Fluctuation Relations for Diffusion Processes

Introduction

Nonequilibrium statistical mechanics attempts a statistical description of closed and open systems evolving under the action of time-dependent conservative forces or under time-independent or time dependent non-conservative ones. Fluctuation relations are robust identities concerning the statistics of entropy production or performed work in such systems. They hold arbitrarily far from thermal equilibrium. Close to equilibrium, they reduce to Green-Kubo or fluctuation-dissipation relations, usually obtained in the scope of linear response theory . Historically, the study of fluctuation relations originated in the numerical observation of Evans, Cohen and Morriss of a symmetry in the distribution of fluctuations of microscopic pressure in a thermostatted particle system driven by external shear. The symmetry related the probability of occurrence of positive and negative time averages of pressure over sufficiently long time intervals and predicted that the former is exponentially suppressed with respect to the latter. Ref. attempted to explain this observation by a symmetry, induced by the time-reversibility, of the statistics of partial sums of finite-time Lyapunov exponents in dissipative dynamical systems. This was further elaborated in where an argument was given explaining such a symmetry in a transient situation when one starts with a simple state which evolves under dynamics, see also . In refs. , Gallavotti and Cohen provided a theoretical explanation of the symmetry observed numerically in employing the theory of uniformly hyperbolic dynamical systems. In this theory, the stationary states correspond to invariant measures of the SRB type and the entropy production is described by phase-space contraction . The authors of established a fluctuation theorem about the rate function describing the statistics of large deviations of the phase-space contraction in the time-reversible dynamics. To relate to the behavior of realistic systems, they formulated the chaotic hypothesis postulating that many such systems behave, for practical purposes, as the uniformly hyperbolic ones. They interpreted the numerical observations of ref. as a confirmation of this hypothesis. The difference between the fluctuation relations for a transient situation analyzed in and the stationary one discussed in was subsequently stressed in . The debate about the connection between the transient and stationary fluctuation relations still continues, see e.g. and .

In another early development, Jarzynski established in a simple transient relation for the statistics of fluctuations of work performed on a system driven by conservative time-dependent forces. This relation is now known under the name of Jarzynski equality. A similar observation, but with more limited scope, was contained in the earlier work , see for a recent comparison. The simplicity of the Jarzynski equality and its possible applications to measurements of free-energy landscape for small systems attracted a lot of attention, see and the references therein.

The first studies of fluctuation relations dealt with the deterministic dynamics of finitely-many degrees of freedom. Such dynamics may be also used to model systems interacting with environment or with heat reservoirs. To this end, one employs simplified finite-dimensional models of reservoirs forced to keep their energy constant . This type of models was often used in numerical simulations and in discussing fluctuation relations, see e.g. . A more realistic treatment of reservoirs would describe them as infinite systems prepared in the thermal equilibrium state. Up to now, only infinite systems of non-interacting particles could be treated effectively, see . A less realistic description of interaction with environment or with reservoirs consists of replacing them by a random noise, usually shortly correlated in time. This leads to Markovian stochastic evolution equations. Stochastic models are often easier to control than deterministic ones and they became popular in modeling nonequilibrium dynamics.

In , Jarzynski generalized his relation to time-dependent Markov processes with the instantaneous generators satisfying the detailed balance relation. At almost the same time, Kurchan has shown in that the stationary fluctuation relations hold for the stochastic Langevin-Kramers evolution. His result was extended to more general diffusion processes by Lebowitz and Spohn in . In , Maes has traced the origin of fluctuation relations to the Gibbsian nature of the statistics of the dynamical histories, see a recent discussion of fluctuation relation from this point of view in . Searles and Evans generalized there transient fluctuation relation to the stochastic setup in . Finally, within the stochastic approach, the scope of the transient fluctuation relations was further extended due to the works of Crooks , Jarzynski , Hatano and Sasa , Speck and Seifert and Chernyak, Chertkov and Jarzynski , just to cite only the papers that influenced most the present authors. It is worth stressing that the general transient fluctuation relations do not impose the time reversibility of the dynamics but compare the fluctuation statistics of the original process and of its time reversal. Such an extension of the scope of fluctuation relations is a possibility in the stationary case as well, but it becomes a necessity in many transient situations. Within the theory of the hyperbolic dynamical systems, the stationary fluctuation theorem of was recently generalized to the random dynamics in .

In , Balkovsky Falkovich and Fouxon noticed another robust relation concerning the large deviations of finite-time Lyapunov exponents in the context of homogeneous hydrodynamic flows. It was remarked in , that this observation, which we shall call, following , the multiplicative fluctuation relation, provides an extension of the previously known fluctuation relations for the phase-space contraction. The simple argument presented in dealt with a transient situation. It was very similar to the original Evans-Searles argument as formulated later in . The multiplicative fluctuation relation was explicitly checked in the Kraichnan model of hydrodynamic flows .

The theoretical work on fluctuation relations has established most of them as mathematical identities holding within precisely defined models, but concerning statistics of events that are rare, especially for macroscopic systems. The relevance of such identities to numerical simulations and, even more, to real experiments, required a confirmation. Numerical (see e.g. ) and experimental testing of the fluctuation relations (see e.g. ) has attracted over years a lot of attention, inspiring further developments. It will probably remain an active field in the future. It is not, however, the topic of the present paper.

The growing number of different fluctuation relations made urgent a development of a unifying approach. Several recent reviews partially provided such a unification from different points of view, see ref. . In the present paper, we attempt another synthesis, with the aim of supplying a uniform derivation of most of the known fluctuation relations, including the multiplicative ones. We shall work in the setup of (possibly non-autonomous) diffusion processes in finite-dimensional spaces, somewhat similar, but more general that the one adopted in . The systems considered include, as special cases, the deterministic dynamics, the Langevin stochastic equation, and the Kraichnan model of hydrodynamic flow. This is certainly not the most general setup possible for discussing fluctuation relations (for example, the discrete-time dynamics, the stochastic dynamics with jumps, or non-Markovian evolutions are not covered), but it is general enough for a unified discussion of a variety of aspects of fluctuation relations. Most of our considerations are simple extensions of arguments that appeared earlier in usually more constrained contexts. There are two basic ideas that we try to exploit to obtain a larger flexibility than in the previous discussions of fluctuation relations. The first one concerns the possible time-reversed processes that we admit. This idea appeared already in , where two different time inversions were used for the Langevin dynamics with non-conservative forces, leading to two different backward processes and two different fluctuation relations. We try to exploit the freedom of choice of the time-inversion in a more systematic way. The second idea, which seems original to us, although it is similar in spirit to the first one, is to obtain new fluctuation relations by considering new diffusion processes derived from the original one. In particular, we show that the multiplicative fluctuation relations for general diffusion processes may be obtained by writing a more standard relation for the tangent diffusion process describing a simultaneous evolution of infinitesimally close trajectories of the original process. The same idea may be used to explain additional fluctuation relations, like the one for the rate function of the difference of finite-time Lyapunov exponents “along unstable flag” that was observed in for the anisotropic Kraichnan model.

The present paper is organized as follows. In Sect. 2, we define the class of diffusion processes that will be discussed and list four special cases. Sect. 3 recalls the notions of transition probabilities and generators of a diffusion process, as well as the detailed balance relation. In Sect. 4, we introduce the tangent diffusion process induced form the original one and define the phase-space contraction. Time inversions leading to different backward processes are discussed in Sect. 5, with few important examples listed in Sect. 6. A formal relation between the expectations in the forward and in the backward process is introduced in Sect. 7. As examples, we discuss the case of tangent process in the homogeneous Kraichnan flow, a simple generalization of the detailed balance relation and the $1^{st}$ law of thermodynamics for the Langevin dynamics. Sect. 8 is devoted to a general version of the Jarzynski equality, whose different special cases are reviewed, and Sect. 9 to a related equality established by Speck and Seifert in . We formulate the Jarzynski equality as a statement that for a certain functional $\,{\cal W}\,$ of the diffusion process, the expectation value of $\,{\rm e}^{-{\cal W}}\,$ is normalized. In Sect. 10, the functional $\,{\cal W}\,$ is related to the entropy production and the positivity of its expectation value is interpreted as the $2^{\rm nd}$ law of thermodynamics for the diffusive processes. In Sect. 11, we show how the general Jarzynski equality reduces in the linear response regime to the Green-Kubo and Onsager relations for the transport coefficient and to the fluctuation-dissipation theorem. In Sect. 12, we discuss briefly a peculiar one-dimensional Langevin process in which the equilibrium is spontaneously broken and replaced by a state with a constant flux, leading to a modification of the fluctuation-dissipation relation. The model is well known from the theory of one-dimensional Anderson localization and describes also the separation of infinitesimally close particles with inertia carried by a one-dimensional Kraichnan flow. Sect. 13.3 formulates in the general setup of diffusion processes what is sometimes termed a detailed fluctuation relation , an extension of the Crooks fluctuation relations . Few special cases are retraced in Sect. 14.

