Fluctuation relations in simple examples of non-equilibrium steady states

Introduction

In statistical mechanics, the fluctuation-dissipation theorem (FDT) provides a simple relation in an equilibrium state between the response of the fixed-time averages to small time-dependent perturbations of the Hamiltonian and the dynamical correlations . Let $\,O^{a}(x)\,$ for $\,a=1,\dots,A\,$ be a collection of (classical) observables. With the shorthand notation $\,O^{a}_{t}\,$ for the single-time functions $\,O^{a}(x_{t})\,$ of the dynamical process $\,x_{t}$, the response function and the 2-time correlation function in a steady state are, respectively,

where $\,\langle\,-\,\rangle_{{}_{h}}\,$ denotes the dynamical expectation obtained from the steady state by replacing the time-independent Hamiltonian $\,H(x)\,$ by a slightly perturbed time-dependent one $\,H(x)-h_{t}\hskip 0.71114ptO^{b}(x)$. The FDT asserts that when the unperturbed state is the equilibrium at inverse temperature $\,\beta\,$ then

Such a direct relation between the response and correlation functions is violated in systems out of equilibrium and a lot of interest in the research on non-equilibrium statistical mechanics was devoted to such violations. In particular, they were studied intensively for glassy systems , for colloidal suspensions , for granular matter , and for biophysical systems . In recent years, it has been realized that the FDT, as well as the Green-Kubo relation, another linear response law of the equilibrium regime, are special cases of more general fluctuation relations that hold also far from equilibrium. Such relations pertain either to non-stationary transient situations or to stationary regimes . In particular, the so called Jarzynski equality for the dynamics with a time dependent Hamiltonian reduces to the FDT for tiny time variations .

In the present paper, we revisit the violations of the FDT in simple examples of non-equilibrium steady states (NESS) for systems with few degrees of freedom evolving according to the Langevin equation possibly including non-conservative forces, see e.g. . For such systems, we show a modified fluctuation-dissipation theorem (MFDT) that may be written in the form

similar to the equilibrium relation, somewhat in the spirit of ref. . Above, $\,{\cal R}^{ab}_{{}_{L}}(t,s)\,$ and $\,{\cal C}^{ab}_{{}_{L}}(t,s)\,$ denote the response function and the dynamical correlation function in the Lagrangian frame moving with the mean local velocity $\,\nu_{{}_{0}}(x)\,$ of the NESS. The Lagrangian-frame functions are obtained by replacing in the definitions (1.1) the time-independent observables $\,O^{a}(x)\,$ by the time-dependent ones $\,O^{a}(t,x)\,$ that evolve according to the advection equation

i.e. are frozen in the Lagrangian frame. In the equilibrium, the mean local velocity $\,\nu_{{}_{0}}(x)\,$ vanishes and the MFDT (1.3) becomes the FDT (1.2).

The other goal of the present work is to explain how the MFDT (1.3) may be obtained from more general fluctuation relations by restricting them to the regime close to NESS, similarly as for the case of the equilibrium FDT. Before doing that, we recall different fluctuation relations holding arbitrarily far from stationarity and equilibrium in Langevin systems and their perturbations. Our discussion follows with minor modifications the recent exposition .

The general results presented in the paper apply, in particular, to two types of one-dimensional systems with NESS. The first type describes an exploding Langevin dynamics on the line with a non-Gibbsian invariant measure. This case arises, for example, when one studies the tangent process for particles with inertia moving in one-dimensional Kraichnan’s random velocities . Such velocities $\,v_{t}(x)\,$ form a Gaussian ensemble with mean zero and covariance

The evolution of the inertial particles is described by the stochastic differential equation (SDE)

where $\,\tau\,$ is the Stokes time measuring the time-delay of the particles relative to the flow motion. The separation between two infinitesimally close trajectories satisfies then the equations

where one may replace $\,\frac{{}_{1}}{{}^{\tau}}\partial_{x}v_{t}(x)\,$ on the right hand side by a white noise $\,\zeta_{t}\,$ with the covariance

for $\,\beta^{-1}=-\frac{1}{2\tau^{2}}\partial_{x}^{2}D(0)$. For the ratio $\,X=\frac{\delta u}{\delta x}$, one obtains the SDE

which has the form of a one-dimensional overdamped Langevin equation for the Hamiltonian $\,H(X)=\frac{1}{3}X^{3}+\frac{1}{2\tau}X^{2}$. The process $\,X_{t}\,$ solving Eq. (1.9) escapes in finite time to $\,-\infty\,$ but has a realization with trajectories that reappear immediately from $\,+\infty$, see Appendix A. This corresponds to the solutions for $\,(\delta x,\delta u)\,$ where $\,\delta x\,$ passes through zero with a non-vanishing speed, i.e. to the crossing of close particle trajectories with faster particles overcoming slower ones (allowed in this model of a dilute particle suspension with no pressure and no back-reaction on the flow ). The Gibbs density $\,{\rm e}^{-\beta H}\,$ is not normalizable here. The resurrecting process has, however, a non-Gibbsian invariant probability measure with constant probability flux. The top Lyapunov exponent for the random dynamical system (1.6) is obtained as the mean value in this measure of $\,X\,$ (which is the temporal logarithmic derivative of $\,|\delta x|$) . The above is a variation of a much older story of the one-dimensional Anderson localization in the stationary Schrödinger equation

with a $\,\delta$-correlated potential $\,V(x)$. For $\,Y=(\frac{d}{dx}\psi)/\psi$, one obtains the equation

that may be viewed as a stochastic evolution equation if $\,x\,$ is interpreted as time. The invariant measure with constant flux for such an SDE was already used in . The substitution $\,Y=X+\frac{1}{2\tau}$, $\,E=-\frac{1}{4\tau^{2}}$, $\,V=\zeta\,$ turns Eq. (1.11) to Eq. (1.9) (provided that one replaces $\,x\,$ by $\,t$).

