb{P}_{\lambda}^{\phi}$ is absolutely continuous with respect to $\mathbb{P}_{\lambda}$.</footnote>

<footnote>$^6$ Here $\langle \cdot, \cdot \rangle$ denotes the Euclidean inner product on $\mathbb{R}^d$.</footnote>

<footnote>$^7$ If $d > 1$, continuity only holds in the relative interior of $\text{Dom}(\Lambda)$, and in particular in its interior, see [Roc70, Theorem 10.1].</footnote>
then the **Gärtner-Ellis theorem** [DZ00, Theorem 2.3.6] asserts that the family $(\xi_{\lambda})_{\lambda \in \mathcal{L}}$ satisfies the LDP w.r.t. $(\mathbb{P}_{\lambda})_{\lambda \in \mathcal{L}}$ with scale $(r_{\lambda})_{\lambda \in \mathcal{L}}$ and good rate $I$ given by the *Legendre-Fenchel transform* of $\Lambda$,

$$I(x) = \sup_{\alpha \in \mathbb{R}^d} (\langle \alpha, x \rangle - \Lambda(\alpha)).$$

If $\text{Dom}(\Lambda) \neq \mathbb{R}$ and Condition (2.4) fails, then the same LDP holds locally on the set $\mathcal{X}$ of all $x \in \mathbb{R}^d$ for which there exists $\alpha$ in the interior of $\text{Dom}(\Lambda)$ such that

$$I(y) > I(x) + \langle \alpha, y - x \rangle$$

holds for all $y \neq x$. Such a $x \in \mathbb{R}^d$ is said to be an *exposed point* of $I$, and the corresponding $\alpha$ is the associated *supporting hyperplane*.

We say that two families $(\xi_{\lambda})_{\lambda \in \mathcal{L}}$ and $(\eta_{\lambda})_{\lambda \in \mathcal{L}}$ of random variables on $(\Omega_{\lambda}, \mathcal{F}_{\lambda})$ taking their values in the same Polish space $\mathcal{X}$ are **physically equivalent** whenever

$$\lim_{\lambda \in \mathcal{L}} \frac{1}{r_{\lambda}} \sup_{\omega \in \Omega_{\lambda}} d(\xi_{\lambda}(\omega), \eta_{\lambda}(\omega)) = 0.$$

This implies that the laws of $r_{\lambda}^{-1}\xi_{\lambda}$ and $r_{\lambda}^{-1}\eta_{\lambda}$ under $\mathbb{P}_{\lambda}$ are exponentially equivalent, i.e., that

$$\limsup_{\lambda \in \mathcal{L}} \frac{1}{r_{\lambda}} \log \mathbb{P}_{\lambda} \{d(\xi_{\lambda}, \eta_{\lambda}) > \varepsilon r_{\lambda}\} = -\infty$$

holds for all $\varepsilon > 0$. It follows [DZ00, Theorem 4.2.13] that if $(\xi_{\lambda})_{\lambda \in \mathcal{L}}$ satisfies the LDP w.r.t. $(\mathbb{P}_{\lambda})_{\lambda \in \mathcal{L}}$ with scale $(r_{\lambda})_{\lambda \in \mathcal{L}}$ and good rate $I$, so does $(\eta_{\lambda})_{\lambda \in \mathcal{L}}$. In addition, if $(\xi_{\lambda})_{\lambda \in \mathcal{L}}$ and $(\eta_{\lambda})_{\lambda \in \mathcal{L}}$ are physically equivalent, then replacing $\xi_{\lambda}$ with $\eta_{\lambda}$ in (2.3) does not change the function $\Lambda$.

## 2.2 Transient Fluctuation Relation

In this section we describe a minimal abstract framework allowing for the formulation of transient fluctuation relations. Our point here is that such relations are tautological consequences of a simple regularity property of a pair of probability measures.

To a pair $(\mathbb{P}_{\lambda}, \hat{\mathbb{P}}_{\lambda})$ of equivalent probability measures on $(\Omega_{\lambda}, \mathcal{F}_{\lambda})$ we associate the *entropy production observable*

$$\sigma_{\lambda} := \log \frac{d\mathbb{P}_{\lambda}}{d\hat{\mathbb{P}}_{\lambda}}. \qquad (2.5)$$

This is a real-valued random variable, and we denote by $P_{\lambda}$ its law under $\mathbb{P}_{\lambda}$. Recall that the family $(\mathbb{P}_{\lambda}, \hat{\mathbb{P}}_{\lambda})_{\lambda \in \mathcal{L}}$ is *in involution* whenever, for each $\lambda \in \mathcal{L}$, $\hat{\mathbb{P}}_{\lambda} = \mathbb{P}_{\lambda} \circ \Theta_{\lambda}$ where $\Theta_{\lambda}: \Omega_{\lambda} \to \Omega_{\lambda}$ is a measure space automorphism satisfying $\Theta_{\lambda} \circ \Theta_{\lambda} = \text{Id}_{\Omega_{\lambda}}$.
The very definition of $\sigma_\lambda$ implies a number of simple, yet important properties.

**Non-vanishing.** $\sigma_\lambda$ vanishes $\mathbb{P}_\lambda$-a.e. iff $\hat{\mathbb{P}}_\lambda = \mathbb{P}_\lambda$. From our point of view this is a trivial and uninteresting situation. In the following we assume that $\hat{\mathbb{P}}_\lambda \neq \mathbb{P}_\lambda$. In cases where $\mathbb{P}_\lambda$ and $\hat{\mathbb{P}}_\lambda$ are in involution, this means that the $\mathbb{Z}_2$-symmetry $\Theta_\lambda$ is broken.

**Alternating.** The entropy production of the pair $(\hat{\mathbb{P}}_\lambda, \mathbb{P}_\lambda)$ is given by $\hat{\sigma}_\lambda = -\sigma_\lambda$. If $\mathbb{P}_\lambda$ and $\hat{\mathbb{P}}_\lambda$ are in involution, then $\hat{\sigma}_\lambda = \sigma_\lambda \circ \Theta_\lambda$, and hence $\sigma_\lambda$ is odd under the $\mathbb{Z}_2$-action. In particular, the range of $\sigma_\lambda$ is symmetric w.r.t. 0.

**Mean entropy production.** Positive values of $\sigma_\lambda$ are preferred under $\mathbb{P}_\lambda$. To see this, recall that the relative entropy$^8$ of two equivalent probability measures $\mathbb{P}$ and $\mathbb{Q}$ is defined by

$$ \mathrm{Ent}(\mathbb{P} | \mathbb{Q}) := \int \log \left( \frac{d\mathbb{P}}{d\mathbb{Q}} \right) d\mathbb{P}. \qquad (2.6) $$

By Jensen's inequality, one has

$$ \mathrm{Ent}(\mathbb{P} | \mathbb{Q}) = - \int \log \left( \frac{d\mathbb{Q}}{d\mathbb{P}} \right) d\mathbb{P} \geq - \log \int \frac{d\mathbb{Q}}{d\mathbb{P}} d\mathbb{P} = 0, $$

with equality iff $\mathbb{P} = \mathbb{Q}$. It follows that the mean entropy production

$$ \mathrm{ep}_{\lambda} := \mathrm{E}_{\lambda}[\sigma_{\lambda}] = \mathrm{Ent}(\mathbb{P}_{\lambda} | \hat{\mathbb{P}}_{\lambda}) \qquad (2.7) $$

is non-negative and vanishes iff $\mathbb{P}_\lambda = \hat{\mathbb{P}}_\lambda$. Expressing this fact in terms of the law $P_\lambda$ of $\sigma_\lambda$ under $\mathbb{P}_\lambda$ yields the inequality

$$ \mathrm{ep}_{\lambda} = \int_{\mathbb{R}} s P_{\lambda}(ds) \geq 0. \qquad (2.8) $$

Thus, if $\mathbb{P}_\lambda$ and $\hat{\mathbb{P}}_\lambda$ are in involution, then $\mathrm{ep}_\lambda$ plays the role of an order parameter for the $\mathbb{Z}_2$-symmetry $\Theta_\lambda$: Inequality (2.8) becomes strict whenever this symmetry is broken.

**Jarzynski's identity.** A more quantitative statement can be obtained by considering the *Rényi relative $\alpha$-entropy* defined by

$$ \mathrm{Ent}_{\alpha}(\mathbb{P} | \mathbb{Q}) := \log \int \left( \frac{d\mathbb{Q}}{d\mathbb{P}} \right)^{\alpha} d\mathbb{P}, \quad (\alpha \in \mathbb{R}). \qquad (2.9) $$

In terms of entropy production, we have

$$ e_{\lambda}(\alpha) := \mathrm{Ent}_{\alpha}(\mathbb{P}_{\lambda} | \hat{\mathbb{P}}_{\lambda}) = \log \mathrm{E}_{\lambda}[e^{-\alpha \sigma_{\lambda}}] = \log \int_{\mathbb{R}} e^{-\alpha s} P_{\lambda}(ds). \qquad (2.10) $$

<footnote>$^8$ also called Kullback-Leibler divergence.</footnote>
The function $\mathbb{R} \ni \alpha \mapsto e_{\lambda}(\alpha) \in [-\infty, +\infty]$ is the cumulant generating function of the random variable $-\sigma_{\lambda}$. It is convex, lower semicontinuous, and vanishes for $\alpha \in \{0, 1\}$. Hence, $e_{\lambda}$ is non-positive on $[0, 1]$ and non-negative outside $[0, 1]$. It admits an analytic continuation to the strip $\{z \in \mathbb{C} | 0 < \operatorname{Re} z < 1\}$. The Rényi entropy vanishes identically as a function of $\alpha$ iff $\operatorname{ep}_{\lambda} = 0$ and is strictly negative for $\alpha \in ]0, 1[$ iff $\operatorname{ep}_{\lambda} > 0$. Expressing the relation $e_{\lambda}(1) = 0$ in terms of $P_{\lambda}$, we derive Jarzynski identity [Jar00]

$$ \int_{\mathbb{R}} e^{-s} P_{\lambda}(ds) = 1. \qquad (2.11) $$

Invoking Markov's inequality, we derive the following exponential bound on the tail of $P_{\lambda}$

$$ P_{\lambda}(|-\infty, -s|) = \mathbb{P}_{\lambda}\{\sigma_{\lambda} \le -s\} \le e^{-s}. \qquad (2.12) $$

To formulate the last properties, let us define

$$ \hat{e}_{\lambda}(\alpha) := \operatorname{Ent}_{\alpha}(\hat{\mathbb{P}}_{\lambda}|\mathbb{P}_{\lambda}) $$

and denote by $\hat{P}_{\lambda}$ the law of $\sigma_{\lambda}$ under $\hat{\mathbb{P}}_{\lambda}$.

**Proposition 2.1 (Transient Fluctuation Relation)**

(1) The following relations hold:

$$ \hat{e}_{\lambda}(\alpha) = e_{\lambda}(1 - \alpha) \quad \text{for } \alpha \in \mathbb{R}, \qquad (2.13) $$

$$ \frac{dP_{\lambda}}{d\hat{P}_{\lambda}}(s) = e^{s} \quad \text{for } s \in \mathbb{R}. \qquad (2.14) $$

(2) If the family $(\mathbb{P}_{\lambda}, \hat{\mathbb{P}}_{\lambda})_{\lambda \in \mathcal{L}}$ is in involution, then

$$ \hat{e}_{\lambda} = e_{\lambda}, \qquad (2.15) $$

$$ \hat{P}_{\lambda} = P_{\lambda} \circ \theta, \qquad (2.16) $$

where $\theta$ denotes the reflection $s \mapsto -s$.

Relations (2.13) and (2.14) are in fact equivalent: the validity of one of them implies the other. We shall refer to them as the *transient FR*. It refines (and implies) Inequality (2.8) and its basic appeal is its universal form. In applications to non-equilibrium physics, the transient FR with $e_{\lambda}$ not identically vanishing or, equivalently, $P_{\lambda}$ not concentrated at $s = 0$, is a manifestation of time-reversal symmetry breaking and emergence of the 2nd Law of Thermodynamics.

**Proof of Proposition 2.1.** (1) Relation (2.13) is a simple consequence of the definition of Rényi's entropy,
$$
\begin{align}
e_{\lambda}(1-\alpha) &= \mathrm{Ent}_{1-\alpha}(\mathbb{P}_{\lambda} | \hat{\mathbb{P}}_{\lambda}) \nonumber \\
&= \log \mathbb{E}_{\lambda}[e^{-(1-\alpha)\sigma_{\lambda}}] = \log \hat{\mathbb{E}}_{\lambda}[e^{\alpha\sigma_{\lambda}}] \tag{2.17} \\
&= \log \hat{\mathbb{E}}_{\lambda}[e^{-\alpha\hat{\sigma}_{\lambda}}] = \mathrm{Ent}_{\alpha}(\hat{\mathbb{P}}_{\lambda} | \mathbb{P}_{\lambda}) = \hat{e}_{\lambda}(\alpha). \nonumber
\end{align}
$$

To prove (2.14), we exponentiate (2.17) and rewrite the result in terms of $P_\lambda$ and $\hat{P}_\lambda$ to get

$$
\int_{\mathbb{R}} e^{-(1-\alpha)s} P_{\lambda}(ds) = \int_{\mathbb{R}} e^{\alpha s} \hat{P}_{\lambda}(ds).
$$

Using now the analyticity of $e_\lambda(z)$ in the strip $0 < \mathrm{Re} z < 1$ and the fact that the
characteristic function uniquely defines the corresponding probability measure,
we deduce that $e^{-s}P_\lambda(ds)$ and $\hat{P}_\lambda(ds)$ coincide. This is equivalent to (2.14).

(2) Relation (2.15) is a direct consequence of the facts that $\hat{P}_{\lambda} = \mathbb{P}_{\lambda} \circ \Theta_{\lambda}$ and $\hat{\sigma}_{\lambda} = -\sigma_{\lambda} = \sigma_{\lambda} \circ \Theta_{\lambda}$, which give that

$$
\begin{align}
\hat{e}_{\lambda}(\alpha) &= \log \hat{\mathbb{E}}_{\lambda}[e^{-\alpha \hat{\sigma}_{\lambda}}] = \log \hat{\mathbb{E}}_{\lambda}[e^{\alpha \sigma_{\lambda}}] \tag{2.18} \\
&= \log \mathbb{E}_{\lambda}[e^{\alpha \sigma_{\lambda} \circ \Theta_{\lambda}}] = \log \mathbb{E}_{\lambda}[e^{-\alpha \sigma_{\lambda}}] = e_{\lambda}(\alpha). \nonumber
\end{align}
$$

Exponentiating the third equality in (2.18) yields

$$
\int e^{\alpha s} \hat{P}_{\lambda}(ds) = \int e^{-\alpha s} P_{\lambda}(ds) = \int e^{\alpha \theta(s)} P_{\lambda}(ds) = \int e^{\alpha s} P_{\lambda} \circ \theta(ds)
$$

and the same characteristic function argument as in the proof of (1) yields (2.16).

<figure><img src="image_6.png" /></figure>

2.3 Fluctuation Theorem

Going back to the abstract setup of Section 2.2, we proceed beyond the transient regime by selecting a scale function $\mathcal{L} \ni \lambda \mapsto r_\lambda$. In a specific context, the choice of $r_\lambda$ is naturally linked to the structure of $\mathcal{L}$. For example, if $\mathcal{L} = \mathbb{Z}_+$ or $\mathbb{R}_+$, its elements being interpreted as time instances, then $r_\lambda = \lambda$.

**Definition 2.2**

(1) We say that the *Fluctuation Theorem* holds for the family $(\mathbb{P}_\lambda, \hat{\mathbb{P}}_\lambda)_{\lambda \in \mathcal{L}}$ with scale $(r_\lambda)_{\lambda \in \mathcal{L}}$ and rate $(I, \hat{I})$ whenever the following two conditions are satisfied:

(i) $(r_{\lambda}^{-1}\sigma_{\lambda})_{\lambda \in \mathcal{L}}$ satisfies the LDP w.r.t. $(\mathbb{P}_{\lambda})_{\lambda \in \mathcal{L}}$ with scale $(r_{\lambda})_{\lambda \in \mathcal{L}}$ and rate $I$.

(ii) $(r_{\lambda}^{-1}\sigma_{\lambda})_{\lambda \in \mathcal{L}}$ satisfies the LDP w.r.t. $(\hat{\mathbb{P}}_{\lambda})_{\lambda \in \mathcal{L}}$ with scale $(r_{\lambda})_{\lambda \in \mathcal{L}}$ and rate $\hat{I}$.

(2) For each $\lambda \in \mathcal{L}$, let $\tilde{\sigma}_{\lambda}$ be a real-valued random variable on $\Omega_{\lambda}$. We say that $(\tilde{\sigma}_{\lambda})_{\lambda \in \mathcal{L}}$ satisfies the *Fluctuation Theorem* w.r.t. the family $(\mathbb{P}_{\lambda}, \hat{\mathbb{P}}_{\lambda})_{\lambda \in \mathcal{L}}$ with scale $(r_{\lambda})_{\lambda \in \mathcal{L}}$ and rate $(I, \hat{I})$ whenever the above conditions (i) and (ii) are satisfied with $\sigma_{\lambda}$ replaced by $\tilde{\sigma}_{\lambda}$.
(3) If the two LDPs mentioned in conditions (i) and (ii) only hold locally on the interval $\mathcal{J} \ni 0$, then we say that the corresponding Fluctuation Theorem holds on $\mathcal{J}$. This Fluctuation Theorem is called global when it holds on $\mathcal{J} = \mathbb{R}$, local otherwise.

**Remark 1.** Recall that $P_\lambda/\hat{P}_\lambda$ denotes the law of $\sigma_\lambda$ under $\mathbb{P}_\lambda/\mathbb{P}_\lambda$. Thus, the LDP estimates expressing the FT can be rewritten as

$$ -\inf_{s \in \bar{S}} I(s) \le \liminf_{\lambda \in \mathcal{L}} r_{\lambda}^{-1} \log P_{\lambda}(r_{\lambda} S) \le \limsup_{\lambda \in \mathcal{L}} r_{\lambda}^{-1} \log P_{\lambda}(r_{\lambda} S) \le -\inf_{s \in \bar{S}} I(s), \quad (2.19) $$

$$ -\inf_{s \in \bar{S}} \hat{I}(s) \le \liminf_{\lambda \in \mathcal{L}} r_{\lambda}^{-1} \log \hat{P}_{\lambda}(r_{\lambda} S) \le \limsup_{\lambda \in \mathcal{L}} r_{\lambda}^{-1} \log \hat{P}_{\lambda}(r_{\lambda} S) \le -\inf_{s \in \bar{S}} \hat{I}(s), \quad (2.20) $$

for any Borel set $S \subset \mathbb{R}$.

**Remark 2.** Suppose that the FT holds on $\mathcal{J}$ for the family $(\mathbb{P}_\lambda, \hat{\mathbb{P}}_\lambda)_{\lambda \in \mathcal{L}}$ with scale $(r_\lambda)_{\lambda \in \mathcal{L}}$ and rate $(I, \hat{I})$. Let $(\sigma_\lambda)_{\lambda \in \mathcal{L}}$ be the associated entropy production. Then, any family $(\tilde{\sigma}_\lambda)_{\lambda \in \mathcal{L}}$ of real valued random variables which is physically equivalent to $(\sigma_\lambda)_{\lambda \in \mathcal{L}}$ satisfies the FT on $\mathcal{J}$ w.r.t. $(\mathbb{P}_\lambda, \hat{\mathbb{P}}_\lambda)_{\lambda \in \mathcal{L}}$ with scale $(r_\lambda)_{\lambda \in \mathcal{L}}$ and rate $(I, \hat{I})$.

## 2.4 Asymptotic Fluctuation Relation

In Section 2.2, the transient FR was obtained as a trivial consequence of the definition of entropy production. Here, we show how the Fluctuation Theorem, together with the transient FR, yield an asymptotic FR, a symmetry relation for the rate functions $(I, \hat{I})$.

**Proposition 2.3** Suppose that the FT holds on $\mathcal{J}$ for the family $(\mathbb{P}_\lambda, \hat{\mathbb{P}}_\lambda)_{\lambda \in \mathcal{L}}$. Then the corresponding rate functions satisfy the relation

$$ \hat{I}(s) = I(s) + s \quad (2.21) $$

for all $s \in \mathcal{J}$. Moreover, if the family $(\mathbb{P}_\lambda, \hat{\mathbb{P}}_\lambda)_{\lambda \in \mathcal{L}}$ is in involution, then

$$ \hat{I}(s) = I(-s) \quad (2.22) $$

for all $s \in \mathcal{J} \cap (-\mathcal{J})$.

**Proof.** In view of the transient FR (2.14) we have

$$ e^{r_{\lambda} \inf S} \hat{P}_{\lambda}(r_{\lambda} S) \le P_{\lambda}(r_{\lambda} S) \le e^{r_{\lambda} \sup S} \hat{P}_{\lambda}(r_{\lambda} S) $$

for any Borel set $S \subset \mathcal{J}$. Using (2.19) and (2.20) we obtain
$$-\inf_{s \in \hat{S}} I(s) \le \liminf_{\lambda \in \mathcal{L}} r_{\lambda}^{-1} \log P_{\lambda}(r_{\lambda} S)$$

$$\le \limsup_{\lambda \in \mathcal{L}} r_{\lambda}^{-1} \log(e^{r_{\lambda} \sup S} \hat{P}_{\lambda}(r_{\lambda} S)) \le \sup S - \inf_{s \in \hat{S}} \hat{I}(s),$$

$$\inf S - \inf_{s \in \hat{S}} \hat{I}(s) \le \liminf_{\lambda \in \mathcal{L}} r_{\lambda}^{-1} (e^{r_{\lambda} \inf S} \log \hat{P}_{\lambda}(r_{\lambda} S))$$

$$\le \limsup_{\lambda \in \mathcal{L}} r_{\lambda}^{-1} \log P_{\lambda}(r_{\lambda} S) \le -\inf_{s \in \hat{S}} I(s).$$

Taking $S = ]b - \varepsilon, b + \varepsilon|$ with $b \in \mathcal{J}$ and $\varepsilon > 0$ small enough, we derive

$$\inf_{|s-b|<2\varepsilon} \hat{I}(s) \le \inf_{|s-b|\le\varepsilon} \hat{I}(s) \le b + \varepsilon + \inf_{|s-b|<\varepsilon} I(s), \qquad (2.23)$$

$$b - \varepsilon + \inf_{|s-b|<2\varepsilon} I(s) \le b - \varepsilon + \inf_{|s-b|\le\varepsilon} I(s) \le \inf_{|s-b|<\varepsilon} \hat{I}(s).$$

Since the functions $I$ and $\hat{I}$ are lower semicontinuous, we have

$$I(b) = \liminf_{\varepsilon \downarrow 0} \inf_{|s-b|<\varepsilon} I(s), \quad \hat{I}(b) = \liminf_{\varepsilon \downarrow 0} \inf_{|s-b|<\varepsilon} \hat{I}(s).$$

Passing to the limit in (2.23) we get

$$\hat{I}(b) \le b + I(b) \le \hat{I}(b)$$

for any $b \in \mathcal{J}$, which proves (2.21).

If the family $(\mathbb{P}_{\lambda}, \hat{\mathbb{P}}_{\lambda})_{\lambda \in \mathcal{L}}$ is in involution, then (2.16) gives $\hat{P}_{\lambda}(r_{\lambda} S) = P_{\lambda}(-r_{\lambda} S)$, and the uniqueness of the rate function yields (2.22). $\square$

**Remark 3.** If the family $(\mathbb{P}_{\lambda}, \hat{\mathbb{P}}_{\lambda})_{\lambda \in \mathcal{L}}$ is in involution, then (2.16) implies the following assertions:

* If the LDP of condition (i) (resp. (ii)) in Definition 2.2 holds with rate function $I$ (resp. $\hat{I}$), then the LDP of condition (ii) (resp. (i)) holds with rate $\hat{I}$ (resp. $I$) obeying (2.21) and (2.22).

* If the LDP of condition (i) (resp. (ii)) holds locally on $\mathcal{J}$, then the LDP of condition (ii) (resp. (i)) holds locally on $-\mathcal{J}$. In particular the FT holds on $\mathcal{J} \cap (-\mathcal{J})$.

* In Part (3) of Definition 2.2, the set $\mathcal{J}$ can always be taken to be symmetric around 0, since when both conditions hold on $\mathcal{J}$, they also hold on $\mathcal{J} \cup (-\mathcal{J})$.

However, if the two families $(\mathbb{P}_{\lambda}, \hat{\mathbb{P}}_{\lambda})_{\lambda \in \mathcal{L}}$ are not assumed to be in involution, then conditions (i) and (ii) in Definition 2.2 are not equivalent in general. Assuming that $I$ and $\hat{I}$ obey (2.21), one can still show the following: If the LDP of condition (i) (resp. (ii)) holds, then the leftmost bound in (2.20) (resp. (2.19)) holds for every Borel set $S$, and the rightmost bound holds for every *precompact* Borel set $S$. LDPs whose upper bound is restricted to precompact sets are called the weak
LDPs in the literature. While some of the results presented below could be formu-
lated in terms of weak LDPs, we will refrain from discussing such technicalities.