Up to this point of the paper, the discussion is centered on the transient evolution where the system is initially prepared in a state that changes under the dynamics. In Sect. 15, we discuss the relation of the transient fluctuation relations to the stationary ones which pertain to the situation where the initial state is preserved by the evolution. The stationary relations are usually written for the rate function of large deviations of entropy production observed in the long-time regime. In our case, they describe the long time asymptotics of the statistics of $\,{\cal W}$. The Gallavotti-Cohen relation was the first example of such identities. We show how the fluctuation relation for the tangent process in the homogeneous Kraichnan flow discussed in Sect. 7 leads to a generalization of the Gallavotti-Cohen relation that involves the large-deviations rate function of the so called stretching exponents whose sum describes the phase-space contraction. In Sect. 16, we extend such a multiplicative fluctuation relation to the case of general diffusion processes. Sect. 17 contains speculation about possible versions of fluctuation relations for multi-point motions and Sect. 18 collects our conclusions. Few simple but more technical arguments are deferred to Appendices in order not to overburden the main text, admittingly already much more technical than most of the work on the subject. Some of the technicalities are due to a rather careful treatment of the intricacies related to the conventions for the stochastic differential equations that are usually omitted in the physical literature. The aim at generality, even without pretension of mathematical rigor, places the stress on the formal aspects and makes this exposition rather distant from physical discourse, although we make an effort to include many examples that illustrate general relations in more specific situations. The physical content is, however, more transparent in examples to such examples which are scarce in the present text but which abound in the existing literature to which we often refer. Certainly, the paper will be too formal for many tastes, and we take precautions to warn the potential reader who can safely omit the more technical passages.

Acknowledgements. The authors are grateful to S. Ciliberto, G. Falkovich, I. Fouxon, G. Gallavotti and P. Horvai for discussions.

Forward process

As mentioned in Introduction, the present paper deals with non-equilibrium systems modeled by diffusion processes of a rather general type. More concretely, the main objects of our study are the stochastic processes $\,{\mathsf{x}}_{t}\,$ in $\boldsymbol{R}^{d}$ (or, more generally, on a $d$-dimensional manifold), described by the differential equation

where $\,\dot{x}\equiv\frac{dx}{dt}$ and, on the right hand side, $\,u_{t}(x)\,$ is a time-dependent deterministic vector field (a drift), and $\,v_{t}(x)\,$ is a Gaussian random vector field with mean zero and covariance

Due to the white-noise nature of the temporal dependence of $\,v_{t}\,$ (typical $\,v_{t}\,$ are distributional in time), Eq. (2.1) is a stochastic differential equation (SDE). We shall consider it with the Stratonovich conventionThe choice of the Stratonovich convention guarantees that $\,u_{t}\,$ and $\,v_{t}\,$ transform as vector fields under a change of coordinates. , keeping for the Stratonovich SDEs the notation of the ordinary differential equations (ODEs). Examples of systems described by Eq. (2.1) include four special cases that we shall keep in mind.

Here $\,v_{t}(x)\equiv 0\,$ and $\,D_{t}^{ij}(x,y)\equiv 0\,$ so that Eq. (2.1) reduces to the ODE

Example 2. Lagrangian flow in the Kraichnan model

This is a process used in modeling turbulent transport. The SDE (2.1), where one usually takes $\,u_{t}(x)\equiv 0$, describes the motion of tracer particles in a stationary Gaussian ensemble of velocities $\,v_{t}(x)\,$ white in time. Such an ensemble, with an appropriate time-independent spatial covariance $\,D^{ij}(x,y)$, was designed by Kraichnan to mimic turbulent velocities. In particular, homogeneous flows are modeled by imposing the translation invariance $\,D^{ij}(x,y)=D^{ij}(x-y)\,$ and isotropic ones by assuming that $\,D^{ij}(x,y)\,$ is rotation-covariant. In this paper, we shall consider only the case when $\,D^{ij}(x,y)\,$ is smooth. A discussion of the case with $\,D^{ij}(x,y)\,$ non-smooth around the diagonal, pertaining to the fully developed turbulence, may be found in , or, on the mathematical level, in .

Here Eq. (2.1) takes the formWe use throughout the paper the summation convention.

where $\,\Gamma\,$ is a constant non-negative matrix and $\,\Pi\,$ an antisymmetric one, the Hamiltonian $\,H_{t}$ is a, possibly time dependent, function, $\,G_{t}\,$ is an additional force, and $\,\zeta_{t}\,$ is the $\,d$-dimensional white noise with the covariance

In this example, the white noise $\,\zeta_{t}\,$ that plays the role of the (space-independent) random vector field $\,v_{t}\,$ so that $\,D_{t}^{ij}(x,y)=2\beta^{-1}\Gamma^{ij}$. For $\,G_{t}\equiv 0\,$ and a time independent Hamiltonian $\,H_{t}\equiv H$, the Langevin dynamics is used to model the approach to thermal equilibrium at inverse temperature $\,\beta$ . The deterministic vector field $\,-\Gamma^{ij}\partial_{j}H\,$ drives the solution towards the minimum of $\,H\,$ (if it exists) whereas the Hamiltonian vector field $\,\Pi^{ij}\,\partial_{j}H\,$ preserves $\,H$. The noise $\,\zeta_{t}\,$ generates the thermal fluctuations of the solution. Note that its spatial covariance is aligned with the matrix $\,\Gamma\,$ appearing in the dissipative force $\,-\Gamma^{ij}\partial_{j}H\,$ (such an alignment, known from Einstein’s theory of Brownian motion, is often called the Einstein relation). Inclusion of the Hamiltonian vector field permits to model systems where the noise acts only on some degrees of freedom, e.g. the ones at the ends of a coupled chain, with the rest of the degrees of freedom undergoing a Hamiltonian dynamics. The introduction of a time-dependence and/or of the force $\,G_{t}\,$ permits to model nonequilibrium systems. In the particular case of vanishing $\,\Gamma$, the SDE (2.4) reduces to the ODE

describing a deterministic Hamiltonian dynamics in the presence of an additional force $\,G_{t}$.

This is a special case of the Langevin dynamics that takes place in the phase space of $n$ degrees of freedom with $\,x=(q,p)\,$ and

where $\,\gamma\not=0\,$ is a non-negative $n\times n$ matrix, $\,m^{-1}\,$ a positive one, and $\,1\,$ the unit one. Here, Eq. (2.4) reduces to the standard relation $\,p_{i}=m_{ij}\dot{q}^{j}\,$ between momenta and velocities, where $\,m\,$ is the mass matrix, and to the second order SDE

that we shall call Langevin-Kramers equation, with the $n$-dimensional white noise $\,\zeta\,$ such that

The Langevin-Kramers equation has the form of the Newton equation with the friction $\,-\gamma\dot{q}\,$ and white-noise $\,\zeta_{t}\,$ forces supplementing the conservative one $\,-\nabla V_{t}\,$ and the additional one $\,f_{t}$. It was discussed in in a very similar context. In the limit of strongly overdamped system when the friction term becomes much larger then the second order one, the Langevin-Kramers equation (2.7) reduces to the first order SDE

which, if $\,\gamma>0$, may be cast again into the form (2.4) but with $\,\Gamma=\gamma^{-1}$, $\,\Pi=0\,$ and $\,H_{t}=V_{t}$. One should keep in mind this change when applying the results described below for the Langevin dynamics (2.4) to the overdamped Langevin-Kramers dynamics.

Transition probabilities and detailed balance

Let us recall some basic facts about the diffusion processes in order to set the notations. We shall denote by $\,\boldsymbol{E}^{t_{0}}_{x}\,$ the expectation of functionals of the Markov process $\,{\mathsf{x}}_{t}\,$ solving the SDE (2.1) with the initial condition $\,{\mathsf{x}}_{t_{0}}=x$. For $t\geq t_{0}$, the relation

defines the transition probabilities $\,P_{t_{0},t}(x,dy)\,$ of the process $\,{\mathsf{x}}_{t}\,$ and the operator $\,P_{t_{0},t}$. The transition probabilities satisfy the normalization condition $\,\int P_{t_{0},t}(x,dy)=1\,$ and the Chapman-Kolmogorov chain rule

The evolution of the expectation values is governed by the second-order differential operators $L_{t}$ defined by the relation

The explicit form of $\,L_{t}\,$ is found by a standard argument that involves the passage from the Stratonovich to the Itô convention. For reader’s convenience, we give the details in Appendix A. The result is:

Due to the relation (3.1), Eq. (3.2) may be rewritten as the operator identity $\,\partial_{t}P_{t_{0},t}=P_{t_{0},t}L_{t}$. Together with the initial condition $\,P_{t_{0},t_{0}}=1$, it implies that $P_{t_{0},t}$ is given by the time-ordered exponential

In particular, $\,P_{t_{0},t}={\rm e}^{(t-t_{0})L}\equiv P_{t-t_{0}}\,$ in the stationary case with $\,u_{t}\equiv u\,$ and $\,D_{t}\equiv D$. The operator $\,L_{t}\equiv L\,$ is then called the generator of the process.