The second particular type of systems with NESS covered by our discussion is obtained by adding a non-conservative force to the Langevin dynamics. More specifically, we shall consider a particle which moves on a circle according to the SDE

where, as before, $\,\zeta_{t}\,$ is the white noise with covariance (1.8). The above dynamics pertains again to the overdamped regime where it is the particle velocity rather than the particle acceleration that is proportional to the force. The angular coordinate $\,x\,$ will be taken modulo $\,2\pi$. We shall assume that $\,H(x+2\pi)=H(x)\,$ and $\,G(x+2\pi)=G(x)\,$ but $\,\int_{0}^{2\pi}G(x)dx\not=0\,$ so that the force $\,G\,$ is not a gradient and it drives the system out of equilibrium. Eq. (1.12) was used, for example, to describe the motion of a colloidal particle in an optical trap . It was discussed recently in in a context similar to the one of this work.

The present paper is organized as follows. Sec. 2, returns to the discussion of stationary Langevin diffusion processes, presenting more details on the one-dimensional systems with explicit non-Gibbsian invariant measures . For such systems, we examine in Sec. 3 the simplest fluctuation-response relation that describes the change of the invariant measure under a small time-independent variation of the Hamiltonian. In Sec. 4, we prove the MFDT (1.3) that holds around NESS of the Langevin-type dynamics, in particular, in the one-dimensional cases with explicit invariant measures. Sec. 5 is devoted to a brief presentation of general fluctuation relations for SDE’s . These are specified for the Langevin systems under consideration in Sec. 6. In particular, we describe the Crooks detailed fluctuation relation and the Hatano-Sasa version of the Jarzynski equality , both holding arbitrarily far from stationarity. In Sec. 7, we return to the MFDT, showing that it may be viewed as a limiting case around the stationary situation of the Crooks transient fluctuation relation or, in a special case, of the Jarzynski-Hatano-Sasa equality. Finally, after brief Conclusions, we collect in Appendix A some facts about the one-dimensional processes with explosion and illustrate in Appendix B the MFDT by an explicit calculation for the Langevin particle driven by a constant force along a circle.

Acknowledgements. The work of G.F. was supported in part by the US National Science Foundation under Grant No. PHY05-51164 and by the Minerva Foundation.

NESS in Langevin processes

The general stationary dynamics that we consider is described by the Langevin equation in $d$-dimensions with an external force:

where $\,\Gamma\,$ is a constant non-negative matrix and $\,\Pi\,$ an antisymmetric one, $\,H\,$ is the Hamiltonian, $\,G\,$ the external force and the white noise $\,\zeta_{t}\,$ has the covariance

The deterministic force $-\Gamma\nabla H$ decreases the energy, driving the solution towards the minimum of $\,H$, if it exists, whereas the noise mimics the effect of a thermal bath. We have added the Hamiltonian force $\Pi\nabla H\,$ that preserves the energy in order to cover systems governed by Langevin-Kramers equations or Fermi-Pasta-Ulam chains . The generator $\,L\,$ of the process $\,x_{t}\,$ satisfying the SDE (2.1) is defined by the relation

It is a second order differential operator:

in the vector notation, with the formal adjoint

The transition probabilities $\,P_{t}(x,dy)\,$ of the process satisfy the evolution equation

with the currentFor convenience, we have included into the current the term $\,\beta^{-1}\Pi\nabla\varrho$.

Following , let us introduce the mean local velocity $\,\nu_{t}=\varrho_{t}^{-1}j_{t}$. With the use of the velocity field, the above continuity equation may be rewritten in the hydrodynamical form as the advection equation for the density $\,\varrho_{t}\hskip 0.71114pt$,

stating that $\,\rho_{t}\,$ is annihilated by the convective derivative or, in other words, that it evolves as the density of Lagrangian particles whose trajectories obey the ordinary differential equation

For an invariant density, the corresponding current is conserved: $\,\nabla\cdot j=0$. If for the density $\,\varrho\,$ the current $\,j\,$ itself vanishes then one says that the dynamics (2.2) satisfies the detailed balance relative to $\,\varrho$. The detailed balance holds relative to the Gibbs density $\,{\rm e}^{-\beta H}\,$ if $\,G=0\,$ (this was assured by the addition of the term $\,\beta^{-1}\Pi\nabla\varrho\,$ to the current). When $\,G\not=0$, the invariant density is not known explicitly, in general, even if it exists. There are, however, special cases of processes satisfying the SDE (2.2) where one may obtain an analytic formula for a non-Gibbsian invariant density.

The first example, that we shall call type 1 below, is obtained for the Langevin equation on the line. In this case, any force is a gradient so that, upon setting for simplicity $\,\Gamma=1$, the dynamical equation becomes

where $\,Z\,$ is the (positive) normalization constant. The density $\,\varrho_{{}_{H}}\,$ of $\,\mu\,$ behaves as $\,(Za\beta kx^{k-1})^{-1}\,$ when $\,x\to\pm\infty$, see the estimate (A.18) in Appendix A. It corresponds to the constant current $\,j=-(\beta Z)^{-1}\,$ with the flux towards negative $\,x$. The situation provides one of the simplest examples of NESS. In particular, the inertial particle in the one-dimensional Kraichnan flow and the Anderson localization in the one-dimensional $\,\delta$-correlated potential lead naturally to the resurrecting processes corresponding to $\,k=3$, as discussed in the introduction.

2 NESS for forced diffusions on circle

The second model with an explicit analytic expression for the invariant non-Gibbsian measure, that we shall call type 2 below, is the perturbed one-dimensional Langevin equation (1.12) on the unit circle. Now, the unique invariant probability measure is given by the formula

Also here the density $\,\varrho_{{}_{H}}\,$ of the measure $\mu\,$ corresponds to a constant probability current

In the following, we shall see how the presence of the probability flux in NESS deforms the usual fluctuation relations.