**2.5 Entropic Pressure and Rate**

**Definition 2.4** Assume that the FT holds for the family $(\mathbb{P}_{\lambda}, \hat{\mathbb{P}}_{\lambda})_{\lambda \in \mathcal{L}}$ with scale $(r_{\lambda})_{\lambda \in \mathcal{L}}$ and rate $(I, \hat{I})$. The associated entropic pressure is the function defined by

$$
\mathbb{R} \ni \alpha \mapsto e(\alpha) := \sup_{s \in \mathbb{R}} (\alpha s - I(-s)). \qquad (2.24)
$$

We note that the entropic pressure is nothing but the *Legendre–Fenchel transform* of the function<fnref n="9" /> $s \mapsto I(-s)$. We shall see that it is closely related to various notions of pressure and other thermodynamic potentials. We use the name “entropic pressure” to clearly distinguish it from these other notions and to emphasize its connection with entropy production.

From the perspective of the theory of convex functions, it follows from its de-
finition that the entropic pressure is a *closed, proper convex function*<fnref n="10" /> [Roc70,
Theorem 12.2]. Invoking the LDP yields $e(0) = 0$. Setting

$$
\hat{e}(\alpha) := \sup_{s \in \mathbb{R}} (\alpha s - \hat{I}(s)),
$$

the asymptotic FR (2.21) further implies $\hat{e}(\alpha) = e(1 - \alpha)$, and the same reasoning yields that $e(1) = \hat{e}(0) = 0$. It immediately follows that $e(\alpha) \le 0$ for $\alpha \in [0, 1]$ and $e(\alpha) \ge 0$ for $\alpha \in \mathbb{R} \setminus [0, 1]$. Note that $\hat{e}$ is another closed proper convex function. Moreover, if the pair $(\mathbb{P}_\lambda, \hat{\mathbb{P}}_\lambda)_{\lambda \in \mathcal{L}}$ is in involution, then (2.22) further gives that $\hat{e}(\alpha) = e(\alpha)$.

Suppose that

$$
\bar{e}(\alpha) := \limsup_{\lambda \in \mathcal{L}} \frac{1}{r_{\lambda}} e_{\lambda}(\alpha) < \infty
$$

for all $\alpha \in \mathbb{R}$. Then Varadhan's lemma implies that

$$
e(\alpha) = \bar{e}(\alpha) = \lim_{\lambda \in \mathcal{L}} \frac{1}{r_{\lambda}} e_{\lambda}(\alpha) \quad (2.25)
$$

for all $\alpha \in \mathbb{R}$. In this case, all the above properties of the entropic pressure are
inherited from the corresponding properties of $e_\lambda$ described in Section 2.2. We
also note that if $(\tilde{\sigma})_{\lambda \in \mathcal{L}}$ is physically equivalent to entropy production, then

$$
e(\alpha) = \lim_{\lambda \in \mathcal{L}} \frac{1}{r_{\lambda}} \log \mathbb{E}_{\lambda} [e^{-\alpha \tilde{\sigma}_{\lambda}}]
$$

<footnote n="9">⁹ The choice of the – sign here is often a nuisance, but we shall keep it for historical reasons.</footnote>

<footnote n="10">¹⁰ A convex function $F: \mathbb{R}^d \to [-\infty, +\infty]$ is proper if it never takes the value $-\infty$ and is not everywhere $+\infty$. A proper, convex function is closed when its epigraph $\{(x, y) \in \mathbb{R}^d \times \mathbb{R} \mid y \ge F(x)\}$ is closed, which is equivalent to $F$ being lower semicontinuous.</footnote>
holds for all $\alpha \in \mathbb{R}$.

Recalling that the rate function $I: \mathbb{R} \to [0, +\infty]$ is lower semicontinuous, whenever it is convex, a well-known property of the Legendre–Fenchel transform [Roc70, Theorem 12.2] leads to the dual identity

$$I(-s) = \sup_{\alpha \in \mathbb{R}} (\alpha s - e(\alpha)) \qquad (2.26)$$

which holds for all $s \in \mathbb{R}$. Denoting by$^{11}$ $\partial^{\pm}e(\alpha)$ the right/left derivative of $e$ at $\alpha \in \mathbb{R}$ and by $\partial e(\alpha) = [\partial^{-}e(\alpha), \partial^{+}e(\alpha)] \cap \mathbb{R}$ its subdifferential, another well-known property of the Legendre–Fenchel transform [Roc70, Theorem 23.5] yields

$$s \in \partial e(\alpha) \implies I(-s) = \alpha s - e(\alpha). \qquad (2.27)$$

If $I$ is not convex on $\mathbb{R}$, then (2.26) and (2.27) hold with $I$ replaced by its convex hull defined by

$$\tilde{I}(s) := \sup_{\varphi \in \text{conv}(I)} \varphi(s), \qquad (2.28)$$

where $\text{conv}(I)$ denotes the set of all convex functions $\varphi: \mathbb{R} \to [-\infty, +\infty]$ such that $\varphi(s) \le I(s)$ for all $s \in \mathbb{R}$. Although there are some intriguing examples where the rate function $I$ is strictly concave on some interval,$^{12}$ in most cases of interest $I$ is convex.

The above discussion can be turned around. Suppose that the function

$$\tilde{e}(\alpha) = \limsup_{\lambda \in \mathcal{L}} \frac{1}{r_{\lambda}} \log \mathbb{E}_{\lambda} [e^{-\alpha \sigma_{\lambda}}]$$

is finite on a neighborhood of $[0, 1]$, then the LDP upper bounds hold (see the proofs of [DZ00, Theorems 2.3.6 and 4.5.3]). Moreover, if the limit

$$e(\alpha) = \lim_{\lambda \in \mathcal{L}} \frac{1}{r_{\lambda}} e_{\lambda}(\alpha)$$

exists and defines a differentiable function on some open interval $\tilde{\mathcal{J}}_d \supset [0, 1]$, then the Gärtner–Ellis theorem implies that the FT holds locally on the interval

$$\mathcal{J} = ]a_{-} \cup (-a_{+}), (-a_{-}) \wedge a_{+}[$$

with

$$a_{-} = \inf_{\alpha \in \tilde{\mathcal{J}}_d} e'(\alpha), \quad a_{+} = \sup_{\alpha \in \tilde{\mathcal{J}}_d} e'(\alpha),$$

and the convex rate functions given by$^{13}$

<footnote>$^{11}$ See the discussion following (2.45) below for definitions and conventions.</footnote>

<footnote>$^{12}$ As we shall see in Section 3.2, the mean field lattice gas provides such an example.</footnote>

<footnote>$^{13}$ The function $I$ defined in (2.29) coincides with the right-hand side of (2.26) at least in the interval $[-a_{+}, a_{-}] \cap \mathbb{R}$.</footnote>
$$I(-s) = \sup_{\alpha \in \tilde{\mathcal{J}}_d} (\alpha s - e(\alpha)), \quad \hat{I}(s) = s + I(s). \qquad (2.29)$$

Moreover, if $e$ is steep, i.e., $a_{\pm} = \pm\infty$, then $I$ and $\hat{I}$ are good rates and the global FT holds. This construction provides a practical route to the proof of the Fluctuation Theorem which has been followed by most mathematically rigorous studies since the seminal work [LS99]. The same approach was adopted in [JPR11].

## 2.6 Higher-level Fluctuation Theorems

The main deficiency of the Gärtner-Ellis route to the FT is its failure in the presence of phase transitions, i.e., non-differentiability of the entropic pressure.<fnref n="14" /> The problem is particularly acute in cases where singularities of $e$ occur in the interval $[0, 1]$, since then even the local FT is out of reach. In Chapter 3, we shall give some examples where such a situation occurs and for which it is therefore necessary to develop other routes to the FT.

There are several alternatives to the Gärtner-Ellis theorem in the literature:

* The Donsker-Varadhan theory [DV75a; DV75b] for additive functionals of Markov processes is of particular interest due to the wide usage of the Markov approximation in non-equilibrium statistical mechanics.

* The approach developed by Ruelle and Lanford in [Rue65; Lan73] is well adapted to studying shift spaces over finite alphabets under very general decoupling conditions for which the thermodynamic formalism does not apply. Such decoupling conditions arise naturally in multifractal analysis, in Gibbs states with hard-core interactions, and in the statistics of repeated quantum measurement processes [Cun+19].

* Kifer's criterion [Kif90] can be particularly useful for infinite dimensional systems, see, e.g. [Jak+15; Jak+18].

These alternative methods also provide a way to lift the FT and the FR from level-1, as formulated above, to level-2 or level-3. In the abstract setting of Sections 2.2 and 2.3, such a lifting can be formulated as follows. Suppose that for each $\lambda \in \mathcal{L}$ there exists a random variable $\xi_{\lambda}$ on $(\Omega_{\lambda}, \mathfrak{F}_{\lambda})$, taking values in a Polish space $\mathcal{X}$ and such that the family $(\xi_{\lambda})_{\lambda \in \mathcal{L}}$ satisfies the LDP w.r.t. both $(\mathbb{P}_{\lambda})_{\lambda \in \mathcal{L}}$ and $(\hat{\mathbb{P}}_{\lambda})_{\lambda \in \mathcal{L}}$ with scale $(r_{\lambda})_{\lambda \in \mathcal{L}}$ and respective rates $\|\|$ and $\hat{\|}$. Suppose also that there is a measurable map $\zeta: \mathcal{X} \to \mathbb{R}$ such that $r_{\lambda}\zeta(\xi_{\lambda})$ is physically equivalent to $\sigma_{\lambda}$. In such a case, we shall say that this family satisfies the FT whenever the lifted FR

$$\hat{\|}(\xi) = \|\|(\xi) + \zeta(\xi) \qquad (2.30)$$

holds for $\xi \in \mathcal{X}$ (see [MNW08; BL08; BC15]). Setting

<footnote n="14">¹⁴ We shall see in our examples that non-differentiability of the entropic pressure is indeed the signature of phase transitions in the usual thermodynamical sense.</footnote>
$$I(s) := \inf\{\|\xi\| \mid \xi \in \mathcal{X}, \zeta(\xi) = s\}, \quad \hat{I}(s) := \inf\{\|\hat{\xi}\| \mid \hat{\xi} \in \mathcal{X}, \zeta(\hat{\xi}) = s\},$$

and observing that (2.30) implies that $(I, \hat{I})$ satisfies the FR (2.21), it follows from the contraction principle that the FT holds for $(\mathbb{P}_\lambda, \hat{\mathbb{P}}_\lambda)_{\lambda \in \mathcal{L}}$ with scale $(r_\lambda)_{\lambda \in \mathcal{L}}$ and rates $(I, \hat{I})$ provided that the rates $\|\cdot\|$ and $\|\cdot\|$ are good and the function $\zeta$ is continuous.

## 2.7 On the Meaning of the Fluctuation Theorem

Steady state FRs reduce to the Einstein–Green–Kubo fluctuation–dissipation formula and Onsager’s reciprocity relations in the limit of weak deviations from thermal equilibrium, as explained in [Gal96; LS99; JPR11]. In this section, we shall derive consequences of the FR which extend beyond the linear response regime. We refer to [MK10; Sei12] for discussions of other consequences of the steady state FRs.

Returning to the level of generality of Sections 2.2 and 2.3, the rate function $I$ and the associated entropic pressure $e$ can be linked to the hypothesis testing error exponents of the family $(\mathbb{P}_\lambda, \hat{\mathbb{P}}_\lambda)_{\lambda \in \mathcal{L}}$. These exponents quantify the rate of separation between measures $\mathbb{P}_\lambda$ and $\hat{\mathbb{P}}_\lambda$ as $\lambda$ moves along the index set $\mathcal{L}$. If the elements of $\mathcal{L}$ are instances of time and $\hat{\mathbb{P}}_\lambda = \mathbb{P}_\lambda \circ \Theta_\lambda$ where $\Theta_\lambda$ implements time reversal, these exponents quantify the emergence of the arrow of time and can be viewed as quantitative forms of the second law of thermodynamics.

**Binary Hypothesis Testing.** Given $\omega \in \Omega_\lambda$, consider the null hypothesis

$$H_0: \omega \text{ was sampled from } \mathbb{P}_\lambda,$$

and the alternative hypothesis

$$H_A: \omega \text{ was sampled from } \hat{\mathbb{P}}_\lambda.$$

To any $\Gamma \in \mathcal{F}_\lambda$, we associate the test according to which

$$\text{if } \omega \in \Gamma, \text{ then we reject } H_A, \text{ otherwise we reject } H_0.$$

The probability of the type-I error—wrongly rejecting the null hypothesis—is $\mathbb{P}_\lambda(\Gamma^c)$, while the type-II error—wrongly rejecting the alternative hypothesis—has probability $\hat{\mathbb{P}}_\lambda(\Gamma)$. Thus, assuming for simplicity that $\mathbb{P}_\lambda$ and $\hat{\mathbb{P}}_\lambda$ have equal a priori probabilities,

$$D(\mathbb{P}_\lambda, \hat{\mathbb{P}}_\lambda, \Gamma) := \mathbb{P}_\lambda(\Gamma^c) + \hat{\mathbb{P}}_\lambda(\Gamma),$$

is twice the total error probability of the test $\Gamma$, and a test $\Gamma_{\text{opt}}$ is optimal whenever

$$D(\mathbb{P}_\lambda, \hat{\mathbb{P}}_\lambda, \Gamma_{\text{opt}}) = D(\mathbb{P}_\lambda, \hat{\mathbb{P}}_\lambda) := \inf_{\Gamma \in \mathcal{F}_\lambda} D(\mathbb{P}_\lambda, \hat{\mathbb{P}}_\lambda, \Gamma).$$
The following variant of the *Neyman-Pearson lemma* identifies an optimal test:

**Proposition 2.5** Assume that $\mathbb{P}_\lambda$ and $\hat{\mathbb{P}}_\lambda$ are equivalent. Then

$$D(\mathbb{P}_\lambda, \hat{\mathbb{P}}_\lambda) = D(\mathbb{P}_\lambda, \hat{\mathbb{P}}_\lambda, \{\sigma_\lambda \ge 0\}), \qquad (2.31)$$

where $\sigma_\lambda$ is given by (2.5).

**Proof.** For later reference, we prove the slightly more general statement

$$\inf_{\Gamma \in \mathcal{F}_\lambda} (\mathbb{P}_\lambda(\Gamma^c) + e^{sr_\lambda} \hat{\mathbb{P}}_\lambda(\Gamma)) = \mathbb{P}_\lambda\{\sigma_\lambda < sr_\lambda\} + e^{sr_\lambda} \hat{\mathbb{P}}_\lambda\{\sigma_\lambda \ge sr_\lambda\}. \qquad (2.32)$$

Indeed, for any $\Gamma \in \mathcal{F}_\lambda$ and $s \in \mathbb{R}$, one has<fnref n="15" />

$$\begin{aligned} \mathbb{P}_\lambda(\Gamma^c) + e^{sr_\lambda} \hat{\mathbb{P}}_\lambda(\Gamma) &= \mathbb{E}_\lambda [1_{\Gamma^c} + e^{sr_\lambda - \sigma_\lambda} 1_\Gamma] \\ &\ge \mathbb{E}_\lambda [\min(1, e^{sr_\lambda - \sigma_\lambda})] \\ &= \mathbb{E}_\lambda [1_{\{\sigma_\lambda < sr_\lambda\}} + e^{sr_\lambda - \sigma_\lambda} 1_{\{\sigma_\lambda \ge sr_\lambda\}}] \\ &= \mathbb{P}_\lambda\{\sigma_\lambda < sr_\lambda\} + e^{sr_\lambda} \hat{\mathbb{P}}_\lambda\{\sigma_\lambda \ge sr_\lambda\}, \end{aligned}$$

which implies the claimed identity. □

Error exponents quantify the exponential decay rate of type-II errors as $\lambda$ increases through $\mathcal{L}$, under various conditions on the probability of type-I errors.

**Stein error exponents.** The lower and upper Stein exponents are defined by

$$\underline{s} := \inf\{\liminf_{\lambda \in \mathcal{L}} r_\lambda^{-1} \log \hat{\mathbb{P}}_\lambda(\Gamma_\lambda) \mid \Gamma_\lambda \in \mathcal{F}_\lambda, \lim_{\lambda \in \mathcal{L}} \mathbb{P}_\lambda(\Gamma_\lambda^c) = 0\},$$

$$\bar{s} := \inf\{\limsup_{\lambda \in \mathcal{L}} r_\lambda^{-1} \log \hat{\mathbb{P}}_\lambda(\Gamma_\lambda) \mid \Gamma_\lambda \in \mathcal{F}_\lambda, \lim_{\lambda \in \mathcal{L}} \mathbb{P}_\lambda(\Gamma_\lambda^c) = 0\}.$$

Given $\gamma \in ]0, 1[$, we also set

$$\mathfrak{p}_\gamma(\mathbb{P}_\lambda, \hat{\mathbb{P}}_\lambda) := \inf\{\hat{\mathbb{P}}_\lambda(\Gamma) \mid \Gamma \in \mathcal{F}_\lambda, \mathbb{P}_\lambda(\Gamma^c) \le \gamma\}.$$

The following result establishes a link between the Stein exponents, the asymptotics of $\mathfrak{p}_\gamma(\mathbb{P}_\lambda, \hat{\mathbb{P}}_\lambda)$, and the weak law of large numbers.

**Proposition 2.6**

(1) Suppose that the net $(r_\lambda^{-1}\sigma_\lambda)_{\lambda \in \mathcal{L}}$ converges in probability to a deterministic limit epr $\epsilon \in [-\infty, +\infty]$.<fnref n="16" /> Then, epr $\ge 0$ and, for any $\gamma \in ]0, 1[$, one has

$$\underline{s} = \bar{s} = \lim_{\lambda \in \mathcal{L}} r_{\lambda}^{-1} \log \mathfrak{p}_{\gamma}(\mathbb{P}_{\lambda}, \hat{\mathbb{P}}_{\lambda}) = -\text{epr}. \qquad (2.33)$$

<footnote n="15">¹⁵ $1_A$ denotes the indicator function of a set A.</footnote>

<footnote n="16">¹⁶ This holds if, for instance, the FT holds with a rate $I$ vanishing at a unique point epr.</footnote>
(2) If the limit (2.25) holds for $\alpha$ in a neighborhood of 0, and the entropic pressure $e$ is differentiable at $\alpha = 0$, then

$$ \lim_{\lambda \in \mathcal{L}} r_{\lambda}^{-1} \mathrm{Ent}(\mathbb{P}_{\lambda} | \hat{\mathbb{P}}_{\lambda}) = -e'(0). \qquad (2.34) $$

If, in addition, the net $(r_{\lambda}^{-1}\sigma_{\lambda})_{\lambda \in \mathcal{L}}$ converges in the mean to a deterministic limit, then the conclusions of Part (1) hold with $epr = -e'(0)$.

**Proof. (1)** For any $\delta > 0$, Equ. (2.12) gives that

$$ \mathbb{P}_{\lambda} \{r_{\lambda}^{-1} \sigma_{\lambda} \leq -\delta\} \leq e^{-\delta r_{\lambda}}, $$

so that $epr \in [0, +\infty)$. To prove (2.33), let us fix $\gamma \in ]0, 1[$. We first show that

$$ \underline{s}_{\gamma} := \liminf_{\lambda \in \mathcal{L}} r_{\lambda}^{-1} \log p_{\gamma}(\mathbb{P}_{\lambda}, \hat{\mathbb{P}}_{\lambda}) \geq -epr, \qquad (2.35) $$

assuming, w.l.o.g., that $epr < \infty$. The assumed convergence in probability gives that for any $s > epr$ there is $\lambda_s \in \mathcal{L}$ such that $\mathbb{P}_{\lambda}\{\sigma_{\lambda} < sr_{\lambda}\} \geq (1+\gamma)/2$ for all $\lambda > \lambda_s$. For any such $\lambda$ and any $\Gamma \in \mathcal{F}_{\lambda}$, by (2.32),

$$ \mathbb{P}_{\lambda}(\Gamma^{c}) + e^{sr_{\lambda}} \hat{\mathbb{P}}_{\lambda}(\Gamma) \geq \mathbb{P}_{\lambda}\{\sigma_{\lambda} < sr_{\lambda}\} + e^{sr_{\lambda}} \hat{\mathbb{P}}_{\lambda}\{\sigma_{\lambda} \geq sr_{\lambda}\} \geq \frac{1+\gamma}{2}, $$

and it follows that

$$ \hat{\mathbb{P}}_{\lambda}(\Gamma) \geq \frac{1-\gamma}{2} e^{-sr_{\lambda}} $$

whenever $\lambda > \lambda_s$ and $\mathbb{P}_{\lambda}(\Gamma^c) \leq \gamma$. Thus, $r_{\lambda}^{-1} \log p_{\gamma}(\mathbb{P}_{\lambda}, \hat{\mathbb{P}}_{\lambda}) \geq -s$ holds for large $\lambda$, and we conclude that $\underline{s}_{\gamma} \geq -s$ for any $s > epr$, which yields (2.35). Next, we prove the upper bound

$$ \bar{s} \leq -epr. \qquad (2.36) $$

It follows from our assumption that, for any $s < epr$, the sets $\Gamma_{\lambda} = \{\sigma_{\lambda} \geq sr_{\lambda}\} \in \mathcal{F}_{\lambda}$ satisfy $\lim_{\lambda \in \mathcal{L}} \mathbb{P}_{\lambda}(\Gamma_{\lambda}^{c}) = \lim_{\lambda \in \mathcal{L}} \mathbb{P}_{\lambda}\{r_{\lambda}^{-1}\sigma_{\lambda} < s\} = 0$. From the estimate

$$ \hat{\mathbb{P}}_{\lambda}(\Gamma_{\lambda}) = \mathbb{E}_{\lambda} [e^{-\sigma_{\lambda}} 1_{\sigma_{\lambda} \geq sr_{\lambda}}] \leq e^{-sr_{\lambda}}, $$

we deduce that $r_{\lambda}^{-1} \log \hat{\mathbb{P}}_{\lambda}(\Gamma_{\lambda}) \leq -s$, and we conclude that $\bar{s} \leq -s$ for any $s < epr$, which yields (2.36). The obvious inequalities $\underline{s}_{\gamma} \leq \bar{s}$, and

$$ \bar{s}_{\gamma} := \limsup_{\lambda \in \mathcal{L}} r_{\lambda}^{-1} \log p_{\gamma}(\mathbb{P}_{\lambda}, \hat{\mathbb{P}}_{\lambda}) \leq \bar{s}, $$

together with (2.35)-(2.36) give (2.33) and the claimed equalities $\underline{s} = \bar{s} = -epr$.
(2) We repeat the argument in the proof of [Ell85, Theorem II.6.3]. One has

$$ \mathrm{Ent}(\mathbb{P}_{\lambda} | \hat{\mathbb{P}}_{\lambda}) = \mathbb{E}_{\lambda}[\sigma_{\lambda}] = -e'_{\lambda}(0). $$

Since the functions $\alpha \mapsto e_{\lambda}(\alpha) = \log \mathbb{E}_{\lambda}[e^{-\alpha \sigma_{\lambda}}]$ are convex, the differentiability of the limiting function

$$ \alpha \mapsto e(\alpha) = \lim_{\lambda \in \mathcal{L}} r_{\lambda}^{-1} e_{\lambda}(\alpha) $$

at $\alpha = 0$ gives, by [Roc70, Theorem 25.7],

$$ e'(0) = \lim_{\lambda \in \mathcal{L}} r_{\lambda}^{-1} e'_{\lambda}(0), $$

from which (2.34) follows. Finally, it follows from the convergence in the mean

$$ \lim_{\lambda \in \mathcal{L}} \mathbb{E}_{\lambda} [ |r_{\lambda}^{-1} \sigma_{\lambda} - \mathrm{epr}| ] = 0, $$

that

$$ \lim_{\lambda \in \mathcal{L}} r_{\lambda}^{-1} \sigma_{\lambda} = \mathrm{epr} = -e'(0) $$

holds in probability, so that Part (1) applies. $\square$

Recall that the *total variation distance* of two probability measures $\mathbb{P}$ and $\mathbb{Q}$ on a measurable space $(\Omega, \mathcal{F})$ is

$$ \| \mathbb{P} - \mathbb{Q} \|_{\mathrm{var}} := \sup_{\Gamma \in \mathcal{F}} (\mathbb{P}(\Gamma) - \mathbb{Q}(\Gamma)). $$

If $\mathbb{Q}$ is absolutely continuous with respect to $\mathbb{P}$, and $\Delta$ stands for the corresponding density, then

$$ \mathbb{P}(\Gamma) - \mathbb{Q}(\Gamma) = 1 - (\mathbb{P}(\Gamma^c) + \mathbb{Q}(\Gamma)) = 1 - \int_{\Omega} (1_{\Gamma^c} + \Delta 1_{\Gamma}) d\mathbb{P} \le 1 - \int_{\Omega} \min(1, \Delta) d\mathbb{P}. $$

Observing that the last inequality is saturated for $\Gamma = \{\Delta \le 1\}$, we get

$$ \| \mathbb{P} - \mathbb{Q} \|_{\mathrm{var}} = 1 - \int_{\Omega} \min(1, \Delta) d\mathbb{P}. $$

Thus, assuming that $\mathbb{P}_{\lambda}$ and $\hat{\mathbb{P}}_{\lambda}$ are equivalent, it follows from Lemma 2.5 that

$$ 1 - \| \mathbb{P}_{\lambda} - \hat{\mathbb{P}}_{\lambda} \|_{\mathrm{var}} = \mathbb{E}_{\lambda} [\min(1, e^{-\sigma_{\lambda}})] = D(\mathbb{P}_{\lambda}, \hat{\mathbb{P}}_{\lambda}). \quad (2.37) $$

**Chernoff error exponents.** The lower and upper Chernoff exponents are

$$ c := \liminf_{\lambda \in \mathcal{L}} r_{\lambda}^{-1} \log(1 - \|\mathbb{P}_{\lambda} - \hat{\mathbb{P}}_{\lambda}\|_{\mathrm{var}}), $$

$$ \bar{c} := \limsup_{\lambda \in \mathcal{L}} r_{\lambda}^{-1} \log(1 - \|\mathbb{P}_{\lambda} - \hat{\mathbb{P}}_{\lambda}\|_{\mathrm{var}}). $$
The following result provides bounds on these exponents in terms of the rate $I$ and the entropic pressure $e$.