The stochastic process $\,{\mathsf{x}}_{t}\,$ may be used to evolve measures. Under the stochastic dynamics, the initial measure $\,\mu_{t_{0}}(dx)\,$ evolves at time $\,t\,$ to the measure

We shall use below the shorthand notation: $\,\mu_{t}=\mu_{0}P_{0,t}$. For measures with densities $\,\mu_{t}(dx)=\rho_{t}(x)\,dx\,$ with respect to the Lebesgue measure $\,dx$, Eq. (3.6) is equivalent to the evolution equation

where $\,L^{\dagger}_{t}\,$ is the (formal) adjoint of the operator $\,L_{t}$. The latter relation may be rewritten as the continuity equation

where $\,\nabla\cdot j\equiv\partial_{i}j^{i}_{t}\,$ is the divergence of the density current $\,j_{t}\,$ corresponding to the measure $\,\mu_{t}\,$ (the probability current, if $\,\mu_{t}\,$ is normalized). In the case with no explicit time dependence when $\,L_{t}\equiv L$, an invariant density $\,\rho$, corresponding to an invariant measure $\,\mu(dx)=\rho(x)\,dx\,$ of the process, satisfies the equation $\,L^{\dagger}\rho=0\,$ which may be rewritten in the form of the current conservation condition $\,\nabla\cdot j=0$. We shall often write the invariant density $\,\rho(x)\,$ in the exponential form as $\,{\rm e}^{-\varphi(x)}$. One says that the process satisfies the detailed balance relation with respect to $\,\varphi\,$ if the density current $\,j\,$ related to the measure $\,\mu(dx)={\rm e}^{-\varphi(x)}dx\,$ vanishes itself, i.e. if

Equivalently, this condition may be written as the relation

for the generator of the process or as the identity

for the transition probabilities. In all these three forms, it implies directly that $\,\mu\,$ is an invariant measure. The converse, however, is not true: there exist stationary diffusion processes with invariant measures that do not satisfy the detailed balance relation.

The generator of the stationary Langevin equation with $\,\Pi=0\,$ and $\,G=0\,$ satisfies the detailed balance relation with respect to $\,\varphi=\beta H\,$ so that the Gibbs density $\,\rho(x)={\rm e}^{-\beta H(x)}$, and, if the latter is normalizable, the Gibbs probability measure $\,\mu^{G}(dx)=Z^{-1}{\rm e}^{-\beta H(x)}dx$, are invariant under such dynamics. The invariance still holds when $\,\Pi\not=0\,$ but, in this case, the detailed balance relation fails. We shall see below how to generalize the latter to catch also the case with conservative forces when $\,\Pi\not=0$.

Tangent process and phase-space contraction

One may generate other processes of a similar nature from the diffusive process (2.1). Such constructions will play an important role in studying fluctuation relations. As the first example, let us consider the separation $\,\delta{\mathsf{x}}_{t}\,$ between the solution $\,{\mathsf{x}}_{t}\,$ of Eq. (2.1) with the initial value $\,{\mathsf{x}}_{0}=x\,$ and another solution infinitesimally close to $\,{\mathsf{x}}_{t}$. Such a separations evolves according to the law

where the matrix $\,{\mathsf{X}}_{t}(x)\,$ with the entries

with the initial condition $\,{\mathsf{X}}_{0}(x)=1$. Together with Eq. (2.1), the SDE (4.2) defines a diffusion process $\,({\mathsf{x}}_{t},{\mathsf{X}}_{t})\,$ that we shall call the tangent process. In particular, the quantity $\,-\ln\det{\mathsf{X}}_{t}\,$ that represents the accumulated phase-space contraction along the trajectory $\,{\mathsf{x}}_{t}$, solves the SDE

The right hand side of Eq. (4.3) is the phase-space contraction rate. We infer that

The second integral on the right hand side should be interpreted with the Stratonovich convention. The phase-space contraction is an important quantity in the study of nonequilibrium dynamics and it will reappear in the sequel.

Backward processes

Among the diffusion processes that may be generated from the original process (2.1) are the ones which may be interpreted as its time reversals. The action of time inversion on space-time will be given by the transformation

that we shall loosely term dissipative and conservative, choosing different time-inversion rules for them. The time-reversed process $\,{\mathsf{x}}^{\prime}_{t}\,$ will be given by the SDE

with the deterministic vector field $\,u^{\prime}_{t}=u^{\prime}_{t,+}+\,u^{\prime}_{t,-}\,$ and the random one $\,v^{\prime}_{t}\,$ defined by the equations

As before, see Eqs. (3.4), we shall denote

Remark 1. Using the chain rule $\,(\partial_{j}{x^{*}}^{i})(x^{*})(\partial_{k}{x^{*}}^{j})(x)=\delta^{i}_{k}$, it is easy to see that the time-inversion transformations (5.5) are involutive.

Let us emphasize that the choice of a time inversion consists of the choice of the involution (5.1) and of the splitting (5.3) of $\,u_{t}$. We shall call the process time-reversible (for a given choice of time inversion) if the deterministic vector fields $\,u\,$ and $\,u^{\prime}\,$ of the forward and of the backward processes coincide and if the respective random vector fields $\,v_{t}\,$ and $\,v^{\prime}_{t}\,$ have the same distribution, i.e. if

Note that the first identity is equivalent to the relations

and can be always achieved by taking such a splitting of $\,u_{t}$. It may be not easy, however, to realize physically the backward process corresponding to the splitting (5.9). The second condition (5.8) is a non-trivial constraint on the distribution of the the white-noise velocity $\,v_{t}$. Nevertheless, if $\,D_{t}\,$ is time-independent, it may be satisfied by choosing the trivial involution $\,x^{*}\equiv x$.

Parallelly to the splitting (5.3) of the drifts $\,u_{t}\,$ and $\,u^{\prime}_{t}$, we shall divide the operators generating the forward and the backward evolution into two parts:

The time-inversion rules become even more transparent when expressed in terms of the split generators. Let $\,R\,$ denote the involution operator acting on the functions by

Proof of Lemma 1, involving a straightforward although somewhat tedious check, is given in Appendix B.

Below, similarly as for the forward process, we shall denote by $\,\boldsymbol{E}^{\prime t_{0}}_{x}\,$ the expectation of functionals of the backward process satisfying the initial condition $\,{\mathsf{x}}^{\prime}_{t_{0}}=x$. For $\,t\geq t_{0}$, the relations

define the operators whose kernels give the transition probabilities of the time-reversed process $\,{\mathsf{x}}^{\prime}_{t}$.

Examples of time-inversion rules

The preceding considerations were very general. Physically, not all time-inversion rules for the diffusive processes (2.1) described above are on the equal footing. In particular situations, some rules may be more natural or easier to implement than the other ones. Let us list here few cases of special time inversions that were discussed in the literature and/or will be used below.

Taking the trivial splitting $\,u_{t,+}=0$, $\,u_{t,-}=u_{t}\,$ combined with an involution $\,x\mapsto x^{*}\,$ leads to the time-inversion rules that produce the backward process with trajectories related by the transformation (5.2) to the ones of the forward process if the pseudo-vector field rule is used when transforming $\,v_{t}$. This is the time inversion usually employed for the deterministic systems but it may be used more generally.

0\,\,\hat{u}_{t,+}=0 Consider the time inversion corresponding to an arbitrary involution $\,x\mapsto x^{*}\,$ and the choice

of the splitting of $\,u_{t}$. Such a time inversion is a slight modification of the natural one to which it reduces in the case of deterministic dynamics (2.3) with $\,v_{t}\equiv 0$. As we show in Appendix C, the backward dynamics corresponding to the splitting (6.1) is given by the relations

where $\,\sigma(x)=\sigma(x^{*})^{-1}\,$ denotes the absolute value $\,|\det(\partial_{j}{x^{*}}^{i})(x)|\,$ of the Jacobian of the involution $\,x\mapsto x^{*}$. The time inversion considered here will be used to obtain fluctuation relations in the limiting case of deterministic dynamics (2.3) when $\,D^{ij}_{t}\,$ is set to zero and the backward dynamics is given by the ODE

obtained from the ODE (2.3) by the natural time inversion.

3 Time inversion in the Langevin dynamics

To explain why the rules of time inversion with non-vanishing $\,u_{t,+}\,$ are more generally needed, we consider the case of the Langevin dynamics that involves the dissipative force $\,-\Gamma\nabla H_{t}$. Let us arbitrarily split the corresponding drift $\,u_{t}\,$ into two parts:

see Eq. (2.4). Recall the relation (2.5) that aligns the matrix $\,\Gamma\,$ with the covariance of the white-noise $\,v_{t}=\zeta_{t}$. It is natural to require the backward dynamics to be also of the Langevin type but for the time-reversed Hamiltonian $\,H^{\prime}_{t}(x)=H_{t^{*}}(x^{*})$. This requires that

and that $\,v^{\prime}_{t}(x)=\zeta^{\prime}_{t}\,$ with the covariance of the white noise $\,\zeta^{\prime}_{t}\,$ aligned with matrix $\,\Gamma^{\prime}\,$ as in Eq. (2.5). Upon restriction to linear involutions $\,x^{*}=rx\,$ with $\,r^{2}=1$, the transformation rules (5.5) become

The condition on the covariance of $\,\zeta^{\prime}_{t}\,$ imposes the relation $\,\Gamma^{\prime}=r\Gamma r^{T}$. Applying $\,r\,$ to the both sides of Eq. (6.5) taken at time $\,t^{*}\,$ and at point $\,rx$, we infer that

The latter identity, together with Eq. (6.4), result in the relations

At least when $\,\Gamma\,$ is strictly positive, $\,H_{t}\,$ is not a constant, and the extra force $\,G_{t}\,$ is absent, one infers that the component $\,u_{t,+}\,$ cannot vanish identically by considering the contraction $\,(\nabla H_{t})\cdot u_{t,+}$. We shall call canonical a choice of the time inversion for the Langevin dynamics for which

Note that such a time inversion treats the force $\,G_{t}\,$ as a part of $\,u_{t,-}\,$ even when this force is of the non-conservative type. The Langevin dynamics is time-reversible under a canonical time inversion if $\,H^{\prime}_{t}=H_{t}\,$ and $\,G^{\prime}_{t}=G_{t}$. For the Langevin-Kramers equation, the standard phase-space involution $\,(q,p)^{*}=r(q,p)=(q,-p)\,$ verifies Eqs. (6.7) and it leads to the particularly simple canonical time-inversion rules with

and to the time-reversibility if $\,V_{t}=V_{t^{*}}\,$ and $\,f_{t}=f_{t^{*}}$.