Modified fluctuation-response relation

As a warmup, let us see what is the form taken by the most elementary fluctuation-response relation in the one-dimensional systems with NESS that we discussed above. The setup of the fluctuation-response relation is as follows. One prepares the system in the far past in the invariant state with probability density $\,\varrho_{{}_{H}}\,$ that is given by Eqs. (2.12) and (2.13) for the type 1 and type 2 systems, respectively. At $\,t=0$, the Hamiltonian $\,H\,$ is perturbed by a small time-independent potential $\,V\,$ (vanishing sufficiently fast when $\,x\to\pm\infty\,$ for type 1), leading to the change $\,H\mapsto H^{\prime}=H+V$. The systems evolves then and converges toward the new steady state with the probability density $\,\varrho_{{}_{H^{\prime}}}$. The fluctuation-response relation compares the initial and the final averages of an observable $\,O_{t}\equiv O(x_{t})\hskip 0.71114pt$:

By a straightforward differentiation of the explicit formulae for the invariant densities, one obtains the identity

up to terms of the second order in $\,V$, where

for the systems of type 1 and type 2, respectively. Eq. (3.2) deforms the usual fluctuation-response relation around the Gibbs state by replacing $\,V\,$ by $\,(V-\widehat{V})\,$ with the averaged potential $\,\widehat{V}\,$ dependent on the initial Hamiltonian $\,H$.

Modified fluctuation-dissipation theorem

Coming back to the general case, let us consider a system prepared at negative times in the steady state of the stationary Langevin dynamics (2.1). This forces the time zero value $\,x_{{}_{0}}\,$ of the corresponding process to be distributed according to the invariant probability measure $\,\mu_{{}_{0}}(dx)=\varrho_{{}_{0}}(x)\,dx$. At $t=0$, one switch on a non-stationary perturbation taking the Hamiltonian for the positive times to be equal to

where $\,h_{a,t}\,$ carry the time dependence and functions $\,O^{a}(x)\,$ (the “observables”) are supposed, for simplicity, to vanish sufficiently fast when $\,|x|\to\infty$. We denote by \,\big{\langle}{\cal F}\big{\rangle}_{{}_{h}}\, the corresponding expectation, with \,\big{\langle}{\cal F}\big{\rangle}_{{}_{0}}\, referring to the non-perturbed case. The expression

defines the response correlations. To shorten further the notations, let us set

for $\,O_{t}\equiv O(x_{t})\,$ and the induced observable

with $\,j_{{}_{0}}\,$ standing for the current (2.8) corresponding to the invariant density $\,\varrho_{{}_{0}}$. Note that for the one-dimensional NESS with constant probability current $\,j_{{}_{0}}$, the observable $\,B^{b}=j_{{}_{0}}\hskip 0.71114pt\varrho_{{}_{0}}^{-1}\partial_{x}O^{b}\,$ has the probability flux as an explicit factor. Remark that, by causality, the response function $\,{\cal R}^{ab}(t-s)\,$ vanishes for $\,s\geq t$. We shall show the following modified version of the fluctuation-dissipation theorem (MFDT) holding for $\,t>s\hskip 0.71114pt$:

The second term on the right hand side of Eq. (4.5) is new as compared to the FDT around the equilibrium steady state. Indeed, in the equilibrium case, the external force $\,G=0\,$ and $\,\varrho_{{}_{0}}=Z^{-1}{\rm e}^{-\beta H}\,$ is the normalized Gibbs factor with $\,j_{{}_{0}}=0\,$ so that $\,B^{b}=0\,$ and the MFDT (4.5) reduces to the standard equilibrium form (1.2).

What the formula (4.5) for the response function means is better understood by rewriting it with the explicit form of the right hand side as

where $\,P_{t}(x,dy)\,$ denotes the stationary transition probabilities of the unperturbed process and we have integrated by parts to obtain the second equality, setting $\,\nu_{{}_{0}}=\varrho_{{}_{0}}^{-1}j_{{}_{0}}$. Note that the time derivative $\,\partial_{s}\,$ of the equilibrium relation has been replaced by the convective derivative $\,\partial_{s}+\nabla\cdot\nu_{{}_{0}}(x)\,$ which acts on the first component of the joint probability density function of the time $\,s\,$ and time $\,t\,$ values of the stationary process $\,x_{t}$. This suggests that the MFDT (4.5) should take the equilibrium form in the Lagrangian frame moving with the stationary mean local velocity $\,\nu_{{}_{0}}(x)$.

To render this interpretation more transparent, let us replace the time-independent observables $\,O^{a}(x)\,$ by the time dependent ones $\,O^{a}(t,x)\,$ evolving according to the advection equation (1.4). We shall define the Lagrangian-frame response function and correlations function by

where \,\big{\langle}\,-\,\big{\rangle}_{{}_{h}}\, denotes now the expectation referring to the Hamiltonian $\,H_{t}(x)=H(x)-\sum\limits_{a}h_{t,a}\,O^{a}(t,x)$. $\,O_{t}(t)\equiv O(t,x_{t})\,$ with the double time-dependence. Writing the MFDT (4.5) with the explicit right hand side given by Eq. (4.1) for the observables $\,O^{a}(t,x)$, one casts this relation into the form

where the last equality follows from the advection equation (1.4). This proves the identity (1.3) announced in the introduction. We should stress that, in spite of the similarity between that relation and the equilibrium FDT (1.2), in general it is not true that the dynamical process $\,x_{t}\,$ viewed in the Lagrangian frame of the velocity field $\,\nu_{0}\,$ is governed by an equilibrium Langevin equation, although this is what happens in the simple example considered in Appendix B.

For the Langevin process on the circle, a fluctuation-dissipation relation for velocities similar to (4.5) was discussed in , see Eq. (11) there, with the interpretation similar in spirit, but not in form, to the above one, see the subsequent discussion there. One of the consequences of the fluctuation-dissipation relation of linking the effective diffusivity and mobility was checked experimentally in , see also .