**Proposition 2.7**

(1) Suppose that $\mathbb{P}_\lambda$ and $\hat{\mathbb{P}}_\lambda$ are equivalent measures for all $\lambda \in \mathcal{L}$, and that the FT holds with the scale $(r_\lambda)_{\lambda \in \mathcal{L}}$ and rates $(I, \hat{I})$. Then<fnref n="17" />

$$-I(0) \le \underline{c} \le \bar{c} \le \inf_{\alpha \in [0,1]} \bar{e}(\alpha). \qquad (2.38)$$

(2) If, in addition, the rate $I$ is convex and (2.25) holds for $\alpha \in [0, 1]$, then the upper and lower Chernoff exponents coincide, and<fnref n="18" />

$$\underline{c} = \bar{c} = \lim_{\lambda \in \mathcal{L}} r_{\lambda}^{-1} \log(1 - \|\mathbb{P}_{\lambda} - \hat{\mathbb{P}}_{\lambda}\|_{\text{var}}) = -I(0) = \inf_{\alpha \in [0,1]} \bar{e}(\alpha). \qquad (2.39)$$

**Proof.** (1) It follows from (2.37) that, for $\alpha \in [0, 1]$,

$$1 - \|\mathbb{P}_{\lambda} - \hat{\mathbb{P}}_{\lambda}\|_{\text{var}} = \mathbb{E}_{\lambda}[\min(1, e^{-\sigma_{\lambda}})] \le \mathbb{E}_{\lambda}[e^{-\alpha\sigma_{\lambda}}] = e^{\lambda(\alpha)}.$$

Thus, taking the limit over $\lambda \in \mathcal{L}$,

$$\bar{c} \le \limsup_{\lambda \in \mathcal{L}} r_{\lambda}^{-1} e_{\lambda}(\alpha) = \bar{e}(\alpha),$$

which yields $\bar{c} \le \inf_{\alpha \in [0,1]} \bar{e}(\alpha)$. Since $\underline{c} \le \bar{c}$, it remains to prove that $\underline{c} \ge -I(0)$ in order to establish (2.38). Observe that for all $\varepsilon > 0$,

$$1 - \|\mathbb{P}_{\lambda} - \hat{\mathbb{P}}_{\lambda}\|_{\text{var}} = D(\mathbb{P}_{\lambda}, \hat{\mathbb{P}}_{\lambda}) = \mathbb{P}_{\lambda}\{\sigma_{\lambda} < 0\} + \hat{\mathbb{P}}_{\lambda}\{\sigma_{\lambda} \ge 0\} \ge e^{-r_{\lambda}\varepsilon}\mathbb{P}_{\lambda}\{\sigma_{\lambda} \le r_{\lambda}\varepsilon\},$$

where we used again Lemma 2.5 and the definition of $\sigma_\lambda$. Hence, invoking the LDP lower bound, we derive $\underline{c} \ge -\inf_{s<\varepsilon} I(s) - \varepsilon \ge -I(0) - \varepsilon$, and since $\varepsilon > 0$ was arbitrary, we obtain $\underline{c} \ge -I(0)$.

(2) Suppose now that (2.25) holds and that $I$ is convex. In view of (2.26), one has $-I(0) = \inf_{\alpha \in \mathbb{R}} e(\alpha)$. Since $e(\alpha)$ is non-positive in $[0, 1]$ and non-negative outside $[0, 1]$, we obtain $-I(0) = \inf_{\alpha \in [0,1]} e(\alpha)$, which, together with (2.38), implies (2.39). $\square$

Thus, if $\bar{c} > 0$, then the measures $\mathbb{P}_\lambda$ and $\hat{\mathbb{P}}_\lambda$ concentrate on the disjoint subsets $\{\sigma_\lambda > 0\}$ and $\{\sigma_\lambda < 0\}$, respectively, and separate with an exponential rate given by, at least, $r_\lambda \bar{c}$, in the sense that:

$$\limsup_{\lambda \in \mathcal{L}} r_{\lambda}^{-1} \log \mathbb{P}_{\lambda}\{\sigma_{\lambda} \le 0\} \le \bar{c}, \quad \limsup_{\lambda \in \mathcal{L}} r_{\lambda}^{-1} \log \hat{\mathbb{P}}_{\lambda}\{\sigma_{\lambda} \ge 0\} \le \bar{c}. \qquad (2.40)$$

<footnote n="17">¹⁷ Under the stated assumptions, a slight refinement of the proof below, using also the LDP upper bound, shows that actually $\underline{c} = \bar{c} = -\min(-\inf_{s \le 0} I(s), -\inf_{s > 0} \hat{I}(s))$.</footnote>

<footnote n="18">¹⁸ Note that if the family $(\mathbb{P}_\lambda, \hat{\mathbb{P}}_\lambda)_{\lambda \in \mathcal{L}}$ is in involution, then $-I(0) = e(1/2)$.</footnote>
The two bounds in (2.40) follow from the definition of $\bar{c}$ and the fact that both $\mathbb{P}_{\lambda}\{\sigma_{\lambda} \le 0\}$ and $\hat{\mathbb{P}}_{\lambda}\{\sigma_{\lambda} \ge 0\}$ are bounded above by the quantity $D(\mathbb{P}_{\lambda}, \hat{\mathbb{P}}_{\lambda}, \{\sigma_{\lambda} \ge 0\})$ in (2.31).

The last result in this section provides more detailed information on the rates at which the measures $\mathbb{P}_{\lambda}$ and $\hat{\mathbb{P}}_{\lambda}$ concentrate on complementary subsets. This information is carried by a third kind of exponents:

**Hoeffding error exponents.** Given $u \in \mathbb{R}$, define

$$ \underline{h}(u) := \inf\left\{ \liminf_{\lambda \in \mathcal{L}} \inf r_{\lambda}^{-1} \log \hat{\mathbb{P}}_{\lambda}(\Gamma_{\lambda}) \middle| \Gamma_{\lambda} \in \mathfrak{F}_{\lambda}, \limsup_{\lambda \in \mathcal{L}} r_{\lambda}^{-1} \log \mathbb{P}_{\lambda}(\Gamma_{\lambda}^{c}) < -u \right\}, $$

$$ \bar{h}(u) := \inf\left\{ \limsup_{\lambda \in \mathcal{L}} r_{\lambda}^{-1} \log \hat{\mathbb{P}}_{\lambda}(\Gamma_{\lambda}) \middle| \Gamma_{\lambda} \in \mathfrak{F}_{\lambda}, \limsup_{\lambda \in \mathcal{L}} r_{\lambda}^{-1} \log \mathbb{P}_{\lambda}(\Gamma_{\lambda}^{c}) < -u \right\}, $$

$$ \underline{h}(u) := \inf\left\{ \lim_{\lambda \in \mathcal{L}} r_{\lambda}^{-1} \log \hat{\mathbb{P}}_{\lambda}(\Gamma_{\lambda}) \middle| \Gamma_{\lambda} \in \mathfrak{F}_{\lambda}, \limsup_{\lambda \in \mathcal{L}} r_{\lambda}^{-1} \log \mathbb{P}_{\lambda}(\Gamma_{\lambda}^{c}) < -u \right\}, $$

where the infimum in the last relation is taken over all families $(\Gamma_{\lambda})_{\lambda \in \mathcal{L}}$ for which the limit exists.

It immediately follows from these definitions that $\underline{h}$, $\bar{h}$, and $h$ are non-decreasing and non-positive functions of $u$ such that

$$ \underline{h}(u) = \bar{h}(u) = h(u) = -\infty \qquad (2.41) $$

for $u < 0$. Moreover, the inequalities

$$ \underline{h}(u) \le \bar{h}(u) \le h(u) \le 0 \qquad (2.42) $$

hold for all $u \ge 0$.

**Proposition 2.8** Suppose that, for all $\lambda \in \mathcal{L}$, $\mathbb{P}_{\lambda}$ and $\hat{\mathbb{P}}_{\lambda}$ are equivalent measures such that the FT holds with the scale $(r_{\lambda})_{\lambda \in \mathcal{L}}$ and convex rates $(I, \hat{I})$. If the associated entropic pressure satisfies

$$ e(\alpha) = \lim_{\lambda \in \mathcal{L}} e_{\lambda}(\alpha) $$

for all $\alpha \in [0, 1]$, then the Hoeffding error exponents coincide and are given by

$$ \underline{h}(u) = \bar{h}(u) = h(u) = -f(u) \qquad (2.43) $$

for all $u \in \mathbb{R}$, where the function $f$ is related to the entropic pressure by

$$ f(u) := \sup_{\alpha \in [0,1]} \frac{-(1-\alpha)u - e(\alpha)}{\alpha}. \qquad (2.44) $$

The proof being quite technical, we postpone it to the final appendix to this chapter.
**Appendix: Proof of Proposition 2.8**

We start with some basic properties of the rate function $I$, its convex hull $\tilde{I}$ (2.28), the corresponding entropic pressure $e$ (2.24), and the function $f$ defined in (2.44). Although convexity is assumed in Proposition 2.8, we do not assume that $I$ is convex in the upcoming technical discussion.

We recall that, as the Fenchel-Legendre transform of the function $s \mapsto I(-s)$, the entropic pressure $e$ is a closed proper convex function on $\mathbb{R}$. It is continuous and admits (possibly infinite) left/right derivatives $\partial^{\mp}e(\alpha)$ at each point $\alpha$ of its essential domain

$$ \mathrm{Dom}(e) := \{\alpha \in \mathbb{R} \mid e(\alpha) \text{ is finite}\} \supset [0, 1], \quad (2.45) $$

see [Roc70, Sections 23-24]. By convention, when $\alpha \notin \mathrm{Dom}(e)$, we set $\partial^{-}e(\alpha) = \partial^{+}e(\alpha) = -\infty$ if $\alpha \le \inf \mathrm{Dom}(e)$ and $\partial^{-}e(\alpha) = \partial^{+}e(\alpha) = \infty$ if $\alpha \ge \sup \mathrm{Dom}(e)$. These derivatives satisfy $\partial^{-}e(\alpha) \le \partial^{+}e(\alpha)$ with equality outside a countable subset of $\mathrm{Dom}(e)$, $\partial^{+}e(\alpha) \le \partial^{-}e(\alpha')$ whenever $\alpha \le \alpha'$, and for all $\alpha \in \mathbb{R}$,

$$ \lim_{\gamma \downarrow \alpha} \partial^{\pm} e(\gamma) = \partial^{+} e(\alpha), \quad \lim_{\gamma \uparrow \alpha} \partial^{\pm} e(\gamma) = \partial^{-} e(\alpha). \quad (2.46) $$

The subdifferential of $e$ at $\alpha \in \mathrm{Dom}(e)$ is defined as the closed interval

$$ \partial e(\alpha) := [\partial^{-} e(\alpha), \partial^{+} e(\alpha)] \cap \mathbb{R}. $$

**Proposition 2.9** Assume that the FT holds for the family $(\mathbb{P}_{\lambda}, \hat{\mathbb{P}}_{\lambda})_{\lambda \in \mathcal{L}}$ with scale $(r_{\lambda})_{\lambda \in \mathcal{L}}$ and rate $(I, \hat{I})$. Let

$$ \begin{gathered} e(\alpha) = \sup_{s \in \mathbb{R}} (\alpha s - I(-s)) \\ \tilde{I}(s) = \sup_{\alpha \in \mathbb{R}} (\alpha s - e(-\alpha)) \quad (2.47) \end{gathered} $$

denote the associated entropic pressure, and

$$ s_* := - \inf_{\alpha \in \mathbb{R}} e(\alpha), \quad \partial e(0) := [-\bar{s}_0, -\underline{s}_0] \cap \mathbb{R}, \quad \partial e(1) := [\underline{s}_1, \bar{s}_1] \cap \mathbb{R}, $$

where, following the above conventions, $\bar{s}_0 = +\infty$ whenever $e(\alpha) = +\infty$ for all $\alpha < 0$ and $\underline{s}_1 = +\infty$ whenever $e(\alpha) = +\infty$ for all $\alpha > 1$.

Then, the following hold:

(1) For any $\alpha \in \mathbb{R}$,

$$ \hat{e}(\alpha) := \sup_{s \in \mathbb{R}} (\alpha s - \hat{I}(s)) = e(1 - \alpha), $$

and in particular
$$s_* = -\inf_{\alpha \in \mathbb{R}} \hat{e}(\alpha), \quad \partial \hat{e}(0) = [-\bar{s}_1, -\underline{s}_1] \cap \mathbb{R}, \quad \partial \hat{e}(1) = [\underline{s}_0, \bar{s}_0] \cap \mathbb{R}.$$

Moreover, the convex hull of $\hat{I}$ is given by

$$J(s) = \sup_{\alpha \in \mathbb{R}} (\alpha s - \hat{e}(\alpha)) = \check{I}(s) + s.$$

(2) Denote by $\text{co}(A)$ the convex hull of the set $A \subset \mathbb{R}$, i.e., the smallest convex subset of $\mathbb{R}$ containing A. If $\underline{s}_0 < \infty$, then

$$\text{co}(I^{-1}(\{0\})) \subset \check{I}^{-1}(\{0\}) = [\underline{s}_0, \bar{s}_0] \cap \mathbb{R}. \qquad (2.48)$$

If $\underline{s}_1 < \infty$, then

$$\text{co}(\hat{I}^{-1}(\{0\})) \subset \check{I}^{-1}(\{0\}) = [-\bar{s}_1, -\underline{s}_1] \cap \mathbb{R}. \qquad (2.49)$$

Moreover, the inclusion in (2.48) and (2.49) becomes an equality whenever its right-hand side is compact.

(3) For all $s \in [-\bar{s}_0, \bar{s}_1] \cap \mathbb{R}$,

$$\check{I}(-s) = \max_{\alpha \in [0,1]} (\alpha s - e(\alpha)). \qquad (2.50)$$

(4) One has

$$0 \le \check{I}(0) = s_* \le \min(\underline{s}_0, \underline{s}_1), \qquad (2.51)$$

where the two inequalities become equalities iff e vanishes identically on $[0, 1]$.

(5) $s_* = 0 \iff \underline{s}_0 = \underline{s}_1 = 0$.

(6) $f$ is a non-increasing closed proper convex function satisfying

$$f(u) = \begin{cases} +\infty & \text{for } u < 0; \\ \underline{s}_0 & \text{for } u = 0; \\ s_* & \text{for } u = s_*; \\ 0 & \text{for } u \ge \underline{s}_1. \end{cases}$$

If $\underline{s}_0 = \infty$ (resp. $\underline{s}_1 = \infty$), then the second (resp. last) equality in this list must be replaced by the relation

$$f(0) = \lim_{u \downarrow 0} f(u) = +\infty, \quad (\text{resp. } \lim_{u \uparrow \infty} f(u) = 0).$$

(7) The function $s \mapsto \check{I}(s)$ is strictly decreasing on $]-\underline{s}_1, \underline{s}_0[$ and maps this interval homeomorphically onto $]0, \underline{s}_1[$. The reciprocal map is given by the restriction to $]0, \underline{s}_1[$ of the function $g$ defined by $g(u) = f(u) - u$. In particular,

$$f \circ \check{I}(s) = J(s)$$

holds for all $s \in ]-\underline{s}_1, \underline{s}_0[$. Moreover, if $\underline{s}_0 < \infty$ (resp. $\underline{s}_1 < \infty$), then $g(0) = \underline{s}_0$ and $\check{I}(g(0)) = 0$ (resp. $g(\underline{s}_1) = -\underline{s}_1$, and $\check{I}(g(\underline{s}_1)) = \underline{s}_1$).
(8) If $(\mathbb{P}_{\lambda}, \hat{\mathbb{P}}_{\lambda})_{\lambda \in \mathcal{L}}$ is in involution, then $\hat{e} = e, \underline{s}_0 = \underline{s}_1, \bar{s}_0 = \bar{s}_1, s_* = -e(1/2)$, and the function $f: [0, \underline{s}_0] \to [0, \underline{s}_0]$ is an involution such that $f \circ \check{I}(s) = \check{I}(-s)$.

**Proof. (1)** Follows directly from the asymptotic FR (2.21).

(2) Since $e$ is convex and vanishes for $\alpha \in \{0, 1\}$, we must have $\underline{s}_0 \ge 0$. Moreover, $s < \underline{s}_0$, i.e., $-s > \partial^+ e(0)$ implies that there exists $\alpha \in ]0, 1[$ such that $-\alpha s > e(\alpha)$. It follows from Relation (2.47) that $\check{I}(s) > 0$ and hence $I(s) > 0$. If $\bar{s}_0 < \infty$, then a completely similar argument applies to $s > \bar{s}_0$. For $-s \in ]-\bar{s}_0, -\underline{s}_0[$, $-\underline{s}_0 \subset \partial e(0)$, Relation (2.27) yields that $\check{I}(s) = 0$, so the second equality in (2.48) holds by continuity.

Since $I(s) \ge \check{I}(s) \ge 0$, one has $I^{-1}(\{0\}) \subset \check{I}^{-1}(\{0\})$, and the convexity of $\check{I}$ further implies that $\check{I}^{-1}(\{0\}) = \check{I}^{-1}(|-\infty, 0])$ is convex, hence $\text{co}(I^{-1}(\{0\})) \subset \check{I}^{-1}(\{0\})$. In cases where $\check{I}^{-1}(\{0\})$ is compact, one has $\check{I}^{-1}(\{0\}) = [a, b]$, with $\check{I}(a) = \check{I}(b) = 0$ and $\check{I}(s) > 0$ for $s \notin [a, b]$. To prove the claimed equality $\text{co}(I^{-1}(\{0\})) = \check{I}^{-1}(\{0\})$, it is enough to show that $I(a) = 0 = I(b)$. We consider only the case of $I(a)$, by symmetry the same reasoning applies to $I(b)$. We argue by contradiction, assuming that $I(a) > 0$. By lower semicontinuity, for some $A > 0$ and $\delta > 0$, $|s - a| < \delta$ implies $I(s) > A$. It follows that there exists a convex function $\bar{I} \le I$ such that $\bar{I}(a) > 0$, which contradicts the definition of $\check{I}$. The figures 2.1 below show how to modify $\check{I}$ to get $\bar{I}$ in the five possible cases:

(1) $a < b, 0 < \check{I}(s) < \infty$ for $a - \delta < s < a$ and $\check{I}(s) = 0$ for $a \le s < a + \delta$;

(2) $a < b, \check{I}(s) = \infty$ for $s < a$ and $\check{I}(s) = 0$ for $a \le s < a + \delta$;

(3) $a = b, 0 < \check{I}(s) < \infty$ for $0 < |s - a| < \delta$;

(4) $a = b, 0 < \check{I}(s) < \infty$ for $a - \delta < s < a$ and $\check{I}(s) = \infty$ for $s > a$;

(5) $a = b, \check{I}(s) = \infty$ for $s \ne a$.

Relations (2.49) follow from the same arguments.

(3) Let $s \in [-\bar{s}_0, \bar{s}_1] \cap \mathbb{R} = [\partial^- e(0), \partial^+ e(1)] \cap \mathbb{R}$. Then, there exists some $\alpha_s \in [0, 1]$ such that $-s \in \partial e(\alpha_s)$, so (2.27), applied here to $\check{I}$, implies that $I(-s) = \alpha_s s - e(\alpha_s)$. Since also $\check{I}(-s) = \sup_{\alpha \in \mathbb{R}} (\alpha s - e(\alpha))$ by (2.47), we obtain (2.50).

(4) By (2.47) we have $\check{I}(0) = s_*$ and by (2.50) we further obtain

$$s_* = - \min_{\alpha \in [0,1]} e(\alpha). \qquad (2.52)$$

Since $e$ is convex and $e(0) = e(1) = 0$, we conclude that $s_* = 0$ iff $e$ vanishes identically on $[0, 1]$. Invoking convexity, one further has $e(\alpha) \ge \max(-\alpha \underline{s}_0, -(1 - \alpha) \underline{s}_1)$ for all $\alpha \in \mathbb{R}$. Using Relation (2.52) we derive

$$s_* = \sup_{\alpha \in [0,1]} (-e(\alpha)) \le \sup_{\alpha \in [0,1]} \min(\alpha \underline{s}_0, (1-\alpha)\underline{s}_1) = \min(\underline{s}_0, \underline{s}_1).$$

Finally, we observe that the right-hand side of the last equality vanishes iff $e$ vanishes identically on $[0, 1]$.
<figure><img src="image_7.png" /><figcaption>Fig. 2.1 The function $\tilde{I}$ (thin lines) and the modification (thick lines) leading to $\tilde{I}$ for the 5 cases listed in the text. Since $I(a) > 0$, lower semicontinuity implies the existence of the dashed box $]a - \delta, a + \delta[ \times [0, A]$ having no intersection with the graph of $I$.</figcaption></figure>

(5) By (2.51), we only have to show that $s_* = 0 \Rightarrow s_0 = s_1 = 0$. By Part (4), $s_* = 0$ implies that $e$ vanishes on $[0, 1]$, which further yields $s_0 = -\partial^+ e(0) = 0$ and $s_1 = \partial^- e(1) = 0$.