4 Reversed protocol

The time inversion corresponding to the choice

and trivial involution $\,x^{*}\equiv x\,$ was termed in a reversed protocol. It may be viewed as consisting of the inversion of the time-parametrization in the vector fields in the SDE (2.1), if the vector-field rule is used to reverse $\,v_{t}$. In the stationary case, where it results in time-reversibility, such a time inversion was employed already in . Here, we shall admit also a possibility of a non-trivial involution $\,x\mapsto x^{*}$. The reversed protocol leads then to the backward process with

5 Current reversal

Suppose that $\,{\rm e}^{-\varphi_{t}}\,$ are densities satisfying $\,L_{t}^{\dagger}{\rm e}^{-\varphi_{t}}=0$. Such densities would be preserved by the evolution if the generator of the process were frozen to $\,L_{t}$. The density current corresponding to $\,{\rm e}^{-\varphi_{t}}\,$ has the form

see Eq. (3.8). It is conserved due to the relation $\,L_{t}^{\dagger}{\rm e}^{-\varphi_{t}}=0$. The time inversion defined by the choice

and an arbitrary involution $\,x\mapsto x^{*}\,$ leads, after an easy calculation using the results of Appendix C, to the backward process with

for $\,\varphi^{\prime}_{t}(x)=(\varphi_{t^{*}}+\ln{\sigma})(x^{*})$. The density current for the backward process corresponding to the densities $\,{\rm e}^{-\varphi^{\prime}_{t}}\,$ is

and is also conserved, as is easy to check. It follows that $\,L^{\prime\dagger}_{t}{\rm e}^{-\varphi^{\prime}_{t}}=0$. We shall term the time inversion corresponding to the choices (6.11) the current reversal. For $\,x^{*}\equiv x\,$ when it just reverses the sign of the current, it was already employed in an implicit way in , and was introduced explicitly (under a different name) in . The latter reference discussed also a simple two-dimensional model for which the inverse protocol and the current reversal led to different backward processes.

6 Complete reversal

Finally, modifying slightly the last scheme, let us suppose the densities $\,\rho_{t}={\rm e}^{-\varphi_{t}}\,$ evolve under the dynamics solving Eq. (3.7). With the same splitting (6.11) as for the current reversal, we obtain the backward process for which Eqs. (6.12) and (6.14) still hold for $\,\varphi^{\prime}_{t}(x)=(\varphi_{t^{*}}+\ln{\sigma})(x^{*})$. We shall call the corresponding time inversion the complete reversal. Unlike in the other examples, it depends also on the choice of the initial density $\,\rho_{0}\,$ and may be difficult to realize physically. The time-reflected densities $\,\rho^{\prime}_{t}={\rm e}^{-\varphi^{\prime}_{t}}\,$ evolve now according to the backward-process version of Eq. (3.7). The current reversal and the complete reversal coincide in the case without explicit time dependence and with the choice of $\,\varphi_{t}\equiv\varphi\,$ such that $\,{\rm e}^{-\varphi}dx\,$ is an invariant measure.

Relation between forward and backward processes

A comparison between the forward and the backward processes will be at the core of fluctuation relations that we shall discuss. To put the processes in the two time directions back-to-back, we shall adapt to the present setup the arguments developed in Sect. 5 of . Let us introduce a perturbed version of the generator $\,L_{t}\,$ of the forward process,

Operator $\,L^{1}_{t}\,$ is related in a simple way to the generator of the backward process:

where $\,R\,$ is defined by Eq. (5.11) and the last equality is a consequence of the relations (5.12). Let us consider the time-ordered exponential of the integral of $\,L^{1}_{t}$. Using the relation $\,L_{t}^{1}=(R\hskip 0.71114ptL^{\prime}_{t^{*}}R)^{\dagger}\,$ that follows from Eq. (7.4), we infer that

Above, the first inversion of the time order from $\,\overrightarrow{\cal T}\,$ to $\,\overleftarrow{\cal T}\,$ was due to the change of integration variables $\,s\mapsto s^{*}=T-s$, and the second one, to the fact that the hermitian conjugation reverses the order in the product of operators. Let us remark that $\,A(y,dx)\,$ is the kernel of the operator $\,A^{\dagger}\,$ and $\,A(x^{*},dy^{*})\,$ of the operator $\,RAR\,$ if $\,A(x,dy)\,$ is the kernel of a real operator $\,A$. Rewriting Eq. (7.6) in terms of the kernels, with these comments in mind, we obtain the identity

Remark 2. The transition probability of the backward process on the right hand side may be replaced by the one of the forward process in the time-reversible case.

Note that the $2^{\rm nd}$ order differential operator $\,L^{1}_{t}\,$ differs from $\,L_{t}\,$ only by lower order terms, see Eq. (7.1). A combination of the Cameron-Martin-Girsanov and the Feynman-Kac formulae permits to express the kernel $\,P^{1}_{t_{0},t}(x,dy)\,$ as a perturbed expectation for the forward process.

Lemma 2. If the matrix \,\big{(}d^{ij}_{t}(x)\big{)}\, is invertible for all $\,t\,$ and $\,x\,$ then

is a (local) functional of the solution ${\mathsf{x}}_{t}$ of the SDE (2.1). The right hand side of Eq. (7.9) uses the vector notation. The first term in the expression for $\,{\cal J}_{t}\,$ has to be interpreted with the Stratonovich convention.

Proof of Lemma 2 is deferred to Appendix D. A combination of the relations (7.8) and (7.7) gives immediately

This is the first fluctuation relation of a series to be considered. It connects the transition probability of the backward process to an expectation in the forward process weighted with an exponential factor. Let us illustrate this relation in a few particular situations related to the examples of the diffusion processes considered in Sect. 2.

Example 5. Tangent process in the stationary homogeneous Kraichnan model

Recall Sect. 4 devoted to the definition of a tangent process. Let us consider the tangent process $\,({\mathsf{x}}_{t},{\mathsf{X}}_{t})\,$ with fixed initial data $\,{\mathsf{x}}_{0}=x\,$ and $\,{\mathsf{X}}_{0}=1\,$ for the homogeneous Kraichnan model. As was discussed in detail in , in this case, the distribution of the process $\,{\mathsf{X}}_{t}\,$ may be obtained by solving, instead of the SDE (4.2) with $\,u_{t}\equiv 0$, a simpler linear Itô SDE

with a matrix-valued white-noise $\,S_{t}\,$ such that

In other words, in Eq. (4.2), we may replace $\,\partial_{k}v^{i}({\mathsf{x}}_{t})\,$ by $\,\partial_{k}v^{i}_{t}(0)\equiv S^{i}_{t\,k}$, if we change the SDE convention to the Itô one at the same time. Consequently, in the homogeneous Kraichnan model, the process $\,{\mathsf{X}}_{t}\,$ may be decoupled from the original process $\,{\mathsf{x}}_{t}$. Let us abbreviate: $\,-\partial_{k}\partial_{l}D_{t}^{ij}(0)=C^{ij}_{kl}$. Remark the symmetries $\,C^{ij}_{kl}=C^{ji}_{lk}=C^{ij}_{lk}=C^{ji}_{kl}$. The Itô SDE (7.11) may be rewritten as the equation

that employs the Stratonovich convention. Upon the use of the notations:

falling within the scope of (stationary) diffusion SDEs (2.1) and defining a Markov process $\,{\mathsf{X}}_{t}$. The covariance of the white-noise “velocity” $\,V_{t}(X)\,$ is

As in the general case (3.4), we shall denote:

Let us apply the reversed-protocol time inversion discussed in Sect. 6.4 to the forward SDE (7.13). It corresponds to the trivial splitting of $\,U\,$ with $\,U_{+}=U\,$ and $\,U_{-}=0\,$ and to an involution $\,X\mapsto X^{*}\,$ that we shall also take trivial: $\,X^{*}\equiv X$. The backward evolution is then given by the same equation (7.12) with $\,S_{t}\,$ replaced by $\,S^{\prime}_{t}=S_{t^{*}}$, a matrix-valued white noise with the same distribution as $\,S_{t}$. The time-reversibility follows. Suppose that the covariance $\,C\,$ of the white noise $\,S(t)\,$ is invertibleThe assumption about inversibility of $\,C\,$ may be dropped at the end by a limiting argument., i.e. that there exists a matrix $\,(C^{-1})^{ln}_{jm}\,$ such that $\,C^{ij}_{kl}(C^{-1})^{ln}_{jm}=\delta^{i}_{m}\delta^{n}_{k}$. Then the matrix

provides the inverse of $\,d^{ij}_{kl}(X)$. Substituting these data into Eq. (7.9), we obtain