It is sometimes more interesting, especially for applications, to re-express the fluctuation-dissipation relations in terms of the frequency space quantities. Let

Note that $\,\hat{\cal R}^{ab}(\omega)\,$ measures the response to the time-dependent potential of frequency $\,\omega$. The MFDT (4.5) is equivalent to the relation

In general, assuming that the transition probabilities $\,P_{t}(x,dy)\,$ converge at long times to the invariant measure, all three terms of Eq. (4.5) tend to zero when $\,(t-s)\to\infty$. Mimicking the idea employed with success for disordered systems , their relative proportions, or the relative proportions of the corresponding terms in Eq. (4.13), could be used to define dynamical temperatures that would, in general, depend also on the observables involved. We discuss those proportions in a simple case of the Langevin dynamics on a circle with a constant force in Appendix B.

We propose three derivations of the result (4.5): the first one direct, that we shall present now, and the next two ones from the general fluctuation relations that will be discussed in the subsequent sections.

2 Direct derivation

The beginning of the argument is quite standard, see e.g. or Sec. 2.3.2 of . By the definition of the response correlations,

where $\,\varrho_{t}\,$ is the density obtained by the perturbed dynamical evolution (2.7) from $\,\varrho_{{}_{0}}$. Using the explicit form (2.8) of the current, one obtains by the first order perturbation the relation

Now, a straightforward although somewhat tedious algebra shows that

where the adjoint generator $\,L^{\dagger}\,$ is given by Eq. (2.5) and in the last equality we have used the conservation of the current $\,j_{{}_{0}}$. Substituting this identity to Eq. (4.16) and integrating the term with $\,L^{\dagger}\,$ by parts, we obtain the relation

which, together with Eq. (2.6), implies the MFDT (4.5).

General fluctuation relations

In , two of us discussed arbitrary diffusion processes in $d$ dimensions defined by the Stratonovich SDE

where $\,u_{t}(x)\,$ is a time-dependent deterministic vector field (a drift) and $\,v_{t}(x)\,$ is a Gaussian random vector field with mean zero and covariance

The Langevin equation (2.1) provides a special example of such an SDE. For the processes solving Eq. (5.1), we showed, combining the Girsanov and the Feynman-Kac formulae, a detailed fluctuation relation (DFR)

$\,\mu_{{}_{0}}(dx)=\varrho_{{}_{0}}(x)\,dx\,$ is the initial distribution of the original (forward) process,

$\,\mu_{{}_{0}}^{\prime}(dx)=\varrho_{{}_{0}}^{\prime}(x)\,dx\,$ is the initial distribution of the backward process obtained from the forward process by applying a time inversion (see below),

$\,P_{{}_{T}}(x;dy,dW)$ is the joint probability distribution of the time $\,T\,$ position $\,x_{{}_{T}}\,$ of the forward process starting at time zero at $\,x\,$ and of a functional $\,{\cal W}_{{}_{T}}\,$ of the same process on the interval $\,[0,T]\,$ (described later),

$\,P_{{}_{T}}^{{}^{\prime}}(x;dy,dW)$ is the similar joint probability distribution for the backward process.

The key behind the DFR is the action of a time inversion on the forward system. First, such an inversion acts on time and space by an involution

The above involution may be extended to the action $\,\,x\mapsto\widetilde{x}\,\,$ on trajectories by the formula $\,\widetilde{x}_{t}=x^{*}_{T-t}\,$ and, further, to the action on functionals of trajectories $\,\,{\cal F}\mapsto\widetilde{\cal F}\,$ by setting $\,\widetilde{{\cal F}}[x]={\cal F}[\widetilde{x}]$. Second, to recover a variety of fluctuation relations discussed in the literature , we allow for a non-trivial behavior of the drift $\,u_{t}(x)\,$ under the time inversion, dividing it into two parts, $\,u=u_{+}+u_{-}$, with $\,u_{+}\,$ transforming as a vector field under the space-time involution (5.4) and $\,u_{-}\,$ as a pseudo-vector field, i.e. defining

The random field $\,v_{t}(x)\,$ may be transformed with either rule. By definition, the backward process satisfies then the Stratonovich SDE

and is, in general, different from the naive time inversion $\,\widetilde{x}_{t}\,$ of the forward process. The functional $\,{\cal W}_{{}_{T}}\,$ involved in the DFR depends explicitly on the densities $\,\varrho_{{}_{0}}\,$ and $\,\varrho_{{}_{T}}$, where the latter is defined by the relation $\,\mu^{\prime}_{{}_{0}}(dx^{*})=\varrho_{{}_{T}}(x)\,dx\,$:

with the notation $\,\Delta_{{}_{T}}\ln\varrho\equiv\ln\varrho_{{}_{T}}(x_{{}_{T}})-\ln\varrho_{{}_{0}}(x_{{}_{0}})$. In the above formula,

where $\,d^{ij}_{t}(x)=D^{ij}_{t}(x,x)\,$ and $\,\widehat{u}_{t,+}^{i}(x)=u_{t,+}^{i}(x)-\frac{1}{2}\partial_{y^{j}}D^{ij}_{t}(x,y)|_{y=x}$. The time integral in Eq. (5.7) should be taken in the Stratonovich sense. The functional $\,{\cal W}^{\prime}_{{}_{T}}\,$ for the backward process is defined in the same way, setting $\,\mu_{{}_{0}}(dx^{*})=\varrho^{\prime}_{T}(x)dx$. One has the relation

where the tilde denotes the involution of trajectory functionals introduced before.

The quantity $\,{\cal J}_{t}\,$ has the interpretation of the rate of entropy production in the environment modeled by the thermal noise. When the density $\,\varrho_{{}_{T}}\,$ coincides with the density obtained from $\,\varrho_{{}_{0}}\,$ by the dynamical evolution (2.7), where now the current

then the first contribution $\,-\Delta_{{}_{T}}\ln\varrho\,$ to $\,{\cal W}_{{}_{T}}\,$ may be interpreted as the change in the instantaneous entropy of the process. In this case, the functional $\,{\cal W}_{{}_{T}}\,$ becomes equal to the overall entropy production. Keep in mind that this is a fluctuating quantity which, in general, may take both positive and negative values.