(6) From its definition (2.44) as the pointwise supremum of a family of non-increasing affine functions $(f_\alpha)_{\alpha \in [0,1]}$ vanishing at $\alpha = 1$, we immediately infer that $f$ is a non-negative and non-increasing, lower semicontinuous convex function such that $f(u) = +\infty$ for $u < 0$. If $s_0 < \infty$, then from the fact that $e(\alpha) \ge -\alpha s_0$, we derive that $f(u) \le s_0$ for $u \ge 0$, with equality for $u = 0$. If $s_0 = \infty$, then the convexity of $e$ implies

$$f(0) = \sup_{\alpha \in [0,1]} \frac{-e(\alpha)}{\alpha} = - \inf_{\alpha \in [0,1]} \frac{e(\alpha) - e(0)}{\alpha - 0} = - \lim_{\alpha \downarrow 0} \frac{e(\alpha) - e(0)}{\alpha - 0} = - \partial^+ e(0) = s_0 = \infty,$$

and lower semicontinuity gives

$$\liminf_{u \to 0} f(u) \ge f(0) = \infty.$$
If $\underline{s}_1 < \infty$, then invoking again convexity we derive

$$ \underline{s}_1 = \partial^- e(1) = \lim_{\alpha \uparrow 1} \frac{e(1) - e(\alpha)}{1 - \alpha} = -\lim_{\alpha \uparrow 1} \frac{e(\alpha)}{1 - \alpha}, = - \inf_{\alpha \in [0,1]} \frac{e(\alpha)}{1 - \alpha}, $$

so that, for $\alpha \in ]0, 1[$,

$$ \frac{-(1-\alpha)u - e(\alpha)}{\alpha} = \frac{1-\alpha}{\alpha} \left( -u - \frac{e(\alpha)}{1-\alpha} \right) \le \frac{1-\alpha}{\alpha} (\underline{s}_1 - u), $$

which gives that $f(u) = 0$ for $u \ge \underline{s}_1$. If $\underline{s}_1 = \infty$, then the function

$$ q(\alpha) := \frac{e(\alpha) - e(1)}{\alpha - 1} $$

is continuous, non-decreasing on the interval $]0, 1[$, and $\lim_{\alpha \uparrow 1} q(\alpha) = \underline{s}_1 = \infty$. Setting

$$ \alpha_*(u) := \sup \{\alpha \in ]0, 1[ \mid q(\alpha) < u\}, $$

continuity gives $q(\alpha_*(u)) = u$, so that

$$ \lim_{u \to \infty} \alpha_*(u) = 1 \qquad (2.53) $$

and

$$ f(u) = \sup_{\alpha \in [\alpha_*(u), 1]} \frac{(1-\alpha)(q(\alpha) - u)}{\alpha} \le \frac{1}{\alpha_* (u)} \sup_{\alpha \in [\alpha_*(u), 1]} ((\alpha-1)u - e(\alpha)) = \frac{\Delta(u)}{\alpha_*(u)}, $$

where $\Delta(u) := (\tilde{\alpha}(u) - 1)u - e(\tilde{\alpha}(u))$ is the largest vertical distance between the graphs of $\alpha \mapsto e(\alpha)$ and $\alpha \mapsto (\alpha - 1)u$ over the interval $[\alpha_*(u), 1]$, see Figure 2.2. Since $\alpha_*(u) \le \tilde{\alpha}(u)$, for large enough $u$ one has $\Delta(u) \le -e(\tilde{\alpha}(u)) \le -e(\alpha_*(u))$ and hence, by (2.53),

$$ \lim_{u \to \infty} f(u) \le \lim_{u \to \infty} \frac{-e(\alpha_*(u))}{\alpha_*(u)} = 0. $$

If $s_* = 0$, then Part (4) implies $f(s) = 0$ for $s \ge 0$, and we may assume that $s_* > 0$. By (2.52) one has $e(\alpha) + s_* \ge 0$, with equality for some $\alpha \in ]0, 1[$. We conclude that

$$ f(s_*) = \sup_{\alpha \in [0,1]} \frac{-(1-\alpha)s_* - e(\alpha)}{\alpha} = s_* - \inf_{\alpha \in [0,1]} \frac{e(\alpha) + s_*}{\alpha} = s_* $$

(7) As the sum of the non-increasing closed proper convex function $f$ and a strictly decreasing affine function, $g$ is a strictly decreasing closed proper convex function. In particular, it is injective and, by Part (6), it maps $]0, \underline{s}_1[$ homeomorphically onto $]-\underline{s}_1, \underline{s}_0[$ with

$$ \lim_{u \to 0} g(u) = \underline{s}_0, \quad \lim_{u \uparrow \underline{s}_1} g(u) = -\underline{s}_1. $$
<figure><img src="image_8.png" /><figcaption>Fig. 2.2 As the slope $u$ of the secant $\alpha \to (\alpha - 1)u$ diverges, its point of intersection $(\alpha_*(u), e(\alpha_*(u)))$ with the graph of $e$ approaches $(1, 0)$. Consequently, the maximal vertical distance $\Delta(u)$ vanishes.</figcaption></figure>

For $s \in ]-s_1, s_0[ = [-] \partial^+ e(0), \partial^- e(1]$, there exists $\gamma \in ]0, 1[$ such that $-s \in \partial e(\gamma)$ and Relation (2.27) yields that

$$r := \check{I}(s) = -\gamma s - e(\gamma).$$

By convexity, one further has $e(\alpha) \ge e(\gamma) + (\gamma - \alpha)s$ and it follows that for $\alpha \in ]0, 1]$

$$\frac{-r - e(\alpha)}{\alpha} = \frac{\gamma s + e(\gamma) - e(\alpha)}{\alpha} \le s,$$

with equality for $\alpha = \gamma$. Thus, we can conclude that

$$g \circ \check{I}(s) = \sup_{\alpha \in ]0,1]} \frac{-r - e(\alpha)}{\alpha} = s.$$

If $s_0 < \infty$, then Part (2) gives $\check{I}(s_0) = 0$ and hence

$$g(\check{I}(s_0)) = g(0) = f(0) = s_0.$$

If $s_1 < \infty$, then the same argument gives $\check{I}(-s_1) = s_1$ so that

$$g(\check{I}(-s_1)) = g(s_1) = f(s_1) - s_1 = -s_1.$$
It follows in particular that $\check{I}$ is strictly decreasing on $]-\underline{s}_1, \underline{s}_0[$. Finally, we note that

$$f \circ \check{I}(s) = \check{I}(s) + g \circ \check{I}(s) = \check{I}(s) + s = J(s), \qquad (2.54)$$

and that the last assertions are direct consequences of Parts (2) and (6).

(8) In the involutive case, an immediate consequence of (2.22) is that $\hat{e} = e$, and hence that $e(1 - \alpha) = \hat{e}(\alpha) = e(\alpha)$ for all $\alpha \in \mathbb{R}$. It follows that

$$\underline{s}_1 = \partial^- e(1) = -\partial^+ e(0) = \underline{s}_0, \quad \bar{s}_1 = \partial^+ e(1) = -\partial^- e(0) = \bar{s}_0,$$

and by symmetry $s_* = \inf_{\alpha \in [0,1]} e(\alpha) = e(1/2)$. Moreover, $\check{I}(-s) = \check{I}(s) + s$ for any $s \in \mathbb{R}$, and hence Relation (2.54) gives $\check{I}(-s) = f(\check{I}(s))$, so that

$$f \circ f(\check{I}(s)) = f(\check{I}(-s)) = \check{I}(s).$$

It follows that $f \circ f = \mathrm{Id}$ on $[0, \underline{s}_0] = \check{I}([-\underline{s}_0, \underline{s}_0])$. $\square$

We now proceed with the proof of Proposition 2.8, and for this we assume that $I$ and $\hat{I}$ are convex, i.e., that $I = \check{I}$ and $\hat{I} = J$. We use the notation of Proposition 2.9, and follow the arguments of [Jak+12, Section 6.4]. The rate functions $I$ and $\hat{I}$ are depicted in Figure 2.3.

<figure><img src="image_9.png" /><figcaption>Fig. 2.3 The graph of $I$ and $\hat{I}$ in a situation where $I$ is convex (so $I = \check{I}$ and $\hat{I} = J$), and where $-\infty < -\bar{s}_1 < -\underline{s}_1 < 0 < \underline{s}_0 < \bar{s}_0 < \infty$, in the notation of Proposition 2.8.</figcaption></figure>

First, we notice that (2.43) is immediate when $u < 0$, in view of (2.41) and Proposition 2.9 (6). We now establish (2.43) in the case $u \ge 0$.

Consider a family $(\Gamma_\lambda)_{\lambda \in \mathcal{L}} \subset \mathcal{F}$. For all $s \in \mathbb{R}$ and $\varepsilon > 0$, using first (2.32), then the definition of $\sigma_\lambda$, and finally the LDP lower bound, we find
$$
\begin{align*}
& \liminf_{\lambda \in \mathcal{L}} r_{\lambda}^{-1} \log (\mathbb{P}_{\lambda}(\Gamma_{\lambda}^{c}) + e^{sr_{\lambda}} \hat{\mathbb{P}}_{\lambda}(\Gamma_{\lambda})) \\
& \geq \liminf_{\lambda \in \mathcal{L}} r_{\lambda}^{-1} \log (\mathbb{P}_{\lambda}\{\sigma_{\lambda} < sr_{\lambda}\} + e^{sr_{\lambda}} \hat{\mathbb{P}}_{\lambda}\{\sigma_{\lambda} \geq sr_{\lambda}\}) \\
& \geq \liminf_{\lambda \in \mathcal{L}} r_{\lambda}^{-1} \log (e^{-\epsilon r_{\lambda}} \mathbb{P}_{\lambda}\{\sigma_{\lambda} \leq r_{\lambda}(s+\epsilon)\}) \geq - \inf_{s' < s+\epsilon} I(s') - \epsilon \geq -I(s) - \epsilon.
\end{align*}
$$

Since $\epsilon > 0$ is arbitrary, we conclude that

$$
\liminf_{\lambda \in \mathcal{L}} r_{\lambda}^{-1} \log (\mathbb{P}_{\lambda}(\Gamma_{\lambda}^{c}) + e^{sr_{\lambda}} \hat{\mathbb{P}}_{\lambda}(\Gamma_{\lambda})) \geq -I(s). \quad (2.55)
$$

Then, since

$$
\liminf_{\lambda \in \mathcal{L}} r_{\lambda}^{-1} \log (a_{\lambda} + b_{\lambda}) \le \max \left( \limsup_{\lambda \in \mathcal{L}} r_{\lambda}^{-1} \log a_{\lambda}, \liminf_{\lambda \in \mathcal{L}} r_{\lambda}^{-1} \log b_{\lambda} \right)
$$

for any families $(a_\lambda)_{\lambda \in \mathcal{L}}$ and $(b_\lambda)_{\lambda \in \mathcal{L}}$ of non-negative numbers, we conclude
from (2.55) that, if $\limsup_{\lambda \in \mathcal{L}} r_\lambda^{-1} \log \mathbb{P}_\lambda(\Gamma_\lambda^c) < -I(s)$, then

$$
\liminf_{\lambda \in \mathcal{L}} r_{\lambda}^{-1} \log \hat{\mathbb{P}}_{\lambda}(\Gamma_{\lambda}) \geq -I(s) - s = -\hat{I}(s).
$$

This shows that, for all $s \in \mathbb{R}$,

$$
\underline{h}(I(s)) \geq -\hat{I}(s). \tag{2.56}
$$

If $\underline{s}_1 < \infty$, then by Proposition 2.9 (2) we have $\hat{I}(-\underline{s}_1) = 0$, and thus $I(-\underline{s}_1) = \underline{s}_1$.
Inequality (2.56) then yields

$$
\underline{h}(\underline{s}_1) = \underline{h}(I(-\underline{s}_1)) \geq -\hat{I}(-\underline{s}_1) = 0.
$$

By (2.42) and the fact that $\underline{h}$, $\bar{h}$ and $h$ are non-decreasing, we conclude that

$$
\underline{h}(u) = \bar{h}(u) = h(u) = 0
$$

for all $u \ge \underline{s}_1$. By Proposition 2.9 (6), we also have $f(u) = 0$ for all $u \ge \underline{s}_1$ and thus (2.43) holds for all $u \ge \underline{s}_1$ if $\underline{s}_1 < \infty$.

It thus remains to prove (2.43) for $u \in [0, \underline{s}_1]$. Since the latter interval is empty when $\underline{s}_1 = 0$, we assume for the remainder of this proof that $\underline{s}_1 \in ]0, \infty]$. By Proposition 2.9 (4)-(5), we have also $\underline{s}_0 > 0$. By Part (7) of the same proposition, the map

$$
S := ]-\underline{s}_1, s_0[ \cup ]\exists s \mapsto I(s) \in ]0, s_1[
$$

is a decreasing homeomorphism. For $s \in \mathbb{R}$ and $\alpha \in [0, 1]$, we have

$$
\begin{align*}
\mathbb{P}_{\lambda}\{\sigma_{\lambda} < sr_{\lambda}\} + e^{sr_{\lambda}}\hat{\mathbb{P}}_{\lambda}\{\sigma_{\lambda} \ge sr_{\lambda}\} &= \mathbb{E}_{\lambda}[1_{\{\sigma_{\lambda} < sr_{\lambda}\}} + e^{sr_{\lambda}-\sigma_{\lambda}}1_{\{\sigma_{\lambda} \ge sr_{\lambda}\}}] \\
&= e^{\alpha sr_{\lambda}}\mathbb{E}_{\lambda}[e^{-\alpha sr_{\lambda}}1_{\{\sigma_{\lambda} < sr_{\lambda}\}} + e^{(1-\alpha)sr_{\lambda}-\sigma_{\lambda}}1_{\{\sigma_{\lambda} \ge sr_{\lambda}\}}] \\
&\le e^{\alpha sr_{\lambda}}\mathbb{E}_{\lambda}[e^{-\alpha \sigma_{\lambda}}1_{\{\sigma_{\lambda} < sr_{\lambda}\}} + e^{-\alpha \sigma_{\lambda}}1_{\{\sigma_{\lambda} \ge sr_{\lambda}\}}] \\
&= e^{\alpha sr_{\lambda}}\mathbb{E}_{\lambda}[e^{-\alpha \sigma_{\lambda}}] = e^{\alpha sr_{\lambda}+e_{\lambda}(\alpha)}.
\end{align*}
$$
Invoking (2.50) and optimizing over $\alpha$, we further obtain, for all $s \in S$, the upper bound

$$ \limsup_{\lambda \in \mathcal{L}} r_{\lambda}^{-1} \log (\mathbb{P}_{\lambda}\{\sigma_{\lambda} < sr_{\lambda}\} + e^{sr_{\lambda}} \hat{\mathbb{P}}_{\lambda}\{\sigma_{\lambda} \ge sr_{\lambda}\}) \le \inf_{\alpha \in [0,1]} (\alpha s + e(\alpha)) = -I(s). $$

In particular, for all $s \in S$,

$$ \limsup_{\lambda \in \mathcal{L}} r_{\lambda}^{-1} \log \mathbb{P}_{\lambda}\{\sigma_{\lambda} < sr_{\lambda}\} \le -I(s), \quad (2.57) $$

$$ \limsup_{\lambda \in \mathcal{L}} r_{\lambda}^{-1} \log \hat{\mathbb{P}}_{\lambda}\{\sigma_{\lambda} \ge sr_{\lambda}\} \le -I(s) - s = -\hat{I}(s). \quad (2.58) $$

Let $s, s' \in S$ satisfy $s < s'$, so that $-I(s) < -I(s')$. By (2.57), the family $(\Gamma_{\lambda})_{\lambda \in \mathcal{L}}$ given by $\Gamma_{\lambda} = \{\sigma_{\lambda} \ge sr_{\lambda}\}$ satisfies

$$ \limsup_{\lambda \in \mathcal{L}} r_{\lambda}^{-1} \log \mathbb{P}_{\lambda}(\Gamma_{\lambda}^{c}) \le -I(s) < -I(s') $$

and by applying (2.56) to $s'$ and using (2.58), we find

$$ -\hat{I}(s') \le \liminf_{\lambda \in \mathcal{L}} r_{\lambda}^{-1} \log \hat{\mathbb{P}}_{\lambda}(\Gamma_{\lambda}) \le \limsup_{\lambda \in \mathcal{L}} r_{\lambda}^{-1} \log \hat{\mathbb{P}}_{\lambda}(\Gamma_{\lambda}) \le -\hat{I}(s). $$

Taking $s' \downarrow s$, the continuity of $\hat{I}$ gives

$$ \lim_{\lambda \in \mathcal{L}} r_{\lambda}^{-1} \log \hat{\mathbb{P}}_{\lambda}(\Gamma_{\lambda}) = -\hat{I}(s), $$

and hence $\underline{h}(I(s) - \varepsilon) \le -\hat{I}(s)$ holds for $s \in S$ and $\varepsilon > 0$.

If $I(s)$ is a point of continuity of $\underline{h}$, then taking $\varepsilon \downarrow 0$ gives, recalling (2.56),

$$ -\hat{I}(s) \le \underline{h}(I(s)) \le \bar{h}(I(s)) \le \underline{h}(I(s)) \le -\hat{I}(s), $$

and hence

$$ \underline{h}(I(s)) = \bar{h}(I(s)) = \underline{h}(I(s)) = -\hat{I}(s). \quad (2.59) $$

Observing that the function

$$ u \mapsto \underline{h}_{(\Gamma_{\lambda})_{\lambda \in \mathcal{L}}}(u) = \begin{cases} \liminf_{\lambda \in \mathcal{L}} r_{\lambda}^{-1} \log \hat{\mathbb{P}}_{\lambda}(\Gamma_{\lambda}) & \text{if } \limsup_{\lambda \in \mathcal{L}} r_{\lambda}^{-1} \log \mathbb{P}(\Gamma_{\lambda}^{c}) < -u; \\ +\infty & \text{otherwise,} \end{cases} $$

is upper semicontinuous, so is

$$ u \mapsto \underline{h}(u) = \inf_{(\Gamma_{\lambda})_{\lambda \in \mathcal{L}}} \underline{h}_{(\Gamma_{\lambda})_{\lambda \in \mathcal{L}}}(u). $$

Since $\underline{h}$ is non-decreasing, it is also right-continuous. The same reasoning applies to the functions $\bar{h}$ and $\underline{h}$; for the latter, one takes the infimum over all families $(\Gamma_{\lambda})_{\lambda \in \mathcal{L}}$ such that $\lim_{\lambda \in \mathcal{L}} r_{\lambda}^{-1} \log \hat{\mathbb{P}}_{\lambda}(\Gamma_{\lambda})$ exists.
Let $\mathcal{S} \subset S$ be such that $I(\mathcal{S})$ is the set of discontinuity points of $h$ in $I(S)$. One easily shows that $\mathcal{S}$ is countable. Relation (2.59) holds for any $s \in S \setminus \mathcal{S}$, and by right-continuity it holds for all $s \in S$. Hence, we have proved that

$$ \underline{h}(u) = \bar{h}(u) = h(u) = -\hat{I} \circ I^{-1}(u) $$

for all $u \in ]0, \underline{s}_1[$. By Proposition 2.9 (7), the right-hand side is equal to $-f(u)$ and hence we have proved (2.43) for $u \in ]0, \underline{s}_1[$.

In order to complete the proof, it only remains to show that (2.43) holds for $u=0$, i.e., in view of Proposition 2.9 (6), we need to show that

$$ \underline{h}(0) = \bar{h}(0) = h(0) = -\underline{s}_0. \qquad (2.60) $$

If $\underline{s}_0 < \infty$, then right-continuity gives

$$ h(0) = \lim_{u \downarrow 0} h(u) = \lim_{s \uparrow \underline{s}_0} h(I(s)) = -\lim_{s \uparrow \underline{s}_0} \hat{I}(s) = -\hat{I}(\underline{s}_0) = -\underline{s}_0. $$

In the opposite case, monotonicity gives

$$ h(0) \le \liminf_{u \downarrow 0} h(u) = \liminf_{s \uparrow \infty} h(I(s)) = -\limsup_{s \uparrow \infty} \hat{I}(s) \le -\limsup_{s \uparrow \infty} s = -\infty, $$

and hence $h(0) = -\infty = -\underline{s}_0$. The same arguments apply to $\underline{h}$ and $\bar{h}$, and thus (2.60) holds and the proof of Proposition 2.8 is complete. $\square$

## References

[BC15] A. C. BARATO AND R. CHETRITE: A formal view on level 2.5 large deviations and fluctuation relations. J. Stat. Phys. **160** (2015), 1154–1172. DOI: 10.1007/s10955-015-1283-0.

[BL08] T. BODINEAU AND R. LEFEVERE: Large deviations of lattice Hamiltonian dynamics coupled to stochastic thermostats. J. Stat. Phys. **133** (2008), 1–27. DOI: 10.1007/s10955-008-9601-4.

[Cun+19] N. CUNEO, V. JAKŠIĆ, C.-A. PILLET, AND A. SHIRIKYAN: Large deviations and fluctuation theorem for selectively decoupled measures on shift spaces. Rev. Math. Phys. **31** (2019), 1950036. DOI: 10.1142/S0129055X19500363.

[DV75a] M. D. DONSKER AND S. R. S. VARADHAN: Asymptotic evaluation of certain Markov process expectations for large time. I. Comm. Pure Appl. Math. **28** (1975), 1–47. DOI: 10.1002/cpa.3160280102.

[DV75b] ———, Asymptotic evaluation of certain Markov process expectations for large time. II. Comm. Pure Appl. Math. **28** (1975), 279–301. DOI: 10.1002/cpa.3160280206.

[DZ00] A. DEMBO AND O. ZEITOUNI: *Large Deviations Techniques and Applications*. Springer, Berlin, 2000. DOI: 10.1007/978-3-642-03311-7.

[Ell85] R. S. ELLIS: *Entropy, large deviations, and statistical mechanics*. Vol. 271. Grundlehren der mathematischen Wissenschaften. Springer, New York, 1985, xiv+364. DOI: 10.1007/978-1-4613-8533-2.
[Gal96] G. GALLAVOTTI: Chaotic hypothesis: Onsager reciprocity and fluctuation-dissipation theorem. J. Stat. Phys. **84** (1996), 899–925. DOI: **10.1007/BF02174123**.

[Hol00] F. DEN HOLLANDER: *Large Deviations*. Fields Institute Monographs. American Mathematical Society, Providence, RI, 2000.

[Jak+12] V. JAKŠIĆ, Y. OGATA, C.-A. PILLET, AND R. SEIRINGER: Quantum hypothesis testing and non-equilibrium statistical mechanics. Rev. Math. Phys. **24** (2012), 1230002, 67. DOI: **10.1142/S0129055X12300026**.

[Jak+15] V. JAKŠIĆ, V. NERSESYAN, C.-A. PILLET, AND A. SHIRIKYAN: Large deviations from a stationary measure for a class of dissipative PDE's with random kicks. Comm. Pure Appl. Math. **68** (2015), 2108–2143. DOI: **10.1002/cpa.21568**.

[Jak+18] ———, Large deviations and mixing for dissipative PDE's with unbounded random kicks. Nonlinearity **31** (2018), 540–596. DOI: **10.1088/1361-6544/aa99a7**.

[Jar00] C. JARZYNSKI: Hamiltonian derivation of a detailed fluctuation theorem. J. Stat. Phys. **98** (2000), 77–102. DOI: **10.1023/A:1018670721277**.

[JPR11] V. JAKŠIĆ, C.-A. PILLET, AND L. REY-BELLET: Entropic fluctuations in statistical mechanics: I. Classical dynamical systems. Nonlinearity **24** (2011), 699–763. DOI: **10.1088/0951-7715/24/3/003**.

[Kif90] Y. KIFER: Large deviations in dynamical systems and stochastic processes. Trans. AMS **321** (1990), 505–524. DOI: **10.2307/2001571**.

[Lan73] O. LANFORD: *Entropy and equilibrium states in classical statistical mechanics*. Lecture Notes in Physics. Vol. 20. Springer, New York, 1973, 1–113. DOI: **10.1007/BFb0112756**.

[LS99] J. L. LEBOWITZ AND H. SPOHN: A Gallavotti-Cohen-type symmetry in the large deviation functional for stochastic dynamics. J. Stat. Phys. **95** (1999), 333–365. DOI: **10.1023/A:1004589714161**.

[MK10] N. MERHAV AND Y. KAFRI: Statistical properties of entropy production derived from fluctuation theorems. J. Stat. Mech. Theor. Exp. (2010), P12022. DOI: **10.1007/s11742-010-0022-9**.

[MNW08] C. MAES, K. NETOCKNÝ, AND B. WYNANTS: Steady state statistics of driven diffusions. Phys. A **387** (2008), 2675–2689. DOI: **10.1016/j.physa.2008.01.097**.

[Roc70] R. T. ROCKAFELLAR: *Convex analysis*. Princeton Mathematical Series, No. 28. Reprint of the 1970 original, Princeton Paperbacks. Princeton University Press, Princeton, N.J., Princeton, NJ, 1970, xviii+451.

[Rue65] D. RUELLE: Correlation functionals. J. Math. Phys. **6** (1965), 201–220. DOI: **10.1063/1.1704272**.

[Sei12] U. SEIFERT: Stochastic thermodynamics, fluctuation theorems and molecular machines. Rep. Prog. Phys. **75** (2012), 126001. DOI: **10.1088/0034-4885/75/12/126001**.
<figure><img src="image_10.png" /></figure>
## Chapter 3
# Illustrations

**Abstract** This chapter provides illustrations of the various aspects of Fluctuation Relations and Fluctuation Theorems. We revisit some paradigmatic models of equilibrium and nonequilibrium statistical mechanics, deriving in each case the Large Deviation Principle giving rise to a fluctuation theorem and discussing its physical meaning and its particular features: convexity properties, smoothness of the entropic pressure, phase transitions, etc.

Let us introduce some standard notation to be repeatedly used in the following. For $a, b \in \mathbb{Z}$, $\llbracket a, b \rrbracket := [a, b] \cap \mathbb{Z}$, $\llbracket a, b \rrbracket := [a, b[\cap \mathbb{Z}]$, etc. $\mathcal{P}(\Omega)$ denotes the set of Borel probability measures on a Polish space $\Omega$. Given a continuous map $\varphi: \Omega \to \Omega$, we further denote by $\mathcal{P}_{\varphi}(\Omega)$ the subset of $\varphi$-invariant measures. $C(\Omega)$ and $C_b(\Omega)$ denote the space of real-valued continuous functions on $\Omega$ and its subspace of bounded elements. For $\mu \in \mathcal{P}(\Omega)$ and $f \in C_b(\Omega)$ we write

$$ \langle f, \mu \rangle := \int_{\Omega} f d\mu. $$

Unless otherwise stated, $\mathcal{P}(\Omega)$ is equipped with the (metrizable and separable) topology of weak convergence. We recall that whenever $\Omega$ is compact, so is $\mathcal{P}(\Omega)$.