The relation (7.10) applied to the case at hand leads to the identity

where $\,P_{t}(X_{0},dX)\,$ denotes the transition probability of the forward process $\,{\mathsf{X}}_{t}\,$ solving the SDEs (7.11) or (7.12) and $\,dX_{0}\,$ on the left hand side and $\,dX\,$ on the right hand side stand for the Lebesgue measures on the space of $\,d\times d\,$ matrices. We made use of the fact that the backward process has the same law as the forward one. Eq. (7.14) is nothing else a the detailed balance relation with respect to $\,\varphi(X)=\ln|\det X|$. Indeed, note that the density current corresponding to the density $\,\rho(X)=|\det X|^{-d+1}\,$

Integrating the left hand side of the above identity against a function $\,f(X_{0},X)\,$ and using the relation $\,P_{t}(X_{0},dX)=P_{t}(1,d(XX_{0}^{-1}))\,$ that follows from the invariance of the corresponding SDE under the right multiplication of $\,X\,$ by invertible matrices, we obtain the equalities

where we twice changed variables in the iterated integrals. On the other hand, the integration of the right hand side of Eq. (7.14) against $\,f(X_{0},X)\,$ gives

Comparing the two expressions, we infer that

This is a version of the Evans-Searles fluctuation relation for the stationary homogeneous Kraichnan model. In the context of general hydrodynamic flows, it was formulated and proven by a change-of-integration-variables argument in , see also . We shall return in Sect. 15 to the relation (7.17) in order to examine some of its consequences. Subsequently, we shall generalize it in Sect. 16 to arbitrary diffusion processes of the type (2.1).

Example 6. Generalized detailed balance relation

Consider the complete-reversal rules discussed in Sect. 6.6 and corresponding to the choice (6.11). Since, by virtue of the assumption that the densities $\,{\rm e}^{-\varphi_{t}}\,$ evolve under the dynamics, see Eq. (3.7),

the last two terms in the definition (7.9) reduce to $-\,(\partial_{t}\varphi_{t})({\mathsf{x}}_{t})\,$ in this case so that

Upon integration over time, this produces boundary terms and Eq. (7.10) implies the generalized detailed balance relation:

for $\,\mu_{t}(dx)={\rm e}^{-\varphi_{t}(x)}dx$. Note that Eq. (7.20) holds for any choice of the involution $\,x\mapsto x^{*}$. Upon integration over $\,x$, Eq. (7.20) assures that the measures $\,\mu_{t}\,$ stay invariant under the dynamics, what was assumed from the very beginning. In the case with no explicit time dependence, i.e. when $\,L_{t}\equiv L$, Eq. (7.20) holds, in particular, for $\,\varphi_{t}\equiv\varphi\,$ such that $\,\mu={\rm e}^{-\varphi}dx\,$ is an invariant measure. In that case, the generalized detailed balance relation reduces to the detailed balance one (3.9) if $\,u_{-}\,$ in the splitting (6.11) vanishes and $\,x^{*}\equiv x$. This was the case in Example 5. Below, we shall see examples where the invariant measure $\,\mu\,$ is known and the generalized detailed balance relation holds but where the detailed balance itself fails. Some of those cases fall under the scope of the Langevin dynamics. Let us discuss them first.

Example 7. $1^{\rm st}$ law of thermodynamics and generalized detailed balance for the Langevin dynamics

For the Langevin dynamics with the splitting (6.8) of the drift, a direct substitution yields

Upon the use of the dynamical equation (2.4),

where $\,{\mathsf{Q}}\,$ may be identified with the heat transfered to the environment modeled by the thermal noise. On the other hand, using the original expression for $\,{\cal J}_{t}\,$ together with the (Stranonovich convention) identity $\,\frac{d}{dt}\,H_{t}({\mathsf{x}}_{t})=(\nabla H_{t})({\mathsf{x}}_{t})\cdot\dot{{\mathsf{x}}}_{t}+(\partial_{t}H_{t})({\mathsf{x}}_{t})$, we obtain the relation

where $\,\Delta{\mathsf{U}}=H_{T}({\mathsf{x}}_{T})-H_{0}({\mathsf{x}}_{0})\,$ is the change of the internal energy of the system and

may be interpreted as the work performed on the system. With this interpretations, a comparison of the two expressions for the integral of $\,{\cal J}_{t}\,$ leads to the $\,{\bf 1^{\rm st}}\,$ law of thermodynamics:

This was discussed in a simple example of the forced and damped oscillator in . In the absence of the extra force $\,G_{t}$, the expression for the work reduces to

and represents the so called Jarzynski work introduced first in for deterministic Hamiltonian dynamics. In the stochastic Langevin-Kramers dynamics, the expressions for the heat and the work become:

The second quantity is equal to the sum of the Jarzynski work and of the work of the external force $\,f_{t}$. It was introduced and discussed in . In the stationary case, it reduces to the injected work and, up to the $\beta$-factor, coincides with the “action functional” (for uniform temperature) given by Eq. (6.3) of . Note that the general expression (7.23) for work makes also sense in the case of deterministic dynamics (2.6) obtained from the SDE (2.4) by setting $\,\Gamma=0$, in particular for the deterministic Hamiltonian evolution with $\,G_{t}\equiv 0$.

If $\,G_{t}\equiv 0$, the splitting (6.8) is a special case of the splitting used for the current reversal for $\,\varphi_{t}=\beta H_{t}$, see Eq. (6.11). In particular, if $\,H_{t}\equiv H\,$ then the transition probabilities of the Langevin process satisfy the generalized detailed balance relation (7.20) that takes the form

for $\,\mu(dx)={\rm e}^{-\beta H(x)}dx\,$ and any involution $\,x\mapsto x^{*}=rx$. The latter identity replaces the detailed balance relation (3.9) in the presence of the conservative force $\,\Pi\nabla H\,$ and still assures that the Gibbs density $\,{\rm e}^{-\beta H}\,$ is invariant under such Langevin dynamics. If the involution $\,r\,$ satisfies additionally the relations (6.7) and $\,H(rx)=H(x)$, resulting in the time-reversibility, then one may replace $\,P^{\prime}_{T}\,$ by $\,P_{T}\,$ in the relation (7.20).

where $\,M\,$ is a $\,d\times d\,$ matrix and $\,\zeta_{t}\,$ is the white noise with the covariance (2.5) and matrix $\,\Gamma\,$ strictly positive. We shall be interested in cases when the matrix $\,\Gamma^{-1}M\,$ is non-symmetric. For an elementary discussion of mathematical aspects of such SDEs see e.g. . In the context of nonequilibrium statistical mechanics, examples of such linear equations were considered in as models of a harmonic chain of oscillators interacting with environment of variable temperature or, quite recently, in for modeling coiled polymers in a shearing flow. The diffusion process $\,{\mathsf{x}}_{t}\,$ that solves Eq. (7.28) with the initial value $\,{\mathsf{x}}_{0}=x\,$ is given by the formula

The transition probabilities of this process are Gaussian and have the explicit form

is a strictly positive matrix. Suppose that all the eigenvalues $\,\lambda\,$ of $\,M\,$ have negative real parts. Under this condition, $\,{\rm e}^{tM}\,$ tends to zero exponentially fast when $\,t\to\infty\,$ so that $\,C_{\infty}\equiv C\,$ is finite and

with the right hand side defining the unique invariant probability measure of the process. This Gaussian measure has the form of the Gibbs measure for the quadratic Hamiltonian

the linear SDE (7.28) may be rewritten in the Langevin form (2.4) as

Conversely, the last SDE with $\,H\,$ as in Eq. (7.31) for some $\,C>0\,$ is turned into the form (7.28) upon setting

Note that the last equation implies the relation (7.32) for $\,\Pi$. In Appendix E, we show that $\,M\,$ given by Eq. (7.34) has necessarily all eigenvalues with negative real part and that $\,C\,$ may be recovered from $\,M\,$ as $\,C_{\infty}\,$ given by Eq. (7.30) with $\,t=\infty$. This establishes the equivalence between the SDEs (7.28) and (7.33).

The probability current associated by the formula (3.8) to the Gaussian invariant Gibbs measure $\,\mu^{G}(dx)=Z^{-1}{\rm e}^{-\beta H(x)}dx\,$ is

It vanishes only when $\,\Pi=0$. In the latter case, the transition probabilities (7.29) satisfy the detailed balance relation (3.9) for $\,\varphi=\beta H+\ln{Z}$. If $\,\Pi\not=0\,$ then only a generalized detailed balance relation (7.27) holds for any choice of the linear involution $\,x\mapsto x^{*}=rx$. If moreover $\,r\Gamma r^{T}=\Gamma$, $\,r\Pi r^{T}=-\Pi\,$ and $\,rCr^{T}=C\,$ then $\,P^{\prime}_{T}\,$ on the right hand side of Eq. (7.20) may be replaced by $\,P_{T}$.