The DFR (5.3) holds even if the measures $\,\mu_{{}_{0}}\,$ and $\,\mu_{{}_{0}}^{\prime}\,$ are not normalized, or even not normalizable. For normalized initial measures, we denote by

the averages over the realizations of the forward and the backward process $\,x_{t},\ 0\leq t\leq T,\,$ with $\,x_{{}_{0}}\,$ distributed according to the probability measure $\,\mu_{{}_{0}}\,$ and $\,\mu^{\prime}_{{}_{0}}$, respectively. $\,{\cal M}[dx]\,$ and $\,{\cal M}^{\prime}[dx]\,$ stand for the corresponding measures over the space of trajectories. One of the immediate consequences of the DFR (5.3) is the identity

obtained by the integration of the both sides. This is a generalization of the celebrated Jarzynski equality . The relation (5.12) implies the inequality \,\big{\langle}{\cal W}_{{}_{T}}\big{\rangle}\geq 0\, that has the form of the second law of thermodynamics stating the positivity of the average entropy production. The Jarzynski equality, however, provides also information about an exponential suppression of the events with negative entropy production in non-equilibrium systems for which \,\big{\langle}{\cal W}_{{}_{T}}\big{\rangle}>0.

With a little more work involving a multiple superposition of the FDR (5.3), the latter may be cast into the Crooks form

In terms of the trajectory measures $\,{\cal M}[dx]\,$ and $\,{\cal M}^{\prime}[dx]$, this becomes the identity

where $\,\widetilde{{\cal M}}^{\prime}[dx]={\cal M}^{\prime}[d\widetilde{x}]$. Eq. (5.14) permits to interpret the expectation of $\,{\cal W}_{{}_{T}}\,$ as the relative entropy of the trajectory measures:

in line with the above entropic interpretation of the functional $\,{\cal W}_{{}_{T}}$.

Fluctuation relations in Langevin dynamics

Let us specify the DFR (5.3) to the case of Langevin dynamics (2.2) with, possibly, time-dependent Hamiltonian $\,H\,$ and external force $\,G$. A canonical choice of the time inversion for such a system takes

and a linear involution $\,x^{*}=Rx\,$ such that $\,R\Gamma R^{t}=\Gamma\,$ and $\,R\Pi R^{t}=-\Pi\,$ . For example, for the Langevin-Kramers dynamics in the phase-space, the usual $\,R\,$ changes the sign of momenta. The backward dynamics has now the same form as the forward one, with the time-reversed Hamiltonian $\,H^{\prime}_{t}(x)=H_{T-t}(Rx)\,$ and the time-reversed external force $\,G^{\prime}_{t}(x)=-RG_{T-t}(Rx)$. In this case, we may use the Gibbs densities $\,\varrho_{t}(x)=Z_{t}^{-1}{\rm e}^{-\beta H_{t}(x)}=\varrho^{\prime}_{T-t}(Rx)\,$ at the initial and final times, with $\,Z_{t}\,$ standing for the partition function $\,\int{\rm e}^{-\beta H_{t}(x)}dx\,$ if the integral is finite and $\,Z_{t}=1\,$ otherwise. A straightforward calculation gives:

For normalizable Gibbs factors this is often called the “dissipative work”. $\,{\cal W}^{\prime}_{{}_{T}}\,$ is given by the same expression with $\,H_{t}\,$ and $\,G_{t}\,$ replaced by $\,H^{\prime}_{t}\,$ and $\,G^{\prime}_{t}$.

Another useful choice of the time inversion is based on the eventual knowledge of the densities $\,\varrho_{t}\,$ corresponding to the conserved currents with $\,\nabla\cdot j_{t}=0$. Note that such densities would be left invariant by the evolution (2.7) if the time-dependence of the Hamiltonian and of the external force were frozen to the instantaneous values $\,H_{t}\,$ and $\,G_{t}$. One takes

With the linear involution $x^{*}=Rx\,$ as above, the backward process has

where $\,\varrho^{\prime}_{t}(x)=\varrho_{T-t}(Rx)\,$ and $\,H^{\prime}\,$ and $\,G^{\prime}\,$ are as before. The current corresponding to the density $\,\varrho^{\prime}_{t}\,$ satisfies

It is also conserved. Such a time inversion (for $\,R=1$) was considered in and, more explicitly, in . In , it was called the current reversal. The DFR (5.3) holds now for

and $\,{\cal W}^{\prime}_{{}_{T}}\,$ given by the same expression with $\,\varrho_{t}\,$ replaced by $\,\varrho^{\prime}_{t}$. The Jarzynski equality (5.12) for this case (assuming that $\,\varrho_{t}\,$ are normalized) was first proven by Hatano and Sasa in . Note that if $\,G=0\,$ then the current corresponding to the densities $\,\varrho_{t}=Z_{t}^{-1}{\rm e}^{-\beta H_{t}}\,$ is conserved and with this choice of $\,\varrho_{t}\,$ the two time inversions coincide. In particular, for $\,G=0\,$ and the time-independent $\,H_{t}\equiv H$, the functional $\,{\cal W}_{{}_{T}}\,$ identically vanishes and the DFR reduces to the equality

which is a more global version of the detailed balance relation. On the right hand side, one may replace $\,P^{\prime}_{T}\,$ by $\,P_{{}_{T}}\,$ for $\,\Pi=0\,$ and $\,x^{*}\equiv x\,$ because in that case, the forward and backward processes have the same distribution..