### 3.1 Finite State Markov Chains

#### 3.1.1 Setup

For a Markov chain on the state space $\mathcal{A} := \llbracket 1, N \rrbracket$ we take $\mathcal{L} = \mathbb{N}$ as the index set and, for each $t \in \mathbb{N}$, denote by $\Omega_t = \mathcal{A}^t$ the $t$-step path space of the process. Assuming that the transition matrix $P = [P_{xy}]_{x,y \in \mathcal{A}}$ admits an invariant probability $p = [p_x]_{x \in \mathcal{A}}$ (i.e., that $p = pP$), the path space measure of the corresponding
stationary Markov process (p, P) is given by

$$ \Omega_t \ni \mathbf{x} = (x_1, \dots, x_t) \mapsto \mathbb{P}_t(\mathbf{x}) := p_{x_1} P_{x_1 x_2} \cdots P_{x_{t-1} x_t}. $$

Given a second stationary Markov process ($\hat{p}, \hat{P}$) on the same state space, the corresponding measure $\hat{P}_t$ is equivalent to $\mathbb{P}_t$ for all $t \in \mathbb{N}$ iff both conditions (i) $p_x = 0 \Leftrightarrow \hat{p}_x = 0$, and (ii) $P_{xy} = 0 \Leftrightarrow \hat{P}_{xy} = 0$ hold for all $x, y \in \mathcal{A}$. Note that if P and $\hat{P}$ are both irreducible,<fnref n="1" /> then the Perron-Frobenius theorem implies that $p_x > 0$ and $\hat{p}_x > 0$ for all $x \in \mathcal{A}$ so that Condition (i) trivially holds. In the following, we shall assume that P and $\hat{P}$ are irreducible and satisfy Condition (ii).

### 3.1.2 Transient Fluctuation Relation

The entropy production random variable of the family $(\mathbb{P}_t, \hat{\mathbb{P}}_t)_{t \in \mathbb{N}}$ is given by

$$ \sigma_t(\mathbf{x}) = \log \frac{\mathbb{P}_t(\mathbf{x})}{\hat{\mathbb{P}}_t(\mathbf{x})} = \log \frac{p_{x_1}}{\hat{p}_{x_1}} + \sum_{s=1}^{t-1} \log \frac{P_{x_s x_{s+1}}}{\hat{P}_{x_s x_{s+1}}}. $$

Simple calculations yield that

$$ \mathrm{ep}_t = \mathrm{Ent}(\mathbb{P}_t|\hat{\mathbb{P}}_t) = \sum_{\mathbf{x} \in \Omega_t} \sigma_t(\mathbf{x}) \mathbb{P}_t(\mathbf{x}) = \mathrm{Ent}(p|\hat{p}) + (t-1) \sum_{\mathbf{x}, y \in \mathcal{A}} p_x P_{xy} \log \frac{P_{xy}}{\hat{P}_{xy}}, \quad (3.1) $$

and

$$ e_t(\alpha) = \mathrm{Ent}_\alpha(\mathbb{P}_t|\hat{\mathbb{P}}_t) = \log \sum_{\mathbf{x} \in \Omega_t} [\mathbb{P}_t(\mathbf{x})]^{1-\alpha} [\hat{\mathbb{P}}_t(\mathbf{x})]^\alpha = \log(q(\alpha)Q(\alpha)^{t-1}\mathbf{1}), \quad (3.2) $$

$$ \hat{e}_t(\alpha) = \mathrm{Ent}_\alpha(\hat{\mathbb{P}}_t|\mathbb{P}_t) = \log \sum_{\mathbf{x} \in \Omega_t} [\mathbb{P}_t(\mathbf{x})]^\alpha [\hat{\mathbb{P}}_t(\mathbf{x})]^{1-\alpha} = \log(q(1-\alpha)Q(1-\alpha)^{t-1}\mathbf{1}), $$

where the row vector $q(\alpha)$, the matrix $Q(\alpha)$, and the column vector $\mathbf{1}$ are given by

$$ q(\alpha) := [p_x^{1-\alpha} \hat{p}_x^\alpha]_{x \in \mathcal{A}}, \quad Q(\alpha) := [P_{xy}^{1-\alpha} \hat{P}_{xy}^\alpha]_{x,y \in \mathcal{A}}, \quad \mathbf{1} := [1, 1, \dots, 1]^T. $$

As an important special case, let us consider the $\mathbb{Z}_2$-action on $\Omega_t$ induced by the time-reversal map

$$ \Theta_t : (x_1, \dots, x_t) \mapsto (x_t, \dots, x_1), $$

and set $\hat{\mathbb{P}}_t = \mathbb{P}_t \circ \Theta_t$. Since $p_x > 0$ for all $x \in \mathcal{A}$, we can write

$$ \hat{\mathbb{P}}_t(\mathbf{x}) = p_{x_t} P_{x_t x_{t-1}} \cdots P_{x_1 x_0} = p_{x_1} \hat{P}_{x_1 x_2} \cdots \hat{P}_{x_{t-1} x_t}, $$

<footnote n="1">¹ P is irreducible iff, for any x, y ∈ A there exists N > 0 such that $(P^N)_{xy} > 0$.</footnote>
with $\hat{P}_{xy} = p_y P_{yx} p_x^{-1}$. It follows at once that $p\hat{P} = p$. Moreover, $\mathbb{P}_t$ and $\hat{\mathbb{P}}_t$ are equivalent whenever $P_{xy} = 0 \Leftrightarrow P_{yx} = 0$, in which case $\hat{P}$ is irreducible. The family $(\mathbb{P}_t, \hat{\mathbb{P}}_t)_{t \in \mathbb{N}}$ is in involution and time-reversal invariance $\mathbb{P}_t = \hat{\mathbb{P}}_t$ holds iff $P = \hat{P}$, which is a disguised form of the detailed balance condition $p_x P_{xy} = p_y P_{yx}$. From (3.1) one derives that the mean entropy production rate

$$ \lim_{t \to \infty} \frac{\mathrm{ep}_t}{t} = \frac{1}{2} \sum_{x,y \in \mathcal{A}} p_x (P_{xy} - \hat{P}_{xy}) \log \frac{P_{xy}}{\hat{P}_{xy}} $$

is strictly positive whenever the time-reversal symmetry is broken.

### 3.1.3 Fluctuation Theorem

We shall use (3.2) to compute the entropic pressure of the family $(\mathbb{P}_t, \hat{\mathbb{P}}_t)_{t \in \mathbb{N}}$. To this end, note that since the entries of $Q(\alpha)$ are non-negative, one has

$$ e^{-c(1+2|\alpha|)} \|Q(\alpha)^n\| \le q(\alpha)Q(\alpha)^n \mathbf{1} \le e^{c(1+2|\alpha|)} \|Q(\alpha)^n\| $$

where the matrix norm is defined as

$$ \|A\| := \sum_{x,y \in \mathcal{A}} |A_{xy}|, $$

and

$$ c := \max_{x \in \mathcal{A}} (|\log p_x|, |\log \hat{p}_x|). $$

It follows that

$$ \frac{1}{t} (\log \|Q(\alpha)^t\| - c(1+2|\alpha|)) \le \frac{1}{t} e_t(\alpha) \le \frac{1}{t} (\log \|Q(\alpha)^t\| + c(1+2|\alpha|)), $$

and taking $t \to \infty$, Gelfand's formula yields

$$ e(\alpha) = \lim_{t \to \infty} \frac{1}{t} e_t(\alpha) = \log r(\alpha) $$

where $r(\alpha)$ denotes the spectral radius of $Q(\alpha)$. One easily shows that for all $\alpha \in \mathbb{R}$ the matrix $Q(\alpha)$, which clearly has non-negative entries, is irreducible. The Perron-Frobenius theorem then asserts that $r(\alpha)$ is the dominant eigenvalue of $Q(\alpha)$. In particular, $r(\alpha)$ is a strictly positive simple eigenvalue of $Q(\alpha)$. It follows from analytic perturbation theory [Kat66, Chapter 2] that the function $\mathbb{R} \ni \alpha \mapsto r(\alpha)$ is real analytic. The same is true of $\mathbb{R} \ni \alpha \mapsto e(\alpha)$ and the Gärtner-Ellis theorem yields a global FT with a rate function given by (2.26).
### 3.1.3.1 Level-3 Fluctuation Relation

We conclude this example with a brief description of the Donsker-Varadhan process-level LDP as an alternative approach to the FT for Markov chains. We refer to [DS89; DZ00] for a more detailed exposition.

Let $\Omega := \mathcal{A}^\mathbb{N}$ be the set of infinite sequences $\mathbf{x} = (x_1, x_2, \dots)$ of elements of $\mathcal{A}$. Equipped with the metric $d(x, y) := 2^{-\inf\{t \in \mathbb{N} | x_t \neq y_t\}}$, $\Omega$ is a compact metric space. We denote the left shift $(x_1, x_2, \dots) \mapsto (x_2, x_3, \dots)$ on $\Omega$ by $\varphi$. The Borel $\sigma$-algebra $\mathcal{B}$ of $\Omega$ is generated by the cylinder sets $Z_I(\Gamma) := \{\mathbf{x} \in \Omega | (x_t)_{t \in I} \in \Gamma\}$, where $I \subset \mathbb{N}$ is finite and $\Gamma \subset \mathcal{A}^I$. The set $\mathcal{P}(\Omega)$, endowed with the weak topology, is a compact metrizable space and $\mathcal{P}_\varphi(\Omega)$ is closed in $\mathcal{P}(\Omega)$. We denote by $\mathbb{P}$ and $\hat{\mathbb{P}}$ the elements of $\mathcal{P}_\varphi(\Omega)$ induced by the sequences $(\mathbb{P}_t)_{t \in \mathbb{N}}$ and $(\hat{\mathbb{P}}_t)_{t \in \mathbb{N}}$.

Analogous constructions apply to $\tilde{\Omega} := \mathcal{A}^\mathbb{Z}$, and we shall also denote by $\varphi$ the left shift on this space. Any $\tilde{Q} \in \mathcal{P}_\varphi(\tilde{\Omega})$ has a unique extension $\tilde{Q} \in \mathcal{P}_\varphi(\Omega)$ which is characterized by

$$ \tilde{Q}(Z_I(\Gamma)) = Q(Z_{I+t}(\Gamma)) $$

for any finite $I \subset \mathbb{Z}$, any $\Gamma \subset \mathcal{A}^I$, and any $t \in \mathbb{N}$ such that $I+t \subset \mathbb{N}$. For $t \in \mathbb{Z}$, we denote by $\tilde{\mathcal{B}}_t$ the $\sigma$-algebra generated by $\{Z_I(\Gamma) | I \subset \mathbb{N}, t \in \mathbb{N}\}$, $I$ is finite, $\Gamma \subset \mathcal{A}^I$ and by $\tilde{Q}_t$ the restriction of $\tilde{Q}$ to $\tilde{\mathcal{B}}_t$. Finally, let $\tilde{Q}_0 \otimes P$ be the probability measure on $\tilde{\mathcal{B}}_1$ defined by

$$ \int_{\tilde{\Omega}} f(\dots, x_0, x_1) \tilde{Q}_0 \otimes P(dx) := \sum_{y \in \tilde{\Omega}} \int_{\tilde{\Omega}} f(\dots, x_0, y) P_{x_0 y} \tilde{Q}_0(dx) $$

for any bounded $\tilde{\mathcal{B}}_1$-measurable function $f: \tilde{\Omega} \to \mathbb{R}$. If the matrix $P$ is irreducible, then the sequence of empirical measures<fnref n="2" />

$$ \Omega \ni \mathbf{x} \mapsto \xi_t := \frac{1}{t} \sum_{s=0}^{t-1} \delta_{\varphi^s(\mathbf{x})}, $$

viewed as $\mathcal{P}(\Omega)$-valued random variables on $(\Omega, \mathcal{B}, \mathbb{P})$, satisfies the LDP

$$ - \inf_{Q \in \tilde{\Gamma}^I} \|Q\| \le \liminf_{t \to \infty} \frac{1}{t} \log \mathbb{P}\{\xi_t \in \Gamma\} \le \limsup_{t \to \infty} \frac{1}{t} \log \mathbb{P}\{\xi_t \in \Gamma\} \le - \inf_{Q \in \tilde{\Gamma}^I} \|Q\| $$

for any Borel set $\Gamma \subset \mathcal{P}(\Omega)$ with the good rate function

$$ \|Q\| := \begin{cases} \mathrm{Ent}(\tilde{Q}_1 | \tilde{Q}_0 \otimes P) & \text{if } Q \in \mathcal{P}_\varphi(\Omega); \\ +\infty & \text{if } Q \in \mathcal{P}(\Omega) \setminus \mathcal{P}_\varphi(\Omega). \end{cases} $$

If $\hat{\mathbb{P}}$ is irreducible, then the same LDP also holds w.r.t. the measure $\hat{\mathbb{P}}$ with the corresponding rate function $\hat{\mathbb{I}}$. Moreover, the map

<footnote n="2">² $\delta_x$ denotes the Dirac mass at $\mathbf{x} \in \Omega$.</footnote>
$$ \mathcal{P}(\Omega) \ni Q \mapsto \zeta(Q) := \langle \sigma, Q \rangle, \quad \sigma(\mathbf{x}) := \log \frac{P_{x_1 x_2}}{\hat{P}_{x_1 x_2}}, $$

is continuous and one easily checks that

$$ t\zeta(\xi_t) = \sum_{s=1}^{t} \log \frac{P_{x_s x_{s+1}}}{\hat{P}_{x_s x_{s+1}}} = \sigma_t + o(t), \quad (t \to \infty). $$

The contraction principle thus yields the FT with the rate functions

$$ I(s) := \inf\{\|\langle Q \rangle\|_Q \in \mathcal{P}(\Omega), \zeta(Q) = s\}, \quad \hat{I}(s) := \inf\{\|\hat{\langle Q \rangle}\|_Q \in \mathcal{P}(\Omega), \zeta(Q) = s\}. $$

Using the simple fact that, for any $\mathbf{x} \in \Omega$,

$$ \frac{d\bar{Q}_0 \otimes P}{d\bar{Q}_0 \otimes \hat{P}}(\mathbf{x}) = \frac{P_{x_0 x_1}}{\hat{P}_{x_0 x_1}}, $$

one derives the process-level or level-3 FR

$$ \begin{aligned} \hat{\mathfrak{I}}(Q) &= \mathrm{Ent}(\bar{Q}_1 | \bar{Q}_0 \otimes \hat{P}) = \int \log \frac{d\bar{Q}_1}{d\bar{Q}_0 \otimes \hat{P}} d\bar{Q}_1 \\ &= \int \left( \log \frac{d\bar{Q}_1}{d\bar{Q}_0 \otimes P} + \log \frac{d\bar{Q}_0 \otimes P}{d\bar{Q}_0 \otimes \hat{P}} \right) d\bar{Q}_1 \\ &= \|\langle Q \rangle\| + \zeta(Q) \end{aligned} $$

for $Q \in \mathcal{P}_{\varphi}(\Omega)$. The same relation holds for $Q \in \mathcal{P}(\Omega) \setminus \mathcal{P}_{\varphi}(\Omega)$, both sides of the resulting identity being infinite.

## 3.2 Mean-field Lattice Gas

### 3.2.1 Setup

For $x = (x_1, \dots, x_d) \in \mathbb{Z}^d$ we set $|x| := \max_i |x_i|$ and denote by $|\lambda|$ the cardinality of a subset $\lambda \subset \mathbb{Z}^d$. Let $\mathcal{L}$ be the set of centered cubic boxes $\{x \in \mathbb{Z}^d \mid |x| \le n\}$ in $\mathbb{Z}^d$ ordered by inclusion and for $\lambda \in \mathcal{L}$ let $\Omega_\lambda = \{0, 1\}^\lambda$ be the configuration space of a lattice gas confined to the box $\lambda$. We shall consider the mean-field Hamiltonian

$$ \Omega_{\lambda} \ni \mathbf{n} = (n_{\mathbf{x}})_{\mathbf{x} \in \lambda} \mapsto H_{\lambda}(\mathbf{n}) := - \frac{2}{|\lambda| - 1} \sum_{\substack{x, y \in \lambda \\ x \neq y}} n_{\mathbf{x}} n_{\mathbf{y}} $$

with attractive interaction. The grand canonical ensemble at inverse temperature $\beta$ and chemical potential $\mu$ is the atomic measure on $\Omega_\lambda$ defined by
$$
\mathbb{P}_{\lambda}(\{\mathbf{n}\}) := e^{-\beta(H_{\lambda}(\mathbf{n}) - \mu N_{\lambda}(\mathbf{n}) + |\lambda|p_{\lambda}(\beta, \mu))},
$$

where $N_{\lambda}(\mathbf{n}) := \sum_{x \in \Lambda} n_x$ is the total number of particles in the box $\lambda$ and

$$
p_{\lambda}(\beta, \mu) := \frac{1}{|\lambda|\beta} \log \sum_{\mathbf{n} \in \Omega_{\lambda}} e^{-\beta(H_{\lambda}(\mathbf{n}) - \mu N_{\lambda}(\mathbf{n}))} \quad (3.3)
$$

is the pressure of the gas. We denote expectation w.r.t. $\mathbb{P}_{\lambda}$ by $\langle \cdot \rangle_{\lambda, \beta, \mu}$.

The particle-hole symmetry

$$
\Theta_{\lambda} : (n_x)_{x \in \lambda} \mapsto (1 - n_x)_{x \in \lambda}
$$

provides a $\mathbb{Z}_2$-action on $\Omega_\lambda$, and we set $\hat{\mathbb{P}}_\lambda = \mathbb{P}_\lambda \circ \Theta_\lambda$.

**3.2.2 Transient Fluctuation Relation**

A simple calculation gives that the entropy production observable of the involu-
tive family $(\mathbb{P}_{\lambda}, \hat{\mathbb{P}}_{\lambda})_{\lambda \in \mathcal{L}}$ is

$$
\sigma_{\lambda} = 2|\lambda|\beta(\mu + 2) \left( \rho_{\lambda} - \frac{1}{2} \right), \qquad (3.4)
$$

where $\rho_{\lambda}(\mathbf{n}) := |\lambda|^{-1} N_{\lambda}(\mathbf{n})$ is the sample density. For $\mu \neq -2$, Inequality (2.8) re-
duces to

$$
\begin{cases}
\langle \rho_{\lambda} \rangle_{\lambda, \beta, \mu} \le \frac{1}{2} & \text{if } \mu < -2; \\
\langle \rho_{\lambda} \rangle_{\lambda, \beta, \mu} \ge \frac{1}{2} & \text{if } \mu > -2,
\end{cases}
$$

which shows that ep$_{λ}$ is an order parameter for the particle-hole symmetry. In-
equality (2.12) yields a quantitative version of this statement with the estimate

$$
\mathbb{P}_{\lambda}\{\rho_{\lambda} - 1/2 < -a\} \leq e^{-2|\lambda|\beta(\mu+2)a},
$$

for $\mu > -2$ (and a similar estimate for $\mu < -2$).

The Rényi entropy of the lattice gas

$$
e_{\lambda}(\alpha) = |\lambda|\beta(p_{\lambda}(\beta, (1-2\alpha)(\mu+2)-2) - p_{\lambda}(\beta, \mu) + \alpha(\mu+2)) \quad (3.5)
$$

vanishes identically for $\mu = -2$. The symmetry (2.13) reduces, in this model, to
the pressure identity

$$
p_{\lambda}(\beta, -2 + \nu) - p_{\lambda}(\beta, -2 - \nu) = \nu.
$$

Differentiation w.r.t. $v$ yields

$$
\langle \rho_{\lambda} \rangle_{\lambda, \beta, -2+v} = 1 - \langle \rho_{\lambda} \rangle_{\lambda, \beta, -2-v},
$$
and in particular $\langle \rho_{\lambda} \rangle_{\lambda, \beta, -2} = 1/2$.

### 3.2.3 Fluctuation Theorem

In terms of the sample density $\rho_{\lambda}$, the pressure (3.3) of the mean-field lattice gas reads

$$p_{\lambda}(\beta, \mu) = \frac{1}{|\lambda|\beta} \log \sum_{\mathbf{n} \in \Omega_{\lambda}} e^{|\lambda|\beta\rho_{\lambda}(\mathbf{n})(2\rho_{\lambda}(\mathbf{n})+\mu)} e^{-\beta\rho_{\lambda}(\mathbf{n})(1-\rho_{\lambda}(\mathbf{n}))^{\frac{2|\lambda|}{|\lambda|-1}}}$$

and since $\rho_{\lambda}(\mathbf{n}) \in [0, 1]$ for all $\mathbf{n} \in \Omega_{\lambda}$, we have

$$p(\beta, \mu) = \lim_{\lambda \in \mathcal{L}} p_{\lambda}(\beta, \mu) = \lim_{\lambda \in \mathcal{L}} \frac{1}{|\lambda|\beta} \log \sum_{\mathbf{n} \in \Omega_{\lambda}} e^{|\lambda|\beta\rho_{\lambda}(\mathbf{n})(2\rho_{\lambda}(\mathbf{n})+\mu)}.$$

Introducing the product $\mathbb{B}^{\lambda} := \otimes_{x \in \lambda} \mathbb{B}$ of the Bernoulli distribution $\mathbb{B}(0) = \mathbb{B}(1) = 1/2$, we can write

$$\sum_{\mathbf{n} \in \Omega_{\lambda}} e^{|\lambda|\beta\rho_{\lambda}(\mathbf{n})(2\rho_{\lambda}(\mathbf{n})+\mu)} = 2^{|\lambda|} \mathbb{B}^{\lambda}[e^{|\lambda|f(\rho_{\lambda})}]$$

with $f(r) := \beta r(2r + \mu)$ and $\rho_{\lambda} = |\lambda|^{-1} \sum_{x \in \lambda} n_x$, the $n_x$ being i.i.d. $\mathbb{B}$-distributed random variables. Combining Cramér's LDP [DZ00, Theorem 2.2.3] with Varadhan's lemma leads to the expression

$$p(\beta, \mu) = \frac{1}{\beta} \sup_{r \in [0,1]} F(\beta, \mu, r), \qquad (3.6)$$

where $F(\beta, \mu, r) := f(r) - r \log r - (1-r) \log(1-r)$. The right-hand side of (3.6) can be evaluated explicitly. This elementary analysis (see [Dor99, Section 30] or [Ell06, Section IV.4]) yields that the set of critical points

$$\Re(\beta, \mu) := \{r \in [0, 1] \mid \partial_r F(\beta, \mu, r) = 0\}$$

is a singleton if either $\beta < \beta_c = 1$ or $\beta \ge \beta_c$ and

$$|1 + \mu/2| > g(\beta) = \sqrt{\beta_c^{-1} - \beta^{-1}} - \beta^{-1} \operatorname{arth}(\sqrt{\beta_c^{-1} - \beta^{-1}}).$$

It contains 3 elements if $\beta > \beta_c$ and $|1 + \mu/2| < g(\beta)$. Moreover, one has

$$p(\beta, \mu) = \frac{1}{\beta} F(\beta, \mu, \varrho(\beta, \mu)),$$

where
$$ \rho(\beta, \mu) := \begin{cases} \min \Re(\beta, \mu) < \frac{1}{2}, & \text{for } \mu < -2; \\ \max \Re(\beta, \mu) > \frac{1}{2}, & \text{for } \mu > -2. \end{cases} $$

It follows that for $\beta > \beta_c$,

$$ \lim_{\mu \uparrow -2} \rho(\beta, \mu) < 1/2 < \lim_{\mu \downarrow -2} \rho(\beta, \mu). $$

The function $(\beta, \mu) \mapsto p(\beta, \mu)$ is real analytic on the domain $(\mathbb{R}_+ \times \mathbb{R}) \setminus (|\beta_c|, \infty[\times \{-2\})$, and since

$$ \rho(\beta, \mu) = \partial_{\mu} p(\beta, \mu), $$

the pressure is non-differentiable as a function of $\mu$ at $\mu = -2$, $\beta > \beta_c$. The mean-field lattice gas thus exhibits a spontaneous breaking of the particle-hole symmetry, see Figure 3.1. By (3.5) the entropic pressure is given by

$$ e(\alpha) = \lim_{\lambda \in \mathcal{L}} \frac{1}{|\lambda|} e_{\lambda}(\alpha) = \beta(p(\beta, (1-2\alpha)(\mu+2)-2) - p(\beta, \mu) + \alpha(\mu+2)) \quad (3.7) $$

which is a real analytic function of $\alpha \in \mathbb{R}$ for $\beta < \beta_c$, but is non-differentiable at $\alpha = 1/2$ for $\beta > \beta_c$ and $\mu \neq -2$. Thus, for $\beta < \beta_c$ the Gärtner-Ellis theorem yields a global FT with rate function given by (2.26). However, due to the singularity appearing at $\alpha = 1/2$, this approach fails for $\beta > \beta_c$.

<figure><img src="image_11.png" /><figcaption>Fig. 3.1 The density $\rho$ (left) and the pressure $p$ (right) of the mean-field lattice gas as functions of $\mu$ near the critical value $\mu = -2$ for various values of $\beta$ (shorter dashes mean lower $\beta$).</figcaption></figure>
To overcome this, we shall now briefly sketch a derivation of the FT which does not rely on the Gärtner-Ellis theorem and which works for any $\beta > 0$. This approach relies on standard techniques of large deviations theory, which easily apply here due to the simple structure of the mean-field lattice gas.