Jarzynski equality

We shall exploit further consequences of the relation (7.10) between the forward and backward processes. In this section we shall derive an identity that generalizes the celebrated Jarzynski equality and shall prepare the ground for obtaining more refined fluctuation relations following the ideas of , and . Let $\,\varphi_{0}\,$ and $\,\varphi_{T}\,$ be two functions generating measures

respectively. In particular, we could take $\,{\rm e}^{-\varphi_{T}(x)}\,$ such that the measure $\,\mu_{T}\,$ is related to $\,\mu_{0}\,$ by the dynamical evolution (3.6), i.e. $\,\mu_{T}=\mu_{0}P_{0,T}$, but we shall not assume such a choice unless explicitly stated. In general, the measures (8.1) may be not normalizable but we shall impose the normalization condition later on. We shall associate to $\,\mu_{0}\,$ and $\,\mu_{T}\,$ the time-reflected measures

Let us modify the functional $\,\int\limits_{0}^{T}{\cal J}_{t}\,dt\,$ introduced in the last section by boundary terms $\,\Delta\varphi\equiv\varphi_{T}({\mathsf{x}}_{T})-\varphi_{0}({\mathsf{x}}_{0})\,$ by setting

The functional $\,{\cal W}\,$ will be the basic quantity in what follows. Its physical interpretation in terms of the entropy production will be discussed in the Sect. 10 below.

for the (unnormalized) expectation of the process $\,{\mathsf{x}}_{t}\,$ with fixed initial and final points, and similarly for the backward process. The following refinement of the relation (7.10) of Proposition 1 holds:

Proof of Proposition 2 is contained in Appendix F. Note that the explicit dependence on the choice of measures $\,\mu_{0}\,$ and $\,\mu^{\prime}_{0}\,$ trivially cancels the one buried in $\,{\cal W}$. In particular, for $\,{\cal F}\equiv 1$, Proposition 2 reduces to Proposition 1 with $\,t_{0}=0\,$ and $\,t=T$. As before, the backward-process expectation $\,\boldsymbol{E}^{\prime}\,$ may be replaced by the forward-process one $\,\boldsymbol{E}\,$ for the time-reversible process.

If the measures $\,\mu_{0}\,$ and $\,\mu^{\prime}_{0}\,$ are normalized then we may use them as the probability distributions of the initial points of the forward and of the backward process, respectively. The corresponding probability measures $\,M(d{\mathsf{x}})\,$ and $\,M^{\prime}(d{\mathsf{x}}^{\prime})\,$ on the space of trajectories on the time-interval $\,[0,T]\,$ are given by the relations

Upon integration over $\,x\,$ and $\,y$, the identity (8.3) induces the following equality between the expectations with respect to the trajectory measures $\,M\,$ and $\,M^{\prime}$:

It was stressed in , and even more explicitly in , that the identity of the type of (8.6), comparing the expectations in the forward and the backward processes, is a source of fluctuation relations. An important special case of Eq. (8.6) is obtained by setting $\,{\cal F}\equiv 1$. It was derived in in the context of the Hamiltonian dynamics and in in the one of Markov processes:

Let us illustrate the meaning of the above relation by considering a few special cases.

With the splitting (6.8) used for the canonical time inversion, upon taking $\,\varphi_{t}=\beta(H_{t}-F_{t})$, where $\,F_{t}=-\beta^{-1}\ln\int{\rm e}^{-\beta H_{t}(x)}dx\,$ denotes the free energy, we infer from Eq. (7.22) that

where $\,\Delta F=F_{T}-F_{0}\,$ is the free energy change and $\,{\mathsf{W}}\,$ is the work given by Eq. (7.23). The difference $\,{\mathsf{W}}-\Delta F\,$ is often called the dissipative work. The Jarzynski equality (8.7) may be rewritten in this case in the original form

in which it has become a tool to compute the differences between free energies of equilibrium states from nonequilibrium processes .

Example 10. The case of deterministic dynamics

Upon splitting the drift $\,u_{t}\,$ as in Eq. (6.1) of Sect. 6.2, the expression (7.9) reduces to $\,{\cal J}_{t}=-(\nabla\cdot\hat{u}_{t})({\mathsf{x}}_{t})$. For the deterministic dynamics where $\,D_{t}^{ij}(x,y)\equiv 0$, one has $\,\hat{u}_{t}=u_{t}\,$ so that

The right hand side represents the phase-space contraction rate along the trajectory $\,{\mathsf{x}}_{t}$, see Eq. (4.3). In this case,

For $\,\varphi_{T}=\varphi_{0}=\varphi$, the last integral in Eq. (8.11) was termed “the integral of the dissipation function” in . In the case of the deterministic dynamics (2.6) obtained from the Langevin equation by setting $\,\Gamma=0$, the expression (8.11) for $\,{\cal W}\,$ reduces to the one of Eq. (8.8) if we take $\,\varphi_{t}=\beta(H_{t}-F_{t})$. In the deterministic case, the Jarzynski equality (8.7) reads

and may be easily proven directly. To this end recall Eq. (4.4) which implies for the deterministic case that $\ \int\limits_{0}^{T}(\nabla\cdot u_{t})({\mathsf{x}}_{t})\,dt=\ln\det{\mathsf{X}}_{T}({\mathsf{x}}_{0})$, were the matrices $\,{\mathsf{X}}_{t}(x)\,$ of the tangent process are given by Eq. (4.1). The equality (8.12) is then obtained by the change of integration variables $\,{\mathsf{x}}_{0}\mapsto{\mathsf{x}}_{T}\,$ whose Jacobian is equal to $\,\det{\mathsf{X}}_{T}({\mathsf{x}}_{0})$.

In the setup of Sect. 6.4 with $\,u_{t,-}=0$,

In the stationary case, the integral $\,\int\limits_{0}^{T}{\cal J}_{t}\,dt$, rewritten with use of the Itô convention, was termed an “action” in , see Eq. (5.3) therein. In , it was considered in the context of the Langevin equation with the extra force $\,G_{t}\,$ (but without the Hamiltonian term $\,\Pi\nabla H_{t}$). It was then identified as $\,\beta{\mathsf{Q}}^{tot}\,$ with the quantity $\,{\mathsf{Q}}^{tot}\,$ interpreted, following , as the total heat produced in the environment. The functional $\,{\cal W}\,$ of the forward process is given here by the formula

In particular, for the Langevin dynamics (2.4), one obtains:

The Jarzynski equality (8.7) was discussed for this case in . Note that $\,{\cal W}^{tot}\,$ is not well defined for the Langevin-Kramers dynamics. On the other hand, for the linear Langevin equation of Example 8 and for $\,\varphi_{t}=\beta(H-F)$,

and it vanishes if $\,\Pi=0$. A long time asymptotics of the probability distribution of a quantity differing from the last one by a boundary term was studied in .

In the current-reversal setup of Sect. 6.5, with the splitting (6.11) of the drift $\,u_{t}\,$ induced by the normalized densities $\,{\rm e}^{-\varphi_{t}}\,$ such that $\,L_{t}^{\dagger}{\rm e}^{-\varphi_{t}}=0$,

since now the last two terms on the right hand side of Eq. (7.9) vanish, compare to Eq. (7.18). Upon integration, this gives:

In , the integral of $\,{\cal J}_{t}\,$ given by Eq. (8.16) was identified in the context of the Langevin equation with the force $\,G_{t}\,$ as equal to $\,\beta{\mathsf{Q}}^{ex}\,$ where $\,{\mathsf{Q}}^{ex}\,$ was termed the excess heat, following . The difference $\,{\mathsf{Q}}^{tot}-{\mathsf{Q}}^{ex}={\mathsf{Q}}^{hk}\,$ was called, in turn, the housekeeping heat and was interpreted as the heat production needed to keep the system in a nonequilibrium stationary state, see again . Using in the definition (8.2) the functions $\,\varphi_{0}\,$ and $\,\varphi_{T}\,$ from the same family, we infer from Eq. (8.17) that

The equality (8.7) for this case was proven by Hatano-Saso , see also . Note that in the stationary case, $\,{\cal W}^{ex}=0$. The Langevin dynamics discussed in Example 9 provides a special instance of the situation considered here if $\,G_{t}\equiv 0$. Consequently, in that case, $\,{\cal W}^{ex}\,$ is equal to the dissipative Jarzynski work (in the $\beta^{-1}$ units) $\,\beta({\mathsf{W}}-\Delta F)\,$ with $\,{\mathsf{W}}\,$ given by Eq. (7.25).