Let us consider the process solving the SDE (2.11), with $\,H_{t}(x)=ax^{k}+o(|x|^{k})\,$ at large $\,|x|\,$ with odd $\,k\geq 3\,$ and $\,a>0$. We admit a mild time-dependence of $\,H_{t}(x)\,$ disappearing when $\,x\to\pm\infty$. The corresponding process still has a resurrecting version, as in the stationary case described in Appendix A. Let us discuss first the canonical time inversion with the trivial involution $\,x^{*}=x\,$ leading to the backward process of the same type with $\,H_{t}(x)\,$ replaced by $\,H^{\prime}_{t}(x)=H_{T-t}(x)$. The definition of the functional $\,{\cal W}_{{}_{T}}\,$ for the resurrecting process requires a little care in order to account for the contributions from the jumps from $\,-\infty\,$ to $+\infty$. This may be done by compactifying the line to a circle writing $\,x=\cot{\theta}\,$ for $\,\theta\,$ modulo $\,\pi$. One has

and the integral diverges to $\,+\infty\,$ whenever $\,\theta_{t}\,$ passes from the negative to the positive values, i.e. whenever $\,x_{t}\,$ jumps from $\,-\infty\,$ to $\,+\infty$. Upon taking $\,\varrho_{t}={\rm e}^{-\beta H_{t}}$, we infer that

and similarly for $\,{\cal W}^{\prime}_{{}_{T}}$. As a result, the contributions of the rebirths to the DFR (5.3) trivially decouple reducing the latter to the identity

between the joint distributions of the endpoints and of the functional $\,\,{\cal W}_{{}_{T}}=\int_{0}^{T}(\partial_{t}H_{t})(x_{t})\,\,$ (usually called the “Jarzynski work”), or of its counterpart $\,{\cal W}^{\prime}_{{}_{T}}$, in the processes without rebirths. In the stationary case, Eq. (6.10) reduces to the detailed balance relation

which assures upon integration over $\,x\,$ that the process without rebirths preserves the infinite measure $\,{\rm e}^{-\beta H(x)}dx$.

On the other hand, one could use for the same SDE with the resurrecting solution the current reversal based on the splitting

of the drift $\,-\partial_{x}H_{t}$, with the density $\,\varrho_{{}_{H_{t}}}\,$ given by Eq. (2.12). The use of the involution $\,x^{*}=-x\,$ leads then to the backward process solving the SDE (2.11) with the Hamiltonian $\,H_{t}(x)\,$ replaced by

From the estimate (A.18) in Appendix A, one infers that $\,H^{\prime}_{t}(x)=ax^{k}+o(x^{k-1})\,$ for large $|x|\,$ (we have used the non-trivial spatial involution to keep $\,a\,$ positive). Hence $\,H^{\prime}_{t}\,$ is of the same type as the Hamiltonian $\,H_{t}\,$ for the forward process. In this case, the functionals $\,{\cal W}_{{}_{T}}\,$ and $\,{\cal W}^{\prime}_{{}_{T}}\,$ are given by Eq. (6.6) with $\,\varrho_{t}(x)\,$ replaced by $\,\varrho_{{}_{H_{t}}}(x)\,$ and $\,\varrho_{{}_{H_{T-t}}}(-x)=\varrho_{{}_{H^{\prime}_{t}}}(x)$, respectively, with no extra contributions from the rebirths. In the stationary case, one has $\,{\cal W}_{{}_{T}}=0={\cal W}^{\prime}_{{}_{T}}\,$ and the DFR (5.3) reduces to the modified detailed balance relation

The latter links the transition probabilities of the resurrecting forward and backward processes and assures upon integration over $\,x\,$ or $\,y\,$ that those processes preserve the probability measures $\,\varrho_{{}_{H}}(x)\,dx\,$ and $\,\varrho_{{}_{H}}(-x)\,dx$, respectively.

2 Fluctuation relations for forced diffusions on circle

Consider the process satisfying the SDE (1.12) with periodic Hamiltonian $\,H(x)=H(x+2\pi)\,$ and external force $\,G(x)=G(x+2\pi)$, both possibly time-dependent. The use of the canonical time inversion with the trivial involution $\,x^{*}=x\,$ leads to the backward process of the same type with $\,H^{\prime}_{t}=H^{\prime}_{T-t}\,$ and $\,G^{\prime}_{t}=-G^{\prime}_{T-t}$. The functionals $\,{\cal W}_{{}_{T}}\,$ and $\,{\cal W}^{\prime}_{T}\,$ are given here by the formula (6.2).

On the other hand, the use of the current reversal with the densities $\,\varrho_{{}_{H_{t}}}$ of Eq. (2.13) and the trivial inversion $\,x^{*}=x\,$ leads to the backward process satisfying the same SDE with $\,H_{t}\,$ replaced by

and $\,G_{t}\,$ by $\,G^{\prime}_{t}\,$ as above. The functionals $\,{\cal W}_{{}_{T}}\,$ and $\,{\cal W}^{\prime}_{{}_{T}}\,$ are given now by Eq. (6.6) with $\,\varrho_{t}(x)\,$ equal to $\,\varrho_{{}_{H_{t}}}(x)\,$ and $\,\varrho_{H^{\prime}_{t}}(x)=\varrho_{{}_{H_{T-t}}}(x)$, respectively. They vanish in the stationary case when the DFR (5.3) reduces again the modified detailed balance relation

Note that, in general, to obtain the DFR for the probability distributions on the circle, one should sum both sides of the relation (5.3) pertaining to the motion on the line, over the shifts of $\,x\,$ or $\,y\,$ by $\,2\pi n\,$ with integer $\,n\,$ (both summations amount to the same). To get the Jarzynski equality, one has to integrate the relation obtained this way over $\,x\,$ and $\,y\,$ from 0 to $2\pi$. Another simple remark is that for the SDE (1.12) on the circle, one may always assume that the external force $\,G\,$ is constant by changing $\,G(x)\,$ to $\,\overline{G}=\frac{1}{2\pi}\int_{0}^{2\pi}G(x)\,dx\,$ and by subtracting $\hskip 0.71114pt\,\int_{0}^{x}(G(y)-\overline{G})\hskip 0.71114ptdy\hskip 0.71114pt\,$ from $\,H(x)$. Such a change does not affect the DFR (5.3) obtained by the current reversal that uses only the invariant densities and it modifies in a simple way the DFR obtained from the canonical time inversion because in the latter we used the Gibbs measures for the initial and final distributions.