Let $\mathcal{A} := \{0, 1\}$. We shall identify without further notice $\mathcal{P}(\mathcal{A})$ with the subset $\{(1-r, r) | r \in [0, 1]\}$ of $\mathbb{R}^2$ and topologize accordingly.

For each $\lambda \in \mathcal{L}$ define the level-2 empirical measure

$$ \Omega_{\lambda} \ni \mathbf{n} = (n_x)_{x \in \lambda} \mapsto \xi_{\lambda} := \frac{1}{|\lambda|} \sum_{x \in \lambda} \delta_{n_x} \in \mathcal{P}(\mathcal{A}), $$

where $\delta_n$ is the unit mass at $n \in \mathcal{A}$. Considering $\xi_{\lambda}$ as a $\mathcal{P}(\mathcal{A})$-valued random variable on $(\Omega_{\lambda}, \mathbb{B}^{\lambda})$, Sanov's theorem [DZ00, Theorem 2.1.10] yields the LDP

$$ - \inf_{Q \in \Gamma} \mathbf{I}(Q) \le \liminf_{\lambda \in \mathcal{L}} \frac{1}{|\lambda|} \log \mathbb{B}^{\lambda} \{\xi_{\lambda} \in \Gamma\} \le \limsup_{\lambda \in \mathcal{L}} \frac{1}{|\lambda|} \log \mathbb{B}^{\lambda} \{\xi_{\lambda} \in \Gamma\} \le - \inf_{Q \in \Gamma} \mathbf{I}(Q) $$

for any Borel subset $\Gamma \subset \mathcal{P}(\mathcal{A})$ with a rate function given by the relative entropy

$$ \mathbf{I}(Q) := \mathrm{Ent}(Q|\mathbb{B}) = \log 2 - S(Q), $$

where

$$ S(Q) := - \sum_{a \in \mathcal{A}} Q(a) \log Q(a) \qquad (3.8) $$

is the Shannon entropy of $Q$. Defining the map $\mathcal{P}(\mathcal{A}) \ni Q \mapsto \mathbf{F}(Q) := \langle G, Q \otimes Q \rangle \in \mathbb{R}$, where

$$ G(a, b) := \beta a(2b + \mu) - \beta p(\beta, \mu), $$

we observe that, for any $\lambda \in \mathcal{L}$,

$$ 2^{|\lambda|} e^{-\frac{1}{2} \frac{|\lambda|}{|\lambda|-1}} e^{|\lambda| \mathbf{F}(\xi_{\lambda})} \le \frac{d \mathbb{P}_{\lambda}}{d \mathbb{B}^{\lambda}} \le 2^{|\lambda|} e^{|\lambda| \mathbf{F}(\xi_{\lambda})}. $$

It follows from Varadhan's lemma (more precisely from its tilted LDP version (2.2)) that the LDP

$$ - \inf_{Q \in \Gamma} \mathbb{I}(Q) \le \liminf_{\lambda \in \mathcal{L}} \frac{1}{|\lambda|} \log \mathbb{P}_{\lambda} \{\xi_{\lambda} \in \Gamma\} \le \limsup_{\lambda \in \mathcal{L}} \frac{1}{|\lambda|} \log \mathbb{P}_{\lambda} \{\xi_{\lambda} \in \Gamma\} \le - \inf_{Q \in \Gamma} \mathbb{I}(Q) $$

holds for $\Gamma \subset \mathcal{P}(\mathcal{A})$ with the rate function

$$ \mathbb{I}(Q) := \mathbf{I}(Q) - \mathbf{F}(Q) - \inf_{Q' \in \mathcal{P}(\mathcal{A})} (\mathbf{I}(Q') - \mathbf{F}(Q')). $$

A simple calculation, using Relation (3.6), shows that

$$ \mathbb{I}(Q) = -\mathbf{F}(Q) - S(Q). $$
Finally, expressing the entropy production random variable (3.4) as $\sigma_{\lambda} = |\lambda|\zeta(\xi_{\lambda})$ with

$$ \zeta(\mathbb{Q}) := \mathbb{Q} \otimes \mathbb{Q}(\sigma), \quad \sigma := G - G \circ \Theta, \qquad (3.9) $$

(here $\Theta(a, b) := (1-a, 1-b)$), the contraction principle shows that the global FT holds with scale $r_{\lambda} = |\lambda|$ and rate function

$$ I(s) = \inf\{\|\mathbb{Q}\| \mid \mathbb{Q} \in \mathcal{P}(\mathcal{A}), \zeta(\mathbb{Q}) = s\}. $$

Observing that the constraint $\zeta(\mathbb{Q}) = s$ actually determines $\mathbb{Q}$, with

$$ \mathbb{Q}(1) = \frac{1}{2} \left( 1 + \frac{s}{\beta(\mu + 2)} \right) := r, $$

another straightforward calculation gives the following expression of the rate function $I$ in terms of the variable $r$,

$$ I(s) = \begin{cases} \beta p(\beta, \mu) - F(\beta, \mu, r), & \text{for } r \in [0, 1]; \\ +\infty, & \text{otherwise.} \end{cases} $$

**Remark 1.** From the simple estimate

$$ |e^{-\alpha \sigma_{\lambda}}| \le e^{\beta |\lambda| ((\mu+2) \operatorname{Re} \alpha)} $$

one easily deduces, invoking Vitali's convergence theorem, that

$$ e'(0) = \lim_{\lambda \in \mathcal{L}} \frac{1}{|\lambda|} e'_{\lambda}(0) = - \lim_{\lambda \in \mathcal{L}} \frac{1}{|\lambda|} \langle \sigma_{\lambda} \rangle_{\lambda, \beta, \mu}. $$

Using (3.7), the left-hand side of this identity is

$$ e'(0) = 2\beta(\mu + 2) \left( \frac{1}{2} - \partial_{\mu} p(\beta, \mu) \right), $$

and recalling (3.4) we conclude that the thermodynamic limit of the mean density is

$$ \lim_{\lambda \in \mathcal{L}} \langle \rho_{\lambda} \rangle_{\lambda, \beta, \mu} = \rho(\beta, \mu), $$

and that of the mean entropy production rate is

$$ \text{epr} := \lim_{\lambda \in \mathcal{L}} \frac{\text{ep}_{\lambda}}{|\lambda|} = 2\beta(\mu + 2) \left( \rho(\beta, \mu) - \frac{1}{2} \right). $$

**Remark 2.** It follows from our explicit formula that $I'(s) = 0$ iff $r \in \mathbb{R}(\beta, \mu)$. Since the latter set is not always a singleton, it may happen that the rate function is not convex. Indeed, an elementary analysis shows that $I$ is convex for $\beta < \beta_c$, while for $\beta > \beta_c$ it is concave on the interval $|s| < |\mu + 2|\sqrt{\beta(\beta/\beta_c - 1)}$ and convex on its complement, see Figure 3.2.
**Remark 3.** The FR, $I(-s) = I(s) + s$, corresponds here to the elementary property

$$F(\beta, \mu, 1-r) = F(\beta, \mu, r) + \beta(\mu+2)(1-2r).$$

**Remark 4.** Setting $\hat{Q}(a) := Q(1-a)$ for $Q \in \mathcal{P}(\mathcal{A})$, it follows from the relation $\xi_{\lambda} \circ \Theta_{\lambda} = \hat{\xi}_{\lambda}$ that $(\xi_{\lambda})_{\lambda \in \mathcal{L}}$ satisfies the LDP w.r.t. $(\hat{P}_{\lambda})_{\lambda \in \mathcal{L}}$ with the scale $r_{\lambda} = |\lambda|$ and rate function $\hat{\mathcal{I}}$ given by $\hat{\mathcal{I}}(Q) := \mathcal{I}(\hat{Q})$. Moreover, rewriting Definition (3.9) as

$$\zeta(Q) = \mathbf{F}(Q) - \mathbf{F}(\hat{Q})$$

immediately leads to the level-2 FR

$$\hat{\mathcal{I}}(Q) = \mathcal{I}(Q) + \zeta(Q),$$

which is (2.30).

<figure><img src="image_12.png" /><figcaption>**Fig. 3.2** The FT rate $I$ as a function of the rescaled variable $\hat{s} = s/\beta(\mu+2)$ for $\beta = 0.9 < \beta_c$ (left) and $\beta = 1.8 > \beta_c$ (right) and $\mu = -2, -1.8, -1.6, -1.4$.</figcaption></figure>
<figure><img src="image_13.png" /><figcaption>Fig. 3.3 The Hoeffding function f (recall Equ. (2.44) and Proposition 2.8) of the mean-field lattice gas for $\beta = 0.8$ (lowest curve), 0.9, 1, 1.1, 1.2 (highest curve) and $\mu = -1.9$.</figcaption></figure>

## 3.3 Ising Model

### 3.3.1 Setup

With the same index set $\mathcal{L}$ as in the previous example, let $\mathcal{A} := \{-1, +1\}$ and $\Omega_{\lambda} = \mathcal{A}^{\lambda}$ be the configuration space of the Ising model in the box $\lambda \in \mathcal{L}$ and denote by

$$ \Omega_{\lambda} \ni \mathbf{s} = (s_x)_{x \in \lambda} \mapsto H_{\lambda}(\mathbf{s}) := -\frac{J}{2} \sum_{\substack{x, y \in \lambda \\ |x-y|=1}} s_x s_y - h \sum_{x \in \lambda} s_x $$

the ferromagnetic ($J > 0$) Ising Hamiltonian with external magnetic field $h \in \mathbb{R}$. The associated canonical ensemble at inverse temperature $\beta > 0$ with free boundary condition is the atomic measure on $\Omega_{\lambda}$ defined by

$$ \mathbb{P}_{\lambda}(\{\mathbf{s}\}) := e^{-\beta(H_{\lambda}(\mathbf{s})+|\lambda|p_{\lambda}(\beta,h))}, $$

where the real analytic function

$$ p_{\lambda}(\beta, h) := \frac{1}{|\lambda|\beta} \log \sum_{\mathbf{s} \in \Omega_{\lambda}} e^{-\beta H_{\lambda}(\mathbf{s})} \qquad (3.10) $$
is the pressure.<fnref n="3" /> We shall denote by $\langle \cdot \rangle_{\lambda, \beta, h}$ the expectation w.r.t. $\mathbb{P}_{\lambda}$. A physically natural $\mathbb{Z}_2$-action on $\Omega_{\lambda}$ is provided by the spin-flip operation $\Theta_{\lambda}: s \mapsto -s$. Defining then $\hat{\mathbb{P}}_{\lambda} = \mathbb{P}_{\lambda} \circ \Theta_{\lambda}$, we obtain by construction that the family $(\mathbb{P}_{\lambda}, \hat{\mathbb{P}}_{\lambda})_{\lambda \in \mathcal{L}}$ is in involution. We note that $\hat{\mathbb{P}}_{\lambda}$ is obtained from $\mathbb{P}_{\lambda}$ by replacing the external field $h$ by $-h$.

### 3.3.2 Transient Fluctuation Relation

The entropy production random variable of the family $(\mathbb{P}_{\lambda}, \hat{\mathbb{P}}_{\lambda})_{\lambda \in \mathcal{L}}$ is given by

$$ \sigma_{\lambda} = \log \frac{d\mathbb{P}_{\lambda}}{d\hat{\mathbb{P}}_{\lambda}} = 2|\lambda|\beta hm_{\lambda}, $$

where

$$ m_{\lambda}(s) := \frac{1}{|\lambda|} \sum_{x \in \lambda} s_x $$

is the specific magnetization. In this setting the general Inequality (2.8) reads

$$ h \langle m_{\lambda} \rangle_{\lambda, \beta, h} \geq 0, $$

and implies that for $h \neq 0$ the mean magnetization is aligned with the external field. Together with Markov's inequality, Jarzynski's identity (2.11), which becomes

$$ \langle e^{-2|\lambda|\beta hm_{\lambda}} \rangle_{\lambda, \beta, h} = 1, $$

allows us to deduce (see (2.12)) the more precise statement

$$ \mathbb{P}_{\lambda}\{hm_{\lambda} \leq -E\} \leq e^{-2|\lambda|\beta E}, $$

showing that the probability of misaligned magnetization becomes exponentially small in large boxes. The Rényi entropy (2.10) is also easily identified with

$$ e_{\lambda}(\alpha) = \log \langle e^{-2|\lambda|\beta \alpha hm_{\lambda}} \rangle_{\lambda, \beta, h} = |\lambda| \beta (p_{\lambda}(\beta, (1-2\alpha)h) - p_{\lambda}(\beta, h)) \quad (3.11) $$

so that the symmetry (2.13) reduces to the pressure identity<fnref n="4" />

$$ p_{\lambda}(\beta, -h) = p_{\lambda}(\beta, h). $$

Differentiation w.r.t $h$ yields the symmetry

$$ \langle m_{\lambda} \rangle_{\lambda, \beta, -h} = -\langle m_{\lambda} \rangle_{\lambda, \beta, h}, $$

<footnote n="3">³ The thermodynamic potential associated to the canonical ensemble $\mathbb{P}_{\lambda}$ is the Helmholtz free energy per spin $f_{\lambda}(\beta, h) = -p_{\lambda}(\beta, h)$. We introduce the pressure here in order to keep a more consistent terminology and notation.</footnote>

<footnote n="4">⁴ Note that this corresponds to Equ. (3.9) in [Rue69, Proposition 5.3.2].</footnote>
and in particular the absence of spontaneous magnetization $\langle m_{\lambda} \rangle_{\lambda, \beta, 0} = 0$.

### 3.3.3 Fluctuation Theorem

The thermodynamic limit of the pressure (3.10)

$$p(\beta, h) := \lim_{\lambda \in \mathcal{L}} p_{\lambda}(\beta, h)$$

exists for all $(\beta, h) \in \mathbb{R}_{+} \times \mathbb{R}$. It follows from (3.11) that the entropic pressure is given by

$$\mathbb{R} \ni \alpha \mapsto e(\alpha) = \beta(p(\beta, (1-2\alpha)h) - p(\beta, h)).$$

In dimension $d=1$, $p$ is given by (see [Hua87, Chapter 14])

$$p(\beta, h) = J + \frac{1}{\beta} \log \left( \cosh(\beta h) + \sqrt{\sinh^2(\beta h) + e^{-4\beta J}} \right).$$

Thus, the entropic pressure

$$e(\alpha) = \log \left( \frac{\cosh((1-2\alpha)\beta h) + \sqrt{\sinh^2((1-2\alpha)\beta h) + e^{-4\beta J}}}{\cosh(\beta h) + \sqrt{\sinh^2(\beta h) + e^{-4\beta J}}} \right)$$

vanishes identically for $h=0$ and is a strictly convex real analytic function for $h \neq 0$. The global FT again follows from the Gärtner-Ellis theorem, the rate function $I$ being given by the Legendre transform (2.26).

In dimension $d > 1$, no explicit formula is known for the pressure $p$. However, it is known that a phase transition occurs at some critical inverse temperature $0 < \beta_c < \infty$, see [Ell06, Chapter V]. For $\beta < \beta_c$, $p(\beta, h)$ is a real analytic function of $h$ and so is the entropic pressure: the global FT still follows from the Gärtner-Ellis theorem in this case. For $\beta > \beta_c$, $p(\beta, h)$ is non-differentiable at $h=0$. More precisely,

$$(\tilde{\partial}_h^\alpha p)(\beta, 0) = -m(\beta) < 0 < m(\beta) = (\partial_h^\alpha p)(\beta, 0),$$

where $m(\beta)$ is the spontaneous magnetization. Consequently, for $h \neq 0$, the entropic pressure is non-differentiable at $\alpha = 1/2$, with

$$(\partial^{-} e)(1/2) = -2\beta h m(\beta) < 0 < 2\beta h m(\beta) = (\partial^{+} e)(1/2),$$

and the Gärtner-Ellis route to the global FT fails. Due to the truly interacting nature of the Ising model, a direct approach along the lines of our analysis of the mean-field lattice gas seems more challenging. However, as we shall now sketch, it is still within the reach of currently available techniques. We refer to [Rue04] for the needed facts on the thermodynamics of lattice systems and to [FV17] for the more specific issues pertaining to the Ising model.
The configuration space of the infinitely extended $d$-dimensional Ising model is $\Omega := \mathcal{A}\mathbb{Z}^d$ endowed with the product topology, a separable compact metrizable space. It comes equipped with its Borel $\sigma$-algebra $\mathcal{B}$. For $\Lambda \subset \mathbb{Z}^d$, set $\Omega_\Lambda := \mathcal{A}^\Lambda$ and let $\mathcal{B}_\Lambda$ be the associated Borel $\sigma$-algebra, which we identify with a subalgebra of $\mathcal{B}$. We denote the restriction of $\mathbf{s} \in \Omega$ to $\Lambda$ by $\mathbf{s}_\Lambda \in \Omega_\Lambda$ and the restriction of $\mathbb{P} \in \mathcal{P}(\Omega)$ to $\mathcal{B}_\Lambda$ by $\mathbb{P}^{(\Lambda)}$. For $\mathbf{s} \in \Omega_\Lambda$, the cylinder subset of $\Omega$ based on $\mathbf{s}$ is

$$[\mathbf{s}] := \{\mathbf{s}' \in \Omega | \mathbf{s}'_\Lambda = \mathbf{s}\}.$$

Let $\tau$ be the natural action of $\mathbb{Z}^d$ on $\Omega$ and let $\mathcal{P}_\tau(\Omega)$ be the set of $\tau$-invariant elements of $\mathcal{P}(\Omega)$. The pressure admits the variational expression

$$\beta p(\beta, h) = \sup_{\mathbb{Q} \in \mathcal{P}_\tau(\Omega)} \left( S(\mathbb{Q}) + \beta \int G_h(s) \mathbb{Q}(ds) \right), \quad (3.12)$$

where

$$\mathcal{P}_\tau(\Omega) \ni \mathbb{Q} \mapsto S(\mathbb{Q}) := \lim_{\lambda \in \mathcal{L}} -\frac{1}{|\lambda|} \sum_{\mathbf{s} \in \Omega_\lambda} \mathbb{Q}([\mathbf{s}]) \log \mathbb{Q}([\mathbf{s}]) = \inf_{\lambda \in \mathcal{L}} -\frac{1}{|\lambda|} \sum_{\mathbf{s} \in \Omega_\lambda} \mathbb{Q}([\mathbf{s}]) \log \mathbb{Q}([\mathbf{s}]),$$

is the mean entropy, a non-negative, upper semicontinuous affine function, and the potential

$$\Omega \ni \mathbf{s} \mapsto G_h(\mathbf{s}) := s_0 \left( h + \frac{J}{2} \sum_{|x|=1} s_x \right)$$

is a continuous function.

A measure $\mathbb{P} \in \mathcal{P}(\Omega)$ is a Gibbs state if it satisfies the DLR equation

$$\mathbb{P}([\mathbf{s}_\lambda]) = \int_{\Omega_\lambda^c} \frac{e^{-\beta(H_\lambda(\mathbf{s}_\lambda) + W_\lambda(\mathbf{s}_\lambda, \mathbf{r}_\lambda^c))}}{\sum_{\mathbf{r}_\lambda \in \Omega_\lambda} e^{-\beta(H_\lambda(\mathbf{r}_\lambda) + W_\lambda(\mathbf{r}_\lambda, \mathbf{r}_\lambda^c))}} \mathbb{P}^{(\lambda^c)}(d\mathbf{r}_\lambda^{(\lambda^c)}) \quad (3.13)$$

for all $\mathbf{s}_\lambda \in \Omega_\lambda$ and all $\lambda \in \mathcal{L}$, where $\lambda^c = \mathbb{Z}^d \setminus \lambda$ and

$$W_\lambda(\mathbf{s}, \mathbf{r}) := -J \sum_{(x,y) \in \partial\lambda} s_x r_y$$

is the interaction energy across the boundary $\partial\lambda := \{(x, y) \in \lambda \times \lambda^c \mid |x - y| = 1\}$. $\mathbb{P}$ is an equilibrium state whenever it is $\tau$-invariant and satisfies

$$S(\mathbb{P}) + \beta \langle G_h, \mathbb{P} \rangle = \beta p(\beta, h).$$

If $\mathbb{P} \in \mathcal{P}_\tau(\Omega)$, then $\mathbb{P}$ is a Gibbs state iff it is an equilibrium state. If $h \neq 0$, then there is a unique Gibbs state $\mathbb{P}_{\beta,h}$. Moreover, $\mathbb{P}_{\beta,h} \in \mathcal{P}_\tau(\Omega)$ so that it is also the unique equilibrium state.

For each $\lambda \in \mathcal{L}$ and $\mathbf{s} \in \Omega$ define the empirical measure
$$ \xi_{\lambda} := \frac{1}{|\lambda|} \sum_{x \in \lambda} \delta_{\tau^{x}(s)} \in \mathcal{P}(\Omega) $$

where $\delta_s$ denotes the unit mass at $s$. It follows from [EKW94, Theorem A] that the family $(\xi_\lambda)_{\lambda \in \mathcal{L}}$ satisfies a level-3 LDP w.r.t. $\mathbb{P}_{\beta,h}$ with scale $r_\lambda = |\lambda|$, i.e.,

$$ - \inf_{Q \in \tilde{\mathcal{R}}} \|Q\| \le \liminf_{\lambda \in \mathcal{L}} \frac{1}{|\lambda|} \log \mathbb{P}_{\beta,h}\{\xi_\lambda \in \Gamma\} \le \limsup_{\lambda \in \mathcal{L}} \frac{1}{|\lambda|} \log \mathbb{P}_{\beta,h}\{\xi_\lambda \in \Gamma\} \le - \inf_{Q \in \tilde{\mathcal{R}}} \|Q\| $$

holds for all Borel sets $\Gamma \subset \mathcal{P}(\Omega)$ with the good rate function

$$ \|Q\| = \begin{cases} \beta p(\beta, h) - \beta \langle G_h, Q \rangle - S(Q), & \text{if } Q \in \mathcal{P}_{\tau}(\Omega); \\ +\infty, & \text{if } Q \in \mathcal{P}(\Omega) \setminus \mathcal{P}_{\tau}(\Omega). \end{cases} \quad (3.14) $$

Let $\Theta$ denote the spin flip on $\Omega$ and set $\sigma(s) := G_h(s) - G_h \circ \Theta(s) = 2hs_0$. One easily checks that

$$ |\lambda|\zeta(\xi_\lambda) = \sigma_\lambda $$

where $\zeta$ is the continuous map

$$ \mathcal{P}(\Omega) \ni Q \mapsto \zeta(Q) := \beta \langle \sigma, Q \rangle. $$

The contraction principle then implies that the LDP

$$ \begin{aligned} - \inf_{s \in \tilde{S}} I(s) &\le \liminf_{\lambda \in \mathcal{L}} \frac{1}{|\lambda|} \log \mathbb{P}_{\beta,h}\{|\lambda|^{-1} \sigma_\lambda \in S\} \\ &\le \limsup_{\lambda \in \mathcal{L}} \frac{1}{|\lambda|} \log \mathbb{P}_{\beta,h}\{|\lambda|^{-1} \sigma_\lambda \in S\} \le - \inf_{s \in \tilde{S}} I(s) \end{aligned} \quad (3.15) $$

holds for all Borel sets $S \subset \mathbb{R}$ with the rate function

$$ I(s) = \inf\{\|Q\| \mid Q \in \mathcal{P}_{\tau}(\Omega), \zeta(Q) = s\}. \quad (3.16) $$

From the DLR equation (3.13) we deduce that

$$ \frac{\mathrm{d}\mathbb{P}^{(\lambda)}}{\mathrm{d}\mathbb{P}_{\lambda}}(\mathbf{s}_{\lambda}) = \int_{\Omega_{\lambda}^{c}} \frac{\sum_{\mathbf{r}_{\lambda} \in \Omega_{\lambda}} e^{-\beta(H_{\lambda}(\mathbf{r}_{\lambda}) + W_{\lambda}(\mathbf{s}_{\lambda}, \mathbf{r}_{\lambda}^{c}))}}{\sum_{\mathbf{r}_{\lambda} \in \Omega_{\lambda}} e^{-\beta(H_{\lambda}(\mathbf{r}_{\lambda}) + W_{\lambda}(\mathbf{r}_{\lambda}, \mathbf{r}_{\lambda}^{c}))}} \mathbb{P}_{\beta,h}^{(\lambda^{c})}(d\mathbf{r}_{\lambda^{c}}), $$

and the estimate

$$ |W_{\lambda}(\mathbf{s}_{\lambda}, \mathbf{r}_{\lambda}^{c}) - W_{\lambda}(\mathbf{r}_{\lambda}, \mathbf{r}_{\lambda}^{c})| \le J \sum_{(x,y) \in \partial \lambda} |(s_x - r_x)r_y| \le 2J|\partial \lambda| $$

implies that

$$ e^{-2\beta J|\partial \lambda|} \le \frac{\mathrm{d}\mathbb{P}^{(\lambda)}}{\mathrm{d}\mathbb{P}_{\lambda}}(\mathbf{s}_{\lambda}) \le e^{2\beta J|\partial \lambda|}. $$
The fact that $\lim_{\lambda \in \mathcal{L}} |\lambda|^{-1} |\partial \lambda| = 0$ immediately yields that (3.15) also holds with $\mathbb{P}_{\beta,h}$ replaced by $\mathbb{P}_{\lambda}$. Thus, the global FT holds. To identify the rate function $I$ we note that, due to the above mentioned properties of the mean entropy, the function $\|\cdot\|$ is affine and lower semicontinuous on the compact set $\mathcal{P}_{\tau}(\Omega)$. Hence, the infimum in (3.16) is achieved by some $Q_s$. Thus, given $s_1, s_2 \in \mathbb{R}$, we can find $Q_{s_1}, Q_{s_2} \in \mathcal{P}_{\tau}(\Omega)$ such that $I(s_i) = \|\langle Q_{s_i} \rangle\|$ and $\zeta(Q_{s_i}) = s_i$. It follows that for any $\alpha \in [0, 1]$ the measure $Q_{|\alpha|} := \alpha Q_{s_1} + (1-\alpha) Q_{s_2} \in \mathcal{P}_{\tau}(\Omega)$ is such that $\|\langle Q_{|\alpha|} \rangle\| = \alpha I(s_1) + (1-\alpha) I(s_2)$ and $\zeta(Q_{|\alpha|}) = \alpha s_1 + (1-\alpha) s_2$. Hence,

$$ 
\begin{aligned}
I(\alpha s_1 + (1-\alpha)s_2) &= \inf \{ \|\langle Q \rangle\| \mid Q \in \mathcal{P}_{\tau}(\Omega), s(Q) = \alpha s_1 + (1-\alpha)s_2 \} \\
&\le \|\langle Q_{|\alpha|} \rangle\| = \alpha I(s_1) + (1-\alpha) I(s_2),
\end{aligned}
$$

i.e., the rate function $I$ is convex. At this point we can either invoke directly the relation between the rate function and the entropic pressure to conclude that $I$ is given by the Legendre transform (2.26), or compute explicitly the right-hand side of (2.25), using (3.16) and (3.14),

$$ 
\begin{aligned}
-\inf_{s \in \mathbb{R}} (s\alpha + I(s)) &= -\inf_{s \in \mathbb{R}} \inf \{s\alpha + \|\langle Q \rangle\| \mid Q \in \mathcal{P}_{\tau}(\Omega), \zeta(Q) = s\} \\
&= -\inf \{\alpha\zeta(Q) + \|\langle Q \rangle\| \mid Q \in \mathcal{P}_{\tau}(\Omega)\} \\
&= \sup_{Q \in \mathcal{P}_{\tau}(\Omega)} (\beta\langle(1-\alpha)G_h + \alpha G_h \circ \Theta, Q\rangle + S(Q)) - \beta p(\beta, h).
\end{aligned}
$$

Since $(1-\alpha)G_h + \alpha G_h \circ \Theta = G_{(1-2\alpha)h}$, it follows from the variational identity (3.12) that

$$ \sup_{s \in \mathbb{R}} (\alpha s - I(-s)) = - \inf_{s \in \mathbb{R}} (s\alpha + I(s)) = \beta(p(\beta, (1-2\alpha)h) - p(\beta, h)) = e(\alpha), $$

and the convex function $I$ is again recovered through (2.26).