Example 13. The case of complete reversal

Recall that for the complete reversal rule of Sect. 6.6 based on the choice of densities $\,{\rm e}^{-\varphi_{t}}\,$ evolving dynamically, $\,{\cal J}_{t}\,$ is the total time derivative, see Eq. (7.19). The use in the definition (8.2) of the functions from the same family annihilates the functional $\,{\cal W}$:

Speck-Seifert equality

Let us consider the two functionals $\,{\cal W}^{tot}\,$ and $\,{\cal W}^{ex}\,$ of the process $\,{\mathsf{x}}_{t}\,$ introduced in Examples 11 and 12. We shall take them with the same functions $\,\varphi_{t}\,$ satisfying $\,L_{t}^{\dagger}{\rm e}^{-\varphi_{t}}=0$. The two Jarzynski equalities \,\Big{\langle}{\rm e}^{-{\cal W}^{tot}}\Big{\rangle}=1=\Big{\langle}{\rm e}^{-{\cal W}^{ex}}\Big{\rangle}\, hold simultaneously. In a third equality of the same type, this time involving the quantity

was established in the context of the Langevin equation where $\,{\cal W}^{hk}=\beta{\mathsf{Q}}^{hk}=\beta{\mathsf{Q}}^{tot}-\beta{\mathsf{Q}}^{ex}\,$ is the housekeeping heat (in the $\beta^{-1}$ units). We shall prove here a generalization of the result of . To this end, let us consider, besides the original process $\,{\mathsf{x}}_{t}\,$ satisfying the SDE (2.1), the Markov process $\,{\mathsf{x}}^{\prime\prime}_{t}\,$ satisfying the same equation but with the drift $\,\hat{u}_{t}\,$ replaced by

We shall denote by \,\big{\langle}\ \cdot\ \big{\rangle}^{\prime\prime}\, the expectation defined by Eq. (8.4) but referring to the process $\,{\mathsf{x}}^{\prime\prime}_{t}$. Note in passing the relations $\,L^{\prime\prime\dagger}_{t}{\rm e}^{-\varphi_{t}}=0$, where the operators $\,L^{\prime\prime}_{t}\,$ are given by Eq. (3.3) with $\,\hat{u}^{\prime\prime}_{t}\,$ replacing $\,\hat{u}_{t}$. In particular, in the stationary case, the processes $\,{\mathsf{x}}_{t}\,$ and $\,{\mathsf{x}}^{\prime\prime}_{t}\,$ have the same invariant measure.

Proof. The above identity may be proven directly with the use of the Cameron-Martin-Girsanov formula, see Appendix D, by comparing the measures of the processes $\,{\mathsf{x}}_{t}\,$ and $\,{\mathsf{x}}^{\prime\prime}_{t}\,$ corresponding to SDEs differing by a drift term. Here we shall give another proof based on applying twice the relation (8.6). First, we use this relation with the functional $\,{\cal F}\,$ replaced by $\,{\cal F}\,{\rm e}^{-{\cal W}^{tot}+2{\cal W}^{ex}}\,$ for the current-reversal time inversion with the trivial involution $\,x^{*}\equiv x\,$ and the vector-field rule for $\,v_{t}$. This results in the equality

where the expectation \,\big{\langle}\ \cdot\ \big{\rangle}^{\prime}\, pertains to the backward dynamics with

see Eqs. (6.12). Now, we observe that the same backward process may be obtained by the reversed-protocol time inversion, again for $\,x^{*}\equiv x$, from the process $\,{\mathsf{x}}^{\prime\prime}_{t}\,$ introduced above. The identity (8.6) applied for the processes $\,{\mathsf{x}}^{\prime\prime}_{t}\,$ and $\,{\mathsf{x}}^{\prime}_{t}\,$ reads:

is the functional $\,{\cal W}\,$ referring to the dynamics with $\,\hat{u}^{\prime\prime}_{t}=\hat{u}^{\prime\prime}_{t,+}\,$ given by Eq. (9.1). The application of Eq. (9.4) to $\,{\cal F}^{\prime\prime}={\cal F}\,{\rm e}^{-{\cal W}^{tot}+2{\cal W}^{ex}}\,$ reduces the right hand side of Eq. (9.3) to the expectation \,\big{\langle}{\cal F}\,{\rm e}^{-({\cal W}^{tot}-2{\cal W}^{ex}+{\cal W}^{\prime\prime})}\big{\rangle}^{\prime\prime}. The equality (9.2) follows by checking that

Setting $\,{\cal F}\equiv 1\,$ in the identity (9.2), we obtain the result that was established by a different argument in in the context of the Langevin equation:

Entropy production

An immediate consequence of the Jarzynski equality (8.7) and of the Jensen inequality (i.e. of convexity of the exponential function) is

Corollary 4 ($2^{\rm nd}$ law of thermodynamics for diffusion processes).

To discuss the relation of the latter inequality to the $2^{\rm nd}$ law of thermodynamics, let us first remark that the quantity on the left hand side has the interpretation of a relative entropy. Recall, that for two probability measures $\,\mu(d{\mathsf{x}})\,$ and $\,\nu(d{\mathsf{x}})={\rm e}^{-w({\mathsf{x}})}\mu(d{\mathsf{x}})$, the relative entropy of $\,\nu\,$ with respect to $\,\mu\,$ is defined by the formula

and is always non-negative. Now, the identity (8.6) may be read as the relation

so that the inequality (10.1) expresses the positivity of the relative entropy.

where the difference $\,S(\mu_{T})-S(\mu_{0})\,$ is the change of entropy of the fixed-time distribution of the process during the time $\,T\,$ and

The latter quantity will be interpreted as the mean entropy production in the environment modeled by the stochastic noise, measured relative to the backward process. The quantity \,\big{\langle}{\cal J}_{t}\big{\rangle}\, represents then the instantaneous mean rate of the entropy production in the environement. The inequality (10.1) states then that the overall entropy production cannot be negative in mean. In this sense, it is a version of the $2^{\rm nd}$ law of thermodynamics for the diffusion processes under consideration. In the stationary case, where $\,\mu_{T}=\mu_{0}$, the overall mean entropy production reduces to the one in the environment $\,\Delta S_{env}$.

Note that $\,\Delta S_{env}\,$ defined above depends on the time inversion employed (more precisely, on the splitting of $\,u_{t}$), and the quantities obtained by employing different time inversions are, in general, different. They may have different physical relevance. For the Langevin equation with the splitting (6.8), $\,\Delta S^{env}=\beta\langle{\mathsf{Q}}\rangle$, where $\,{\mathsf{Q}}\,$ is the heat transfered to the environment given by Eq. (7.21). We may talk about the total mean entropy production in the environment $\,\Delta S_{env}^{tot}\,$ if the reversed protocol of Sect. 6.4 and Example 11 is used or about the excess mean entropy production $\,\Delta S_{env}^{ex}\,$ in the environment for the current reversal of Sect. 6.5 and Example 12. The Speck-Seifert equality (9.5) combined with the Jensen inequality implies that the former does not exceeds the latter. As an illustration, consider the stationary Langevin equation with vanishing additional force where $\,\Delta S^{ex}_{env}=0\,$ although $\,\Delta S^{tot}_{env}\,$ may be non-zero if $\,\Pi\not=0$. In particular, in the linear case studied in Example 8, $\,{\cal W}^{tot}\,$ is given by Eq. (8.15) and

where the second equality was obtained using the SDE (7.28) together with the fact that, for the integral $\ \int\limits_{0}^{T}{\mathsf{x}}_{t}\cdot C^{-1}\Pi\Gamma^{-1}\zeta_{t}\,dt$, the Stratonovich and the Itô conventions coincide so that its expectation vanishes. Finally, note that for the complete reversal, the overall entropy production vanishes because $\,{\cal W}\equiv 0\,$ in this case, see Eq. (8.19). With our flexibility of the choice of the backward process, there are always ones with respect to which there is no entropy production!

In the deterministic case when $\,{\cal J}_{t}\,$ is given by Eq. (8.10), the mean rate of entropy production in the environment is

where $\,\mu_{t}\,$ is obtained by the dynamical evolution from the measure $\,\mu_{0}$, i.e. $\mu_{t}=\mu_{0}P_{0,t}\,$ for $\,P_{0,t}({\mathsf{x}}_{0},dy)=\delta(y-{\mathsf{x}}_{t})dy$. For uniformly hyperbolic dynamical systems without explicit time dependence, the measures $\,\mu_{t}\,$ tend for large $\,t\,$ to the invariant SRB measure $\,\mu_{\infty}\,$ and the mean rate of entropy production in the environment converges to the expectation of the phase-space contraction rate $\,-\nabla\cdot u\,$ with respect to $\,\mu_{\infty}\,$ . A discussion of the relation between of the phase-space contraction to the production of thermodynamic entropy in deterministic dynamics employing models of finite-dimensional thermostats may be found in .

If the measure $\,\mu_{T}\,$ is not obtained by evolving dynamically $\,\mu_{0}\,$ then one has to distinguish between the measures $\,\mu_{T}\,$ and $\,\mu_{0}P_{0,T}$. In this case, the relation (10.2) is modified to

i.e. the left hand side is increased by the relative entropy of the measure $\,\mu_{T}\,$ with respect to the measure obtained from $\,\mu_{0}\,$ by the dynamical evolution. Consequently, the average \,\big{\langle}{\cal W}\big{\rangle}\, is minimal when $\,\mu_{T}=\mu_{0}P_{0,T}$.