Fluctuation relations close to NESS

As promised, we shall show here that the MFDT (4.5), proven directly in Sec. 4.1, may be also derived by reducing the Crooks DFR for the current reversal, and, in a special case, the Jarzynski-Hatano-Sasa equality, to the situations close to NESS.

Let us consider the DFR (5.13) for the Langevin dynamics (2.1) with $\,{\cal F}=O^{a}_{t}\,$ and $\,0<t<T$, the backward dynamics determined by the current reversal, and, for simplicity, the trivial space involution $\,x^{*}\equiv x$, see Sec. 6. It reads:

We shall assume that the time-dependence of the Hamiltonian is given by Eq. (4.1) with $\,h_{a,0}=0=h_{a,T}$, and that the external force $\,G\,$ is time-independent. Let, as above, $\,\varrho_{{}_{0}}(x)\,dx\,$ denote the invariant probability measure of the unperturbed process (assumed to exist). By $\,\varrho_{t}$, we shall denote now the normalized densities whose current $\,j_{t}$, given by Eq. (2.8), is conserved, i.e. such that

is chosen by imposing the orthogonality of $\,\varrho^{a}_{1}\,$ to the constant mode, required by the normalization of $\,\varrho_{t}$. For the current reversal, the functional $\,{\cal W}_{{}_{T}}\,$ is given by Eq. (6.6) so that

where we have integrated ones by parts over $\,t$. The application of the operator $\,\frac{\delta}{\delta h_{b,s}}|_{h=0}\,$ for $\,0<s<t\,$ to the both sides of Eq. (7.1) gives the identity

We have used the fact that, by causality, the right hand side of Eq. (7.1) does not give the contribution because the perturbation is concentrated around time $\,(T-s)>(T-t)$. From the relation (7.8), it follows that

which is Eq. (4.16) above. The rest of the proof of the identity (4.5) goes as before.

2 Jarzynski-Hatano-Sasa equality and MFDT

The standard FDT around the equilibrium Langevin dynamics (2.1) without the external force may be obtained by expanding the Jarzynski equality (5.12) for the Hamiltonian (4.1) up to the second order in $\,h$, see . The Jarzynski equality may then be viewed as an extension of the FDT to the case of Hamiltonians with arbitrary time dependence driving the system far from equilibrium. The natural question is whether this picture may be generalized to the case of the modified FDT (4.5) holding around NESS. We shall show here that the answer is a qualified yes.

Let us expand to the second order in $\,h\,$ the Hatano-Sasa version of the Jarzynski equality (5.12) obtained from the Croocks DFR (7.1) for the current reversal by replacing $\,O^{a}\,$ by $\,1$. We shall need to know the form of the densities $\,\varrho_{t}\,$ with conserved current, i.e. satisfying Eq. (7.2), to the second order in $\,h$. One has

with $\,\varrho_{1a}\,$ as before, see Eq. (7.5), and

Expanding, in turn, the functional $\,{\cal W}_{{}_{T}}\,$ given by Eq. (6.6) to the second order, we obtain

The second term on the right hand side integrates to zero because of the stationarity of the unperturbed expectation and the boundary conditions $\,h_{0,a}=0=h_{T,a}$. Expanding the remaining perturbed expectation in the first term on the right hand side, we infer that

where the last equality follows from the (generalized) Jarzynski equality (5.12). The integration by parts and the causality permit to conclude that for $\,t>s$,

(we used the fact that both sides vanish for $\,t=s$). Note that Eq. (7.21) stays true if we add to $\,\varrho^{a}_{1}\,$ any multiple of $\,\varrho_{0}$, so that we may drop the normalization condition $\,\int\varrho^{a}_{1}(x)\,dx=0$. From Eqs. (7.5) and (4.17), it follows then that we may take

The identity (7.21) becomes now the relation

Integrating by parts in the last term and using the definition (7.23), we finally obtain the identity

which is the MFDT (4.5) with the observable $\,O^{a}\,$ replaced by $\,A^{a}$. It is a consequence of the identity (4.5) but, in general, it does not seem to be equivalent to it, except for the equilibrium case with vanishing external force and $\,\varrho_{{}_{0}}=Z^{-1}{\rm e}^{-\beta H}\,$ when $\hskip 0.71114pt\,j_{{}_{0}}=0\,\,$ and $\,\,A^{a}=O^{a}$.

In the special case of the one-dimensional NESS with constant current $\,j_{{}_{0}}\,$ described above, the dressing (7.23) of the observables coincides with the one given by Eqs. (3.3) for the types 1 and 2, respectively. This may be easily seen by checking that for the latter,

with $\,\varrho_{{}_{0}}=\varrho_{{}_{H}}\,$ given by Eqs. (2.12) and (2.13).

Conclusions

We have discussed different fluctuation relations for the Langevin dynamics. Those included an extension of the fluctuation-dissipation theorem (FDT) (1.2), one of the most important relations of the (close to) equilibrium statistical mechanics, to the case of non-equilibrium steady states (NESS) of Langevin processes. The modified fluctuation-dissipation theorem (MFDT) (4.5) that holds around NESS has a new term containing the probability current but in the Lagrangian frame moving with the mean local velocity determined by the current, it takes the form (1.3) similar to that of the equilibrium FDT. We also pointed out that, similarly to the equilibrium FDT, the MFDT may be viewed as a limiting case of more general fluctuation relations that are valid arbitrarily far from the stationary situation, namely of the Crooks detailed fluctuation relation (5.13) for the backward dynamics with inverted probability current or of the Jarzynski-Hatano-Sasa equality (5.12). The general discussion was illustrated on two examples of one-dimensional systems with explicit non-equilibrium invariant measures.