By Proposition 2.3, we know that the FR, $I(-s) = I(s) + s$, holds for all $s \in \mathbb{R}$. The same conclusion immediately follows from Equ. (2.26) and the symmetry $e(1-\alpha) = e(\alpha)$ of the entropic pressure which is equivalent to the symmetry $p(\beta, -h) = p(\beta, h)$ of the pressure. Finally, the FR can also be recovered from the level-3 fluctuation relation

$$ \|\langle Q \circ \Theta \rangle\| = \|\langle Q \rangle\| + \zeta(Q), $$

which holds for any $Q \in \mathcal{P}(\Omega)$ as a direct consequence of Definition (3.14) and of the obvious fact that $S(Q \circ \Theta) = S(Q)$.

The main lesson of this example is that despite the fact that the entropic pressure is not everywhere differentiable, the global FT holds with a rate function given by the Legendre transform of the entropic pressure. Figure 3.4 shows the typical aspect of the entropic pressure and rate function for the critical Ising Model. Note that, due to the kink of the entropic pressure at $\alpha = 1/2$, the rate function is flat on the interval $[-\partial^+ e(1/2), -\partial^- e(1/2)]$. The points in this interval
<figure><img src="image_14.png" /><figcaption>Fig. 3.4 A typical entropic pressure $e(\alpha)$ (left) and the corresponding rate function $I(s)$ (right) for the Ising model in the phase transition regime $\beta > \beta_c$.</figcaption></figure>

are said to be unexposed and the Gärtner-Ellis theorem fails to provide the LDP lower bound at such points (see [DZ00, Theorem 2.3.6]).

## 3.4 Langevin Dynamics

Our previous examples are illustrations of the use of the contraction principle in the derivation of the FR/FT, starting from the level-2 and/or level-3 LDP. The next example shows that in some cases it may be relatively easy to derive the LDP and prove a FR at level-2 or level-3 using well-known techniques, but that the real difficulty appears when trying to apply the contraction principle.

### 3.4.1 Setup

Let $\mathcal{G} = (\mathcal{V}, \mathcal{E})$ be a finite connected graph. For each vertex $i \in \mathcal{V}$, let $V_i \in C^\infty(\mathbb{R})$ be such that

$$ \inf_{q} V_{i}(q) \geq 0, \quad \lim_{|q| \to \infty} |V'_{i}(q)| = +\infty, \quad \sup_{q} |V''_{i}(q)| < \infty. $$
For each edge $e \in \mathcal{E}$, let $W_e \in C_b^\infty(\mathbb{R})^5$ be non identically vanishing and such that

$$W_e(-q) = W_e(q).$$

Set

$$U(q) := V(q) + W(q) := \sum_{i \in \mathcal{V}} V_i(q_i) + \sum_{\{i,j\} \in \mathcal{E}} W_{\{i,j\}}(q_i - q_j),$$

and consider the Hamiltonian $H(q, p) := \frac{1}{2}|p|^2 + U(q)$ on the phase space

$$\mathcal{X} := T^*\mathbb{R}^V = \mathbb{R}^V \oplus \mathbb{R}^V$$

equipped with its canonical symplectic structure. We set $x := (q, p)$ and denote by $dx = dqdp$ the induced Liouville measure on $\mathcal{X}$. Let $\Gamma$ and $T$ be diagonal matrices acting on $\mathbb{R}^V$ with strictly positive diagonal entries $\Gamma_i$ and $T_i$. The stochastic differential equation (SDE)

$$\begin{aligned} dq(s) &= p(s)ds; \\ dp(s) &= -(\nabla U(q(s)) + T^{-1}\Gamma p(s))ds + (2\Gamma)^{1/2}dw(s), \end{aligned} \qquad (3.17)$$

where $w$ denotes a standard Wiener process in $\mathbb{R}^V$, provides a statistical model for the dynamics of an open network $\mathcal{S}$ whose internal dynamics is governed by the Hamiltonian $H$ and where each node $i \in \mathcal{V}$ is thermostated by a reservoir $\mathcal{R}_i$ in thermal equilibrium at temperature $T_i$. In the physically more interesting case where $\Gamma$ is degenerate, the analysis of the resulting dynamical system becomes quite delicate and relies on the detailed structure of the potential $U$. We refer the reader to [EPR99b; EPR99a; EH00; RT02a; RT02b; EH03; HM09] for results pertaining to boundary driven chains of oscillators, and to [MV03; EZ04; BL08; CE14; JPS17; Cun+18] for more general networks. Here, we shall only sketch the strategy that leads to the FT in the non-degenerate case $\Gamma > 0$. Our discussion heavily relies on the works [Wu01; BL08] to which we refer for more details.

The Markov generator associated to the SDE (3.17) is given by$^6$

$$L = (\nabla_p - T^{-1}p) \cdot \Gamma \nabla_p + \{H, \cdot\}.$$

We denote by

$$L^T = \nabla_p \cdot \Gamma (\nabla_p + T^{-1}p) - \{H, \cdot\}$$

its formal adjoint, and by

$$\gamma(f, g) := \frac{1}{2}(Lfg - fLg - gLf) = \nabla_p f \cdot \Gamma \nabla_p g,$$

$$\gamma(f) := \gamma(f, f) = |\Gamma^{1/2} \nabla_p f|^2,$$

<footnote>$^5$ $C_b^\infty(\mathbb{R})$ denotes the space of smooth, bounded functions whose derivatives of all orders are bounded.</footnote>

<footnote>$^6$ Here $\{H, \cdot\}$ denotes a Poisson bracket.</footnote>
the associated carré du champ operators. We recall the chain rule

$$L(F \circ f) = (F' \circ f)Lf + (F'' \circ f)\gamma(f). \qquad (3.18)$$

It follows that, for $\vartheta > T_{\max} := \max_i T_i$,

$$L e^{H/\vartheta} = \vartheta^{-1} (\mathrm{tr}(\Gamma) - \Gamma^{1/2} p \cdot (T^{-1} - \vartheta^{-1}) \Gamma^{1/2} p) e^{H/\vartheta} \le \vartheta^{-1} \mathrm{tr}(\Gamma) e^{H/\vartheta}.$$

Invoking [Kha12, Theorem 3.5] we can conclude that SDE (3.17) has global strong solutions which, together with the law of the initial condition $x(0)$, assumed to be independent of the driving Wiener process, define a Markov process with continuous path and hence, for each $t \in \mathcal{L} = \mathbb{R}_+$, the law of $\mathbf{x} = (x(s))_{s \in [0, t]}$ is a probability measure on $\Omega_t = C([0, t], \mathcal{X})$. For $\mathbf{x} \in \mathcal{X}$, we shall denote by $\mathbb{P}_{\mathbf{x},t}$ the law of the process started at $\mathbf{x}(0) = \mathbf{x}$ and by $\mathbb{E}_{\mathbf{x},t}$ the corresponding expectation functional. If $\mu \in \mathcal{P}(\mathcal{X})$, then

$$\mathbb{P}_{\mu,t} := \int_{\mathcal{X}} \mathbb{P}_{\mathbf{x},t} \mu(dx)$$

is the law of the process started with $x(0)$ distributed according to $\mu$, and we shall denote by $\mu_s$ the law of $x(s)$ under $\mathbb{P}_{\mu,t}$.

Invoking Hörmander's commutator theorem [Hör07, Theorem 22.2.1], one easily establishes the micro-hypoellipticity of the operators $L$, $L^T$, $\partial_t - L$ and $\partial_t - L^T$. It follows that for all $s > 0$ the Markov transition kernel of the process is strong Feller with a smooth density $P^s(x, y)$ w.r.t. the Liouville measure on $\mathcal{X}$, i.e., for any bounded measurable function $f$ on $\mathcal{X}$,

$$\mathbb{E}_{x,t}[f(x(t))] = \int_{\mathcal{X}} P^t(x, y)f(y)dy,$$

where the function $(t, x, y) \mapsto P^t(x, y)$ is in $C^\infty([0, \infty[ \times \mathcal{X} \times \mathcal{X})$.

The system is controllable, i.e., for any $x, y \in \mathcal{X}$ and any $t > 0$ there exists a control $u \in C([0, t], \mathbb{R}^T)$ such that the solution $x_u = (q_u, p_u)$ of the system

$$\dot{q}(s) = p(s),$$

$$\dot{p}(s) = -(\nabla U(q(s)) + T^{-1}\Gamma p(s)) + u(s),$$

with initial condition $x_u(0) = x$ satisfies $x_u(t) = y$. It follows from the results of [SV72], see also [EPR99b; Cun+18], that for any $(t, x, y) \in ]0, \infty[ \times \mathcal{X} \times \mathcal{X}$ one has $P^t(x, y) > 0$. Moreover, the process has a unique stationary measure $\mu_{st}$ whose density $\rho_{st} = \frac{d\mu_{st}}{dx} \in C^\infty(\mathcal{X})$ is everywhere positive and satisfies $L^T \rho_{st} = 0$.

In the special case where the interaction potential $W$ vanishes identically, the stationary measure is the local equilibrium state

$$v(dx) := Z^{-1}e^{-S(x)}dx, \quad S(x) := \frac{1}{2}p \cdot T^{-1}p + \sum_{i \in \mathcal{V}} T_i^{-1}V_i(q_i), \qquad (3.19)$$
which provides a convenient reference measure. Note that $v$ is invariant under the time-reversal involution of $\mathcal{X}$ given by $\theta : (q, p) \mapsto (q, -p)$. A $\mathbb{Z}_2$-action on $\Omega_t$ is defined by

$$ \Theta_t : \mathbf{x} = (x(s))_{s \in [0, t]} \mapsto \hat{\mathbf{x}} = (\theta x(t-s))_{s \in [0, t]}. $$

In the following, we shall consider the family $(\mathbb{P}_t, \hat{\mathbb{P}}_t)_{t \in \mathbb{R}_+}$ where

$$ \mathbb{P}_t := \mathbb{P}_{\mu_{\text{st}}, t}, \quad \hat{\mathbb{P}}_t := \mathbb{P}_t \circ \Theta_t. $$

$\hat{\mathbb{P}}_t$ describes a stationary process with invariant measure $\hat{\mu}_{\text{st}} := \mu_{\text{st}} \circ \theta$. For reasons which will become clear below, we define a smooth function $\phi : \mathcal{X} \to \mathbb{R}$ by

$$ \phi(x) := \log \frac{d\hat{\mu}_{\text{st}}}{dv}(x). \qquad (3.20) $$

A simple calculation shows that the adjoint $L^*$ of $L$ w.r.t. the inner product of the Hilbert space $L^2(\mathcal{X}, dv)$ satisfies the generalized detailed balance relation

$$ \Theta L^* \Theta = L - \sigma \qquad (3.21) $$

where the operators $\Theta$ and $\sigma$ are defined, respectively, by $f \mapsto f \circ \theta$ and multiplication with the function

$$ \sigma(x) := \{H, S\}(x) = -T^{-1}p \cdot \nabla W(q) = G(x) - G \circ \theta(x), \quad G(x) = \frac{1}{4}|T^{-1/2}(p - \nabla W(q))|^2. $$

An immediate consequence of (3.21) is that the function (3.20) satisfies

$$ L\phi + \gamma(\phi) = \sigma. \qquad (3.22) $$

It follows from well-known results on the time-reversal of diffusion processes (see [And82; HP86]) that $\hat{\mathbb{P}}_t$ is the law of the solutions $\hat{\mathbf{x}} = (\hat{q}(s), \hat{p}(s))_{s \in [0, t]}$ of the SDE

$$ d\hat{q}(s) = \hat{p}(s)ds $$

$$ d\hat{p}(s) = -(\nabla U(\hat{q}(s)) + T^{-1}\Gamma\hat{p}(s) - 2\Gamma(\nabla_p\phi)(\hat{q}(s), \hat{p}(s)))ds + (2\Gamma)^{1/2}d\hat{w}(s) $$

with initial condition $\hat{x}(0)$ distributed according to $\hat{\mu}_{\text{st}}$, independently of the standard Wiener process $\hat{w}$. In particular, the family $(\hat{\mathbb{P}}_t)_{t \in \mathbb{R}_+}$ describes a stationary Markov process with generator $\hat{L} = L + 2\gamma(\phi, \cdot)$ and transition kernel

$$ \hat{P}^t(x, y) = \rho_{\text{st}}(\theta y) P^t(\theta y, \theta x) \rho_{\text{st}}(\theta x)^{-1}. $$

We shall denote by $\hat{\mathbb{P}}_{x,t}$ the probability induced on $\Omega_t$ by conditioning $\hat{\mathbb{P}}_t$ on the starting value $\hat{x}(0) = x$. Setting

$$ \eta(s) := \int_0^s (2\Gamma)^{1/2} \nabla_p \phi(x(r)) \cdot dw(r), \quad [\eta](s) := \int_0^s |(2\Gamma)^{1/2} \nabla_p \phi(x(r))|^2 dr, $$
Itô calculus and the generalized detailed balance, through Relation (3.22), yield

$$ \eta(s) - \frac{1}{2}[\eta](s) = \phi(x(s)) - \phi(x(0)) - \int_{0}^{s} \sigma(x(r))dr. $$

Following the lines of the proof of [JPS17, Proposition 3.9], one shows that the Doléans-Dade stochastic exponential

$$ [0, t] \ni s \mapsto Z(s) := e^{\eta(s) - \frac{1}{2}[\eta](s)} $$

is a $\mathbb{P}_{x,t}$-martingale. It follows from Girsanov's theorem that

$$ \frac{d\mathbb{P}_{x,t}}{d\hat{\mathbb{P}}_{x,t}}(\mathbf{x}) = \exp \left[ \phi(x(0)) - \phi(x(t)) + \int_{0}^{t} \sigma(x(s))ds \right]. $$

### 3.4.2 Transient Fluctuation Relation

Since

$$ \frac{d\mathbb{P}_t}{d\hat{\mathbb{P}}_t}(\mathbf{x}) = \frac{d\mathbb{P}_{\mu_{\text{st}},t}}{d\hat{\mathbb{P}}_{\hat{\mu}_{\text{st}},t}}(\mathbf{x}) = \frac{d\mathbb{P}_{\mu_{\text{st}},t}}{d\hat{\mathbb{P}}_{\hat{\mu}_{\text{st}},t}}(\mathbf{x}) \frac{d\mu_{\text{st}}}{d\hat{\mu}_{\text{st}}}(x(0)) = \frac{d\mathbb{P}_{x(0),t}}{d\hat{\mathbb{P}}_{x(0),t}}(\mathbf{x}) e^{\phi \circ \theta(x(0)) - \phi(x(0))}, $$

the entropy production random variable of the pair $(\mathbb{P}_t, \hat{\mathbb{P}}_t)$ is given by

$$ \sigma_t(\mathbf{x}) = \log \frac{d\mathbb{P}_t}{d\hat{\mathbb{P}}_t}(\mathbf{x}) = \phi \circ \theta(x(0)) - \phi(x(t)) + \int_0^t \sigma(x(s))ds. \quad (3.23) $$

To connect this expression with thermodynamics, we note that by Itô's formula the change in energy is given by

$$ dH(x(t)) = \sum_{i \in \mathcal{V}} \left( (2\Gamma_i)^{1/2} p_i(t) dw_i(t) + \Gamma_i (1 - T_i^{-1} p_i(t)^2) dt \right) = - \sum_{i \in \mathcal{V}} d\Phi_i(t). $$

Interpreting $d\Phi_i(t)$ as the instantaneous flux of energy dissipated in the reservoir $\mathcal{R}_i$, the total thermodynamic entropy dissipated in the environment during the interval $[0, t]$ is given by

$$ \begin{aligned} \mathcal{E}_t(\mathbf{x}) &:= \sum_{i \in \mathcal{V}} \int_0^t \frac{d\Phi_i(s)}{T_i} \\ &= - \int_0^t \left( (2\Gamma)^{1/2} T^{-1} p(s) \cdot dw(s) - \frac{1}{2} |(2\Gamma)^{1/2} T^{-1} p(s)|^2 ds + \text{tr}(\Gamma T^{-1}) ds \right). \end{aligned} $$

Applying Itô's formula to the function $S$ defined in (3.19) we derive

$$ dS(x(t)) = (2\Gamma)^{1/2} T^{-1} p(t) \cdot dw(t) - \left( \frac{1}{2} |(2\Gamma)^{1/2} T^{-1} p(t)|^2 - \text{tr}(\Gamma T^{-1}) - \sigma(x(t)) \right) dt $$
and hence

$$ \mathcal{S}_t(\mathbf{x}) = \log \frac{dv}{dx}(x(t)) - \log \frac{dv}{dx}(x(0)) + \int_0^t \sigma(x(s))ds. \quad (3.24) $$

Comparison with (3.23) allows us to conclude that

$$ \sigma_t(\mathbf{x}) = \log \rho_{\text{st}}(x(0)) - \log \rho_{\text{st}} \circ \theta(x(t)) + \mathcal{S}_t(\mathbf{x}). $$

Thus, up to boundary terms, $\sigma_t$ coincides with the entropy dissipated during the time interval $[0, t]$. To clarify the physical content of this expression, we observe that it leads to the following form of the relative entropy

$$ \mathrm{ep}_t := \mathrm{Ent}(\mathbb{P}_t|\hat{\mathbb{P}}_t) = \mathbb{E}_t[\sigma_t] = \mathrm{Ent}(\mu_{\mathrm{st}}|\hat{\mu}_{\mathrm{st}}) + \Delta S_{\mathcal{R}}(t), $$

where

$$ \Delta S_{\mathcal{R}}(t) = \mathbb{E}_t[\mathcal{S}_t] = t\mu_{\text{st}}(\sigma) $$

is the expected increase in the entropy of the environment $\mathcal{R}$. Inequality (2.8) yields

$$ \mu_{\text{st}}(\sigma) = \lim_{t \to \infty} \frac{\mathrm{ep}_t}{t} \geq 0. $$

Since the configurational entropy of the system $\mathcal{S}$

$$ S_{\mathcal{S}}(t) := - \int \rho_{\text{st}} \log \rho_{\text{st}} dx $$

is constant, we can rewrite the previous inequality as

$$ \Delta S_{\mathcal{S}+\mathcal{R}}(t) = \Delta S_{\mathcal{S}}(t) + \Delta S_{\mathcal{R}}(t) \geq 0, $$

which is a microscopic form of the 2nd law of thermodynamics. Using Identity (3.22), we can further write

$$ \mu_{\text{st}}(\sigma) = \mu_{\text{st}}(L\phi + \gamma(\phi)) = \mu_{\text{st}}(\gamma(\phi)), $$

from which we conclude that $\mu_{\text{st}}(\sigma) = 0$ iff $\nabla_p \phi$ vanishes identically. Assume now that $\mu_{\text{st}}(\sigma) = 0$. Then

$$ \rho_{\text{st}}(q, p) = e^{-\frac{1}{2}p \cdot T^{-1}p + f(q)} \quad (3.25) $$

for some $f \in C^\infty(\mathbb{R}^\mathcal{V})$. Insertion into the equation $L^T \rho_{\text{st}} = 0$ yields

$$ p \cdot (\nabla f(q) + T^{-1}\nabla U(q)) = 0 $$

from which we deduce

$$ \nabla f + T^{-1}\nabla U = 0, $$

and hence

$$ \left( \frac{1}{T_i} - \frac{1}{T_j} \right) \frac{\partial^2 W}{\partial q_i \partial q_j} = 0. \quad (3.26) $$
Under our assumptions on the interaction potential W, this is only possible whenever $T_i = T_j$ for all $\{i, j\} \in \mathcal{E}$, i.e., $T_i = T_*$ for all $i \in \mathcal{V}$. This means that the environment $\mathcal{R}$ is in thermal equilibrium at temperature $T_*$. In this case, the process is stationary and in thermal equilibrium, with

$$ \mu_{\text{st}}(dx) = Z^{-1}e^{-H(x)/T_*}dx, \qquad (3.27) $$

the Gibbs measure at temperature $T_*$. Note that since $H \circ \theta = H$, the measure (3.27) is time-reversal invariant and it follows that the process is reversible, i.e., $\hat{P}_t = P_t$. Reciprocally, if the process is reversible then $\mu_{\text{st}} \circ \theta = \mu_{\text{st}}$ and the relation $L^T \rho_{\text{st}} = 0$ splits into the two equations

$$ \nabla_p \cdot \Gamma(\nabla_p + T_*^{-1}p)\rho_{\text{st}} = 0, \quad \{H, \rho_{\text{st}}\} = 0. $$

The first one implies (3.25) and the second one leads again to (3.26). Thus, the process is reversible iff the environment $\mathcal{R}$ is in thermal equilibrium and in the opposite case the mean entropy production rate

$$ \text{epr} := \lim_{t \to \infty} \frac{ep_t}{t} $$

is strictly positive and the Rényi entropy

$$ e_t(\alpha) = \text{Ent}_\alpha(\mathbb{P}_t | \hat{\mathbb{P}}_t) $$

is non-vanishing for $\alpha \in ]0, 1[$.

### 3.4.3 Fluctuation Theorems

FTs are only known to hold for a few special classes of Langevin dynamical systems. A local FT for boundary driven chains of anharmonic oscillators has been obtained in [RT02a]. The status of the global FT for general harmonic networks has been discussed in detail in [JPS17; DHP20]. Closely related examples are the local FT for Gaussian dynamical systems obtained in [Jak+12; JPS16] and the local FT for strongly dissipative stochastic partial differential equations proved in [Jak+15]. As we shall outline now, the main difficulty one encounters in such derivations lies in the lack of compactness of the phase space $\mathcal{X}$ and in the unboundedness of the entropy production random variable.