Linear response for the Langevin dynamics

As noted in , fluctuations relations may be viewed as extensions to the non-perturbative regime of the Green-Kubo and Onsager relations for the nonequilibrium transport coefficients valid within the linear response description of the vicinity of the equilibrium. Here, for the sake of completeness, we shall show how such relations follow formally from the Jarzynski equality (8.7) for the Langevin dynamics. To this end, we shall consider the latter with a time independent Hamiltonian $\,H_{t}\equiv H\,$ and the additional time-dependent force

where the couplings $\,g_{ta},\,a=1,2$, are arbitrary (regular) functions of time (and the summation over the index $\,a\,$ is understood). In the case at hand, we infer from Eq. (8.8) that

In particular, for the Langevin-Kramers equation (2.7),

is the power injected by the external force $\,f^{a}\,$ (in the $\,\beta^{-1}\,$ units). The quantities $\,J^{a}\,$ are often called fluxes associated to the forces $\,G^{a}$.

Let us denote by \,\big{\langle}{\cal F}\big{\rangle}\, the expectation defined by Eq. (8.4) with $\,\mu_{0}\,$ standing for the Gibbs measure $\,Z^{-1}{\rm e}^{-\beta H}dx\,$ and by \,\big{\langle}{\cal F}\big{\rangle}_{0}\, the same expectation taken for $\,g_{ta}\equiv 0$, i.e. in the equilibrium system. Expanding Eq. (8.7) up to the second order in $\,g_{ta}\,$ and abbreviating $\,J^{a}({\mathsf{x}}_{t})\equiv J^{a}_{t}$, we obtain the identity

where the insertion of the response field $\,{\cal R}^{a}_{t}\,$ is defined by the relation

Note that \,\big{\langle}J^{a}_{t}\,{\cal R}^{b}_{t^{\prime}}\big{\rangle}_{0}=0\, for $\,t^{\prime}>t\,$ because of the causal nature of the stochastic evolution. The vanishing of the term linear in $\,g_{ta}\,$ in Eq. (11.1) implies that the equilibrium expectation of the fluxes $\,J^{a}\,$ vanishes

which is easy to check directly. Stripping the quadratic term in Eq. (11.1) of arbitrary functions $\,g_{ta}$, we infer that

The integration of the latter equation over $\,t^{\prime}\geq 0\,$ results in the relation

where on the left hand side we consider the derivative with respect to the coupling $\,g_{b}\,$ constant in time. In the limit $\,t\to\infty$, we may expect the convergence of the expectation \,\big{\langle}J^{a}_{t}\big{\rangle}\, in the presence of the time-independent force $\,g_{a}G^{a}\,$ (and of its derivatives over $\,g_{b}$) to the nonequilibrium stationary expectation \,\big{\langle}J^{a}_{t}\big{\rangle}_{st}\, (and its derivatives). Let us also assume that the temporal decay of the stationary equilibrium correlation function of the fluxes is sufficiently fast, e.g. exponential. These may be often established for the dynamics governed by the Langevin equation by studying the properties of its generator. With these assumptions, Eq. (11.2) implies

The stationary equilibrium correlation function \,\big{\langle}J^{a}_{t}\,J^{b}_{t^{\prime}}\big{\rangle}_{0}\, depends only on the difference $\,t-t^{\prime}\,$ of times. Besides, if the system is time-reversible, then \,\big{\langle}J^{a}_{t}\,J^{b}_{t^{\prime}}\big{\rangle}_{0}=\big{\langle}J^{b}_{t}\,J^{a}_{t^{\prime}}\big{\rangle}_{0}\, and the Green-Kubo formula may be rewritten in the form

2 Fluctuation-dissipation theorem

Let us consider again the Jarzynski equality for the Langevin dynamics, this time in the absence of the additional force $\,G_{t}\,$ but with a time dependent Hamiltonian

where $\,h_{ta},\,a=1,2$, vanish at $\,t=0\,$ and $\,O^{a}(x)\,$ are functions of $\,x\,$ (“observables”). In this case, Eq. (8.8) reduces to the relation

Expanding the left hand side of the Jarzynski equality (8.7) up to the second order in $\,h_{ta}\,$ and abbreviating $\,O^{a}({\mathsf{x}}_{t})\equiv O^{a}_{t}$, we infer that

where the insertion of the response field $\,R^{a}_{t}\,$ is defined similarly as that of $\,{\cal R}^{a}_{t}\,$ before by

Again, similarly as before, \,\big{\langle}O^{a}_{t}\,R^{b}_{t^{\prime}}\big{\rangle}_{0}=0\, for $\,t^{\prime}>t\,$ because of causality.

The first order equality (11.4) is equivalent to the time-independence of the equilibrium expectation of $\,O^{a}_{t}$. As for the second order relation (11.7), upon expressing $\,h_{ta}\,$ as the integral of $\,\dot{h}_{ta}$, it is turned into the equality

After the change of the order of integration over $\,t^{\prime}\,$ and $\,t^{\prime\prime}\,$ followed by the interchange of those symbols, the right hand side becomes

with the use of causality. Stripping the resulting identity of the integrals against arbitrary functions $\,\dot{h}_{ta}$, we obtain the identity

which is the integrated version of the differential relation between the dynamical 2-point correlation function and the response function:

Proposition 5 (Fluctuation-dissipation theorem). For $\,t>t^{\prime}$,

Note the explicit factor $\,\beta\,$ in this identity. Relations between the dynamical correlation functions and the response functions were used in recent years to extend the concept of temperature to nonequilibrium systems .

One-dimensional Langevin equation with flux solution

Let us consider, as an illustration, the one-dimensional Langevin equation of the form

with \,\big{\langle}\zeta_{t}\,\zeta_{t^{\prime}}\big{\rangle}=2\beta^{-1}\delta(t-t^{\prime})\, (any force is a gradient in one dimension). As before, $\,{\mathsf{x}}_{t}\,$ will represent the Markov process solving the SDE (12.1). First, let us consider the time-independent case with a polynomial Hamiltonian $\,H(x)=ax^{k}+\dots\,$ with $\,a\not=0\,$ and the dots representing lower order terms.

If $\,k=0\,$ then, up to a linear change of variables, $\,{\mathsf{x}}_{t}\,$ is a Brownian motion and does not have an invariant probability measure.

If $\,k=1\,$ then $\,{\mathsf{x}}_{t}+at\,$ is, up to a linear change of variables, a Brownian motion and $\,{\mathsf{x}}_{t}\,$ still does not have an invariant probability measure.

If $\,k\geq 2\,$ and is even then for $\,a>0\,$ the Gibbs measure $\,\mu_{0}(dx)=Z^{-1}{\rm e}^{-\beta H(x)}dx\,$ provides the unique invariant probability measure of the process $\,{\mathsf{x}}_{t}$. It satisfies the detailed balance condition $\,j(x)=0$, where $\,j(x)\,$ is the probability current defined by Eq. (3.8). If $\,a<0$, however, then the Gibbs density $\,{\rm e}^{-\beta H(x)}\,$ is not normalizableThis leads to the breaking of the quantum-mechanical supersymmetry underlying the Fokker-Planck formulation of the Langevin dynamics .. In this case, the process $\,{\mathsf{x}}_{t}\,$ escapes to $\,\pm\infty\,$ in finite time with probability one and it has no invariant probability measure.

If $\,k\geq 3\,$ and is odd then the Gibbs density $\,{\rm e}^{-\beta H(x)}\,$ is not normalizable. The process $\,{\mathsf{x}}_{t}\,$ escapes in finite time to $\,-\infty\,$ if $\,a>0\,$ and to $\,+\infty\,$ if $\,a<0$, but it has a realization with the trajectories that reappear immediately from $\,\pm\infty$. Such a resuscitating process has a unique invariant probability measure

with the density $\,{\rm e}^{-\varphi_{0}(x)}={\cal O}(x^{-k+1})\,$ when $\,x\to\pm\infty\,$ and $\,N\,$ the (positive) normalization constant. The measure $\mu_{0}\,$ corresponds to a constant probability current $\,j(x)=\pm(\beta N)^{-1}\,$ and the model provides the simplest example on a nonequilibrium steady state with a constant flux.

Let us look closer at the last case. Adding the time-dependence and taking $\,\varphi_{t}\,$ as in Eq. (12.2) but with $\,H_{t}\,$ replacing $\,H$, we obtain the Hatano-Sasa version of the Jarzynski equality (8.7) with $\,{\cal W}={\cal W}^{ex}\,$ given by Eq. (8.18). Suppose, in particular, that the time dependence of $\,H_{t}\,$ has the form (11.3) with functions $\,O^{a}\,$ having compact support. Let us introduced also the deformed observables

Expanding the Jarzynski identity (8.7) to the second order in $\,h_{at}\,$ as in Sect. 11.2, one obtains:

Proposition 6 (Deformed fluctuation-dissipation relation). For $\,t>t^{\prime}$,

where $\,P_{t}(x,dy)\,$ is the transition probability in the stationary process and $\,A^{a}=O^{a}-\widehat{O^{a}}$.

Remark 3. It is easy to show directly, that Eq. (12.3) still holds if $\,A^{a}\,$ is replaced by $\,O^{a}$. Note that the term on the right hand side of (12.3) violating the standard fluctuation-dissipation theorem (11.8) contains the constant flux of the probability current $\,j(x)\,$ as a factor. Proof of Proposition 6 and of its version with $\,A^{a}\,$ replaced by $\,O^{a}\,$ will be given in .