Coming back to the setup of the Langevin equation (3.17), our current assumptions imply that the Markov transition kernel $P^t$ and $\hat{P}^t$ are strong Feller. Moreover, the Lyapunov functions described in Lemma 3.1 together with [Wu01, Theorem 3.1] give that the Donsker-Varadhan theory [DV83] applies to both the direct and the time-reversed stationary processes. To formulate these results, we need to introduce some additional objects and notations.
Equip the path spaces $\Omega := C(\mathbb{R}_+, \mathcal{X})$ and $\bar{\Omega} := C(\mathbb{R}, \mathcal{X})$ with the topology of locally uniform convergence and the induced Borel structure. For a subset $I \subset \mathbb{R}$ and $\mathbf{x} = (x(s))_{s \in \mathbb{R}}$, set $\mathbf{x}_I = (x(s))_{s \in I}$ and let $\mathfrak{F}_I$ be the $\sigma$-subalgebra generated by $\mathbf{x}_I$. We denote by $\tau$ the left shift on $\Omega$, i.e., the map

$$ \mathbf{x} = (x(t))_{t \in \mathbb{R}_+} \mapsto \tau^s(\mathbf{x}) := (x(s+t))_{t \in \mathbb{R}_+}. $$

We shall use the same symbol to denote the left shift on $\bar{\Omega}$.

Any $\mathbb{Q} \in \mathcal{P}_{\tau}(\Omega)$ has a unique extension $\tilde{\mathbb{Q}} \in \mathcal{P}_{\tau}(\bar{\Omega})$, the measure determined by $\tilde{\mathbb{Q}}(\Gamma) = \mathbb{Q} \circ \tau^{-t}(\Gamma)$ for $\Gamma \in \mathcal{F}_{[-t, \infty]}$. The map $\tilde{\mathbb{Q}} \mapsto \tilde{\mathbb{Q}}$ is bijective, its inverse being the restriction $\mathcal{P}_{\tau}(\bar{\Omega}) \ni \tilde{\mathbb{Q}} \mapsto \mathbb{Q}|_{\tilde{\mathfrak{F}}_{[0, \infty]}} \in \mathcal{P}_{\tau}(\Omega)$. Let $\theta$ be the time reversal defined on $\bar{\Omega}$ by

$$ \mathbf{x} = (x(t))_{t \in \mathbb{R}} \mapsto \theta(\mathbf{x}) := (\theta x(-t))_{t \in \mathbb{R}}, $$

and denote by $\tilde{\mathbb{Q}} := \mathbb{Q} \circ \theta$ the involution induced on $\mathcal{P}_{\tau}(\bar{\Omega})$. Setting $\tilde{\mathbb{Q}} := \tilde{\mathbb{Q}}|_{\tilde{\mathfrak{F}}_{[0, \infty]}}$ defines an involution on $\mathcal{P}_{\tau}(\Omega)$ such that $\tilde{\mathbb{Q}} = \tilde{\mathbb{Q}}$. In particular, if $\mathbb{P} \in \mathcal{P}_{\tau}(\Omega)$ is the measure induced by the family $(\mathbb{P}_t)_{t \in \mathbb{R}_+}$, then $\tilde{\mathbb{P}}$ coincides with the measure induced by the family $(\tilde{\mathbb{P}}_t)_{t \in \mathbb{R}_+}$.

For $\mathbb{Q} \in \mathcal{P}_{\tau}(\bar{\Omega})$ and $t \in \mathbb{R}$ we set $\mathbb{Q}_{[t]} := \mathbb{Q}|_{\tilde{\mathfrak{F}}_{[-\infty, t]}}$ and define

$$ \tilde{\mathbb{Q}}_{[0]} \otimes \mathbb{P}_t(\mathrm{d}\mathbf{x}) := \tilde{\mathbb{Q}}(\mathrm{d}\mathbf{x}_{[-\infty, 0]})\mathbb{P}_{x(0), t}(\mathrm{d}\mathbf{x}_{[0, t]}). $$

**Level-3 FT and FR.** The process-level empirical measures $\xi_t \in \mathcal{P}(\Omega)$ defined by

$$ \Omega \ni \mathbf{x} \mapsto \xi_t := \frac{1}{t} \int_0^t \delta_{\tau^s(\mathbf{x})} \mathrm{d}s $$

satisfy the LDP w.r.t. the measures $\mathbb{P}$ and $\tilde{\mathbb{P}}$ with scale $r_t = t$, the good rate functions being given by [DV83, Theorems 4.6 and 5.5]

$$ \mathbb{I}(\mathbb{Q}) := \begin{cases} \mathrm{Ent}(\tilde{\mathbb{Q}}_1|\tilde{\mathbb{Q}}_0) \otimes \mathbb{P}_1 & \text{if } \mathbb{Q} \in \mathcal{P}_{\tau}(\Omega); \\ +\infty & \text{otherwise,} \end{cases} \qquad \hat{\mathbb{I}}(\mathbb{Q}) := \begin{cases} \mathrm{Ent}(\tilde{\mathbb{Q}}_1|\tilde{\mathbb{Q}}_0) \otimes \tilde{\mathbb{P}}_1 & \text{if } \mathbb{Q} \in \mathcal{P}_{\tau}(\Omega); \\ +\infty & \text{otherwise.} \end{cases} $$

Since

$$ \log \frac{\mathrm{d}\tilde{\mathbb{Q}}_0|\otimes \mathbb{P}_1}{\mathrm{d}\tilde{\mathbb{Q}}_0|\otimes \tilde{\mathbb{P}}_1} = \log \frac{\mathrm{d}\mathbb{P}_{x(0),1}}{\mathrm{d}\tilde{\mathbb{P}}_{x(0),1}} = \phi(x(0)) - \phi(x(1)) + \int_0^1 \sigma(x(s)) \mathrm{d}s, $$

it immediately follows that the level-3 FR

$$ \hat{\mathbb{I}}(\mathbb{Q}) = \mathbb{I}(\mathbb{Q}) + \int \sigma(x(0)) \mathbb{Q}(\mathrm{d}\mathbf{x}) \qquad (3.28) $$

holds for all $\mathbb{Q} \in \mathcal{P}_{\tau}(\Omega)$ such that both $\phi(x(0))$ and $\sigma(x(0))$ belong to $L^1(\Omega, \mathbb{Q})$. Let us denote by $\phi_{\pm}$ the positive/negative part of $\phi$. We shall see in the next paragraph that $\sigma(x(0))$ and $\phi_{+}(x(0))$ belong to $L^1(\Omega, \mathbb{Q})$ whenever $\mathbb{I}(\mathbb{Q}) < \infty$. Thus,
(3.28) holds for all stationary processes $\mathbb{Q}$ such that $\|\mathbb{Q}\| < \infty$ and$^7$ $\phi_-(x(0)) \in L^1(\Omega, \mathbb{Q})$. Moreover, invoking the uniqueness of the rate function one easily derives the relation

$$ \hat{\mathbb{I}}(\mathbb{Q}) = \|\hat{\mathbb{I}}(\mathbb{Q})\| \quad (3.29) $$

for all $\mathbb{Q} \in \mathcal{P}_T(\Omega)$.

We shall now proceed to lower levels FR/FT via contractions. In order to keep the discussion to a reasonable level of technicality, we will only outline the argument and leave two technical lemmata to the Appendix.

**Level-2 FT and FR.** By contraction, the level-2 empirical measures $\xi_t \in \mathcal{P}(\mathcal{X})$, defined by

$$ \Omega \ni \mathbf{x} \mapsto \xi_t := \frac{1}{t} \int_0^t \delta_{x(s)} \mathrm{d}s, $$

satisfy a LDP w.r.t. the families $(\mathbb{P}_t)_{t \in \mathbb{R}_+}$ and $(\hat{\mathbb{P}}_t)_{t \in \mathbb{R}_+}$ with scale $r_t = t$ and good rate functions

$$ \mathbf{I}(\mathbb{Q}) = \inf \left\{ \|\mathbb{I}(\mathbb{Q})\| \mid \mathbb{Q} \in \mathcal{P}_T(\Omega), \int f(x(0)) \mathbb{Q}(\mathrm{d}\mathbf{x}) = \int f(x) \mathbb{Q}(\mathrm{d}\mathbf{x}) \text{ for all } f \in C_b(\mathcal{X}) \right\}, $$

$\hat{\mathbf{I}}$ being defined similarly. As immediate consequences of this formula, we observe that Relation (3.29) yields that

$$ \hat{\mathbf{I}}(\mathbb{Q}) = \mathbf{I}(\mathbb{Q} \circ \theta), \quad (3.30) $$

and that $\|\mathbb{I}(\mathbb{Q})\| < \infty$ implies $\mathbb{Q} \in \mathcal{P}_T(\Omega)$ with a marginal $\mathbb{Q} \in \mathcal{P}(\mathcal{X})$ satisfying $\mathbf{I}(\mathbb{Q}) \le \mathbf{I}(\mathbb{Q}) < \infty$. The Donsker-Varadhan variational formula (see [DV76] or [DS89, Theorem 4.2.43]),

$$ \mathbf{I}(\mathbb{Q}) = \sup_{\varphi} \int (-L\varphi - \gamma(\varphi)) \mathrm{d}\mathbb{Q}, $$

and its analog for $\hat{\mathbf{I}}$ hold, the supremum being taken over all real-valued $C^2$-functions which are bounded below on $\mathcal{X}$. By [Jai90, Proposition 2.8], $\mathbf{I}(\mathbb{Q}) < \infty$ implies that $\mathbb{Q} \ll \mu_{\mathrm{st}}$ and hence that $\mathbb{Q}$ has a density w.r.t. the local equilibrium measure $\nu$. Invoking [Wu01, Corollary 2.2], it follows from the Lyapunov estimates of Lemma 3.1 that (recall Equ. (3.19))

$$ \int |\nabla S|^2 \mathrm{d}\mathbb{Q} < \infty $$

whenever $\mathbf{I}(\mathbb{Q}) < \infty$ or $\hat{\mathbf{I}}(\mathbb{Q}) < \infty$. A direct consequence is that $\sigma \in L^1(\mathcal{X}, \mathbb{Q})$, and invoking the upper bound (3.42) one can further deduce that $\phi_+ \in L^1(\mathcal{X}, \mathbb{Q})$.<fnref n="8" /> We shall prove in Lemma 3.2 that under the same condition $\gamma(\phi) \in L^1(\mathcal{X}, \mathbb{Q})$ holds and that this implies the level-2 FR

<footnote n="7">⁷ We are not aware of any estimate of $\phi_-$ allowing us to deduce that $\phi_-(x(0)) \in L^1(\Omega, \mathbb{Q})$ from the fact that $\|\mathbb{I}(\mathbb{Q})\| < \infty$.</footnote>

<footnote n="8">⁸ This establishes the claim made in the previous paragraph.</footnote>
$$ \hat{I}(Q) = I(Q) + \langle \sigma, Q \rangle, \qquad (3.31) $$

for all $Q \in \mathcal{P}(\mathcal{X})$ such that either $I(Q) < \infty$ or $\hat{I}(Q) < \infty$.

**Remark.** For $Q \in \mathcal{P}(\mathcal{X})$ such that $Q \ll v$ with

$$ \log \frac{dQ}{dv} \in \mathcal{D} := \left\{ \varphi \in C^2(\mathcal{X}) \mid |\varphi| + |\nabla_p \varphi| \in L^2(\mathcal{X}, dQ) \right\}, \qquad (3.32) $$

one can decompose the level-2 rate function as

$$ I(Q) = K(Q) + I(Q|v) - \frac{1}{2} \langle \sigma, Q \rangle, $$

(see [BL08, Proposition 2]), with

$$ K(Q) = \sup_{\varphi \in \mathcal{D}} \langle \{H, \varphi\} - \gamma(\varphi), Q \rangle, $$

and the relative Fisher information

$$ I(Q|v) = 4 \int \gamma \left( \sqrt{\frac{dQ}{dv}} \right) dv. $$

One easily checks that both $K$ and $I(\cdot|v)$ are even under $\theta$, so that the FR (3.31) follows for $Q$ satisfying (3.32).

**Level-1 FT and FR.** Formally, the FT for our Langevin process follows from (3.23) by contraction of the Donsker-Varadhan level-2 LDP, the rate function being given by

$$ I(s) = \inf \{ I(Q) \mid Q \in \mathcal{P}_T(\mathcal{X}), \langle \sigma, Q \rangle = s \}. \qquad (3.33) $$

The corresponding FR (2.21)-(2.22) are directly inherited from the level-2 relations (3.31)-(3.30) and the fact that $\sigma \circ \theta = -\sigma$. The two obstructions to turning this formal argument into a rigorous proof stem from the lack of compactness of the phase space $\mathcal{X}$:

(1) Due to the unboundedness of $\phi$, the additive functional $\tilde{\sigma}_t$ defined by

$$ \tilde{\sigma}_t(\mathbf{x}) := \int_0^t \sigma(x(s))ds $$

is not physically equivalent to the entropy production $\sigma_t$ given by (3.23). We note that the same problem occurs when trying to derive a fluctuation theorem for the dissipated thermodynamic entropy $\mathcal{S}_t$, the role of $\phi$ being there played by $S$.

(2) The unboundedness of $\sigma$ makes the map

$$ \mathcal{P}(\mathcal{X}) \ni Q \mapsto \langle \sigma, Q \rangle $$

discontinuous for the weak topology of $\mathcal{P}(\mathcal{X})$.
Thus, under the current assumptions, it is not known whether our Langevin process satisfies the FT in the canonical form of Definition 2.2. We shall leave this important question for future investigations. On the positive side, since $|σ(x)| ≤ C|∇S(x)|$ for some constant C, invoking [Wu01, Corollary 2.3] and Lemma 3.1 yields that the functional $\tilde{\sigma}_t$ satisfies the FT with rate function (3.33).

From a more conceptual perspective, one may argue that the level-3 and level-2 FR (3.28) and (3.31), which only depend on time-reversal, are more fundamental than the level-1 FR (2.21)-(2.22) which also depend on the choice of an entropy production random variable. With this point of view, the natural entropy production random variable is $\tilde{\sigma}_t$, and it satisfies the global FT. Rewriting (3.24) as

$$ \tilde{\sigma}_t(\mathbf{x}) = \mathcal{G}_t(\mathbf{x}) + S(x(t)) - S(x(0)), $$

we express the total entropy production as the sum of the entropy dissipated into the environment and the increase in some local entropy. The latter vanishes on average since the system is in a steady state, but this does not prevent fluctuations which contribute to the fluctuations of $\tilde{\sigma}_t$.

It is instructive to compare the situation just described with what happens in purely harmonic networks.<fnref n="9" /> There, one can show that the global FT holds for the entropy production $\sigma_t$ with a rate function satisfying the FR (2.21)-(2.22). The dissipated thermodynamic entropy $\mathcal{G}_t$ also satisfies the global LDP. However, the corresponding rate function only satisfies the FR

$$ \Delta(s) := I(-s) - I(s) = s $$

for $s$ in some finite interval around 0. Outside of this interval, the symmetry function $\Delta$ becomes non-universal. We refer to [JPS17] and references therein for a discussion of this phenomenon which leads to what is sometimes referred to as the extended FT in the physics literature. Invoking [JPS17, Proposition 3.22], one deduces that a similar scenario holds for the additive functional $\tilde{\sigma}_t$. This is illustrated in Figure 3.5. There, we consider a triangular harmonic network with potential

$$ U(q) := \frac{1}{2}(q_1^2 + q_2^2 + q_3^2) - \frac{1}{4}(q_1 q_2 + q_2 q_3 + q_3 q_1). $$

The ratios of the reservoirs temperatures are given by

$$ T_1 : T_2 : T_3 = 2 - \tau : 3 : 1 + \tau, \quad \tau \in ]0, 1/2[, $$

and $\Gamma$ is the unit matrix. On the interval $|1/2 - \kappa_c(\tau)|, 1/2 + \kappa_c(\tau)|$ the entropic pressure is finite. It is infinite on the interior of its complement, and its derivative diverges at its boundary. We have chosen 3 values of $\tau$ which are the largest solutions of $\kappa_c(\tau) \in \{1, 3/2, 2\}$. Figure 3.5 displays the departure of the symmetry function $\Delta(s)$ from the straight line $\Delta(s) = s$ for the rate function of the dissipated

<footnote n="9">⁹ Note that in such systems the potentials $V$ and $W$ are quadratic functions. Hence, $W$ is unbounded. Harmonic networks therefore do not fall in the framework described in this subsection.</footnote>
thermodynamic entropy $\mathcal{S}_t$ (left pane) and of the additive functional $\tilde{\sigma}_t$ (right pane).

<figure><img src="image_15.png" /><figcaption>Fig. 3.5 The symmetry function $\Delta$ for the triangular network described in the text. On the left: the case of the dissipated thermodynamic entropy $\mathcal{S}_t$ in the steady state. On the right: the case of the additive functional $\tilde{\sigma}_t$. The unit on both axes in both figures is the entropy production rate epr.</figcaption></figure>

## 3.5 Reversible Smooth Dynamical Systems

### 3.5.1 Setup

Let $(\varphi^t)_{t \in \mathcal{L}}$ be a group of homeomorphisms of a Polish space $\Omega$, with $\mathcal{L} = \mathbb{Z}$ or $\mathbb{R}$. In the second case, we also assume that $\varphi$ is continuous as a map from $\mathcal{L} \times \Omega$ to $\Omega$. Assume that $\mu \in \mathcal{P}(\Omega)$ is such that $\mu_t := \mu \circ \varphi^{-t}$ is equivalent to $\mu$ for all $t \in \mathcal{L}$. For $\tau \in \mathcal{L}_+ := \mathcal{L} \cap [0, \infty[$, set $\Omega_\tau := C([0, \tau], \Omega)$ if $\mathcal{L} = \mathbb{R}$ and $\Omega_\tau := \Omega^{[0, \tau]}$ if $\mathcal{L} = \mathbb{Z}$. Define the map $\phi_\tau : \Omega \to \Omega_\tau$ by $\phi_\tau(x) := (\varphi^t(x))_{t \in [0, \tau] \cap \mathcal{L}}$ and the path space measure $\mathbb{P}_{\mu, \tau}$ on $\Omega_\tau$ by

$$ \mathbb{P}_{\mu, \tau} := \mu \circ \phi_{\tau}^{-1} = \int \delta_{\phi_{\tau}(x)} \mu(dx). $$

Let $\theta$ be a time reversal, i.e., an involution of $\Omega$ such that $\varphi^t \circ \theta = \theta \circ \varphi^{-t}$. Assume also that $\hat{\mu} := \mu \circ \theta$ is equivalent to $\mu$. Define a $\mathbb{Z}_2$-action on the path space $\Omega_\tau$ by

$$ \mathbf{x} = (x(t))_{t \in [0, \tau] \cap \mathcal{L}} \mapsto \Theta_{\tau}(\mathbf{x}) := (\theta x(\tau - t))_{t \in [0, \tau] \cap \mathcal{L}}. $$
Setting $\hat{\mathbb{P}}_{\mu,\tau} = \mathbb{P}_{\mu,\tau} \circ \Theta_{\tau}$, we obtain a pair $(\mathbb{P}_{\mu,\tau}, \hat{\mathbb{P}}_{\mu,\tau})_{\tau \in \mathcal{L}_+}$ in involution. Its entropy production random variable is

$$ \sigma_{\tau}(\mathbf{x}) = \log \frac{d\mathbb{P}_{\mu,\tau}}{d\hat{\mathbb{P}}_{\mu,\tau}}(\mathbf{x}) = \log \frac{d\mu}{d\hat{\mu}_{\tau}}(x(0)) = \log \frac{d\mu}{d\hat{\mu}}(x(0)) - \log \frac{d\mu_{\tau}}{d\mu}(\theta(x(0))) $$

where $\hat{\mu}_{\tau} = \mu_{\tau} \circ \theta = \hat{\mu} \circ \varphi^{\tau}$. Setting

$$ \kappa_{\tau} := \log \frac{d\mu_{\tau}}{d\mu}, \qquad \theta := \log \frac{d\mu}{d\hat{\mu}}, $$

one easily derives the cocycle relation

$$ \kappa_{\tau+t} = \kappa_{\tau} + \kappa_t \circ \varphi^{-\tau}. $$

It follows that in the case $\mathcal{L} = \mathbb{Z}$,

$$ \sigma_{\tau}(\mathbf{x}) = \vartheta(x(0)) + \sum_{t=0}^{\tau-1} \tilde{\sigma}(x(t)), \qquad \tilde{\sigma}(x) := -\log \frac{d\mu_1}{d\mu} \circ \theta(x), $$

and in the case $\mathcal{L} = \mathbb{R}$, under appropriate regularity assumptions, that

$$ \sigma_{\tau}(\mathbf{x}) = \vartheta(x(0)) + \int_{0}^{\tau} \tilde{\sigma}(x(t))dt, \qquad \tilde{\sigma}(x) := -\left. \frac{d}{dt} \frac{d\mu_t}{d\mu} \right|_{t=0} \circ \theta(x). $$

Assuming that there exists a reference measure $\mathbb{V}$ on $\Omega$ such that $\mathbb{V} \circ \theta = \theta$ and $\mu$ is equivalent to $\mathbb{V}$, one can rewrite the above formulas as

$$ \sigma_{\tau}(\mathbf{x}) = \log \rho(x(0)) - \log \rho \circ \theta(x(\tau)) + \sum_{t=0}^{\tau-1} \sigma(x(t)), \qquad (3.34) $$

with

$$ \sigma := \log \frac{d\mathbb{V} \circ \varphi^{-1}}{d\mathbb{V}} \circ \varphi \qquad \rho := \frac{d\mu}{d\mathbb{V}}, $$

in the case $\mathcal{L} = \mathbb{Z}$, and

$$ \sigma_{\tau}(\mathbf{x}) = \log \rho(x(0)) - \log \rho \circ \theta(x(\tau)) + \int_{0}^{\tau} \sigma(x(t))dt \qquad (3.35) $$

with

$$ \sigma := \left. \frac{d}{dt} \frac{d\mathbb{V} \circ \varphi^{-t}}{d\mathbb{V}} \right|_{t=0}, \qquad \rho := \frac{d\mu}{d\mathbb{V}}, $$

in the case $\mathcal{L} = \mathbb{R}$.

We note that the situation described in this example is similar to the one discussed in [JPR11], with the only difference that we do not require the measure $\mu$ to be $\theta$-invariant. The boundary terms

$$ \log \rho(x(0)) - \log \rho \circ \theta(x(\tau)) \qquad (3.36) $$
in (3.34) and (3.35) reflect the lack of time-reversal invariance of $\mu$. Indeed, if $\mu \circ \theta = \mu$ then we can set $\mathbb{V} = \mu$ and hence $\log \rho \equiv 0$ in these formulas, and we recover the results of [JPR11] for the entropy production random variable.

### 3.5.2 Phase Space Contraction Rate

Continuing with the previous setting, let us further assume that $\Omega$ is equipped with the structure of a smooth Riemannian manifold and that $\varphi$ is a diffeomorphism in the case $\mathcal{L} = \mathbb{Z}$, while the flow $t \mapsto \varphi^t$ is generated by a smooth vector field $X$ in the case $\mathcal{L} = \mathbb{R}$. Denote by $\mathbb{V}$ a Riemannian volume measure on $\Omega$ and assume that $\mathbb{V} \circ \theta = \mathbb{V}$ and that $\mu$ and $\mathbb{V}$ are equivalent. Then, Relations (3.34) and (3.35) hold with

$$ \rho = \frac{d\mu}{d\mathbb{V}}, \quad \sigma = \begin{cases} -\log|\det(\varphi^t)|, & \text{if } \mathcal{L} = \mathbb{Z}; \\ -\text{div}(X), & \text{if } \mathcal{L} = \mathbb{R}. \end{cases} \qquad (3.37) $$

Thus, up to the boundary term (3.36), the entropy production random variable $\sigma_{\tau}$ is directly related to the phase space contraction rate $\sigma$.

### 3.5.3 Transient Fluctuation Relations, Fluctuation Theorem and Steady States

Physically interesting examples falling in the present context are Hamiltonian systems driven out of equilibrium by external forces and coupled to Gaussian thermostats, see [JPR11] and references therein. It is precisely for such systems that *Transient FRs* were first discovered in the study of Second Law violating microstates [ECM93; ES94]. Inequality (2.8) can be written as<fnref n="10" />

$$ \int \left( \frac{1}{\tau} \sum_{t=0}^{\tau-1} \sigma \circ \varphi^t(x) \right) \mu(dx) \geq \frac{1}{\tau} \int \left( \log \rho \circ \theta \circ \varphi^\tau(x) - \log \rho(x) \right) \mu(dx). $$

Assuming that<fnref n="11" />

$$ R = \sup_{x \in \Omega} |\log \rho(x)| < \infty, \qquad (3.38) $$

we derive the following asymptotic inequality for the time averaged phase space contraction rate

$$ \liminf_{\tau \to \infty} \int \left( \frac{1}{\tau} \sum_{t=0}^{\tau-1} \sigma \circ \varphi^t(x) \right) \mu(dx) \geq 0. $$

<footnote n="10">¹⁰ For simplicity we shall only consider the case $\mathcal{L} = \mathbb{Z}$, the other case being similar.</footnote>

<footnote n="11">¹¹ Observe that $R = 0$ whenever $\mu$ is a $\theta$-invariant Riemannian volume.</footnote>