Appendix Appendix A

We return here to the case of the stationary SDE (2.11). The transition probabilities $\,P_{t}(x,dy)\,$ for the diffusion process solving this equation are given by the kernels of the exponential of the generator $\,L=\beta^{-1}\partial_{x}^{2}-(\partial_{x}H)\partial_{x}\,$ of the process:

see Eq. (2.6). One has the following relation

for $\,Q=\partial_{x}+\frac{1}{2}\beta(\partial_{x}H)$, $\,Q^{\dagger}=-\partial_{x}+\frac{1}{2}\beta(\partial_{x}H)$. The Fokker-Planck operator

is a positive self-adjoint Hamiltonian of a super-symmetric quantum mechanics so that the transition probabilities may be defined by the relation

If the Gibbs density is normalizable with $\,Z=\int{\rm e}^{-H(x)}dx<\infty\,$ then $\,\psi_{{}_{0}}(x)=Z^{-1/2}{\rm e}^{-\frac{1}{2}\beta H(x)}\,$ provides the zero-energy groundstate of the Fokker-Planck Hamiltonian. Such a groundstate is supersymmetric: $\,Q\psi_{{}_{0}}=0$. For the Hamiltonian $\,H(x)=ax^{k}+o(|x|^{k})\,$ at large $\,|x|\,$ with either even $\,k\geq 2\,$ and $\,a<0\,$ or odd $\,k\geq 3$, however, the groundstate $\,\psi_{{}_{0}}\,$ of $\,\beta^{-1}Q^{\dagger}Q\,$ is not given by $\,{\rm e}^{-\frac{1}{2}\beta H(x)}\,$ but has positive energy $\,E_{{}_{0}}>0\,$ and breaks the supersymmetry: $\,Q\psi_{{}_{0}}\not=0$. In these cases, the transition probabilities (A.4) are not normalized for $\,t>0\,$ with

The defect \,\big{(}1-\int P_{t}(x,dy)\big{)}\, gives the probability that the diffusion process $\,x_{t}\,$ solving the SDE (2.11) and starting at time zero at $\,x\,$ escapes by time $\,t\,$ to $\,\pm\infty$. Writing $\,P_{t}(x,dy)\equiv P_{t}(x,y)\,dy$, the probability that the escape happens between times $\,s\,$ and $\,s+ds\,$ may be expressed as

with the $\,y=\pm\infty\,$ terms determining the rates of escape to $\pm\infty$, respectively.

Let us concentrate on the case with $\hskip 0.71114pt\,H(x)=ax^{k}+o(|x|^{k})\,\,$ for odd $\,k\geq 3\,$ and $\,a>0$, denoting the transition probabilities of Eq. (A.4) by $\,P^{0}_{t}(x,dy)$. Although they are not given by a closed analytic expression, their time integral, equal to the Green kernel of $\,L$, is:

so that the process $\,x_{t}\,$ escapes here only to $\,-\infty\,$ (this is already true if we ignore the noise in Eq. (2.11)). On the other hand, since the limit of the right hand side of Eq. (A.8) when $\,x\to+\infty\,$ exists, it follows that the transition probabilities from $\,x=+\infty$,

are finite, non-zero measures. One may then define a resurrecting version of the process $\,x_{t}\,$ solving the SDE (2.11). The trajectories of such a process, after almost surely reaching $\,-\infty\,$ reappear immediately at $+\infty$. The resurrecting process is Markov and its transition probabilities are

where $\,P^{n}_{t}(x,dy)\,$ are the transition probabilities with exactly $\,n\,$ jumps from $\,-\infty\,$ to $\,+\infty$. They are given by the recursion relation:

the recursion (A.12) becomes the equality

for $\,n\geq 1$. Re-summing the geometric progression, one obtains for the Laplace transform of the transition probabilities of the resurrecting process the expression

Note that $\,\hat{P}^{0}_{\omega}(x,dy)\,$ is analytic in $\,\omega\,$ for $\,{\rm Re}\hskip 0.71114pt\omega>-E_{{}_{0}}\,$ but $\,\hat{P}_{\omega}(x,dy)\,$ has a pole at $\,\omega=0\,$ with the residue

where $\,Z\,$ is the normalization constant. This is the invariant probability measure (2.12) of the resurrecting process.

Let us finish by estimating the behavior of the density of the invariant measure $\,\mu(dy)\,$ when $\,|y|\to\infty$. We shall show that

To this end, we rewrite the latter integral as

We take $\,H(y)=ay^{k}+h(y)\,$ and assume that $\,h(y)\,$ is smooth and that

if $\,\epsilon>0\,$ is small enough. Next, for $\,0<z<\frac{1}{2}|y|\,$ and $\,|y|\,$ big enough,

for $\,z=uy^{1-k}$. It is also easy to see that for each $\,u>0$,

The estimate (A.18) follows then from the dominant convergence theorem applied to the integral on the right hand side of Eq. (A.19).

Appendix Appendix B

We shall illustrate here the MFDT (4.5) and its version (4.13) in the frequency space on the simple example of the one-dimensional Langevin equation (1.12) on a circle with the Hamiltonian $\,H=0\,$ and a constant force $\,G$. In this case, Eq. (1.12) has, of course, the explicit solution

for the standard Brownian motion $\,W(t)$. Since $\,x\,$ is the angular variable, the above process possesses an invariant probability measure with the constant density $\,\varrho_{{}_{0}}=\frac{1}{2\pi}$. The corresponding current $\,j_{{}_{0}}=\frac{1}{2\pi}G\,$ is constant and so is the local mean velocity $\,\nu_{{}_{0}}=G$. The transition probabilities of the process have the form of the shifted and periodized heat kernel

where $\,\vartheta_{3}(z,q)\,$ is the Jacobi theta function . For the (real) observables $\,O^{a}(x)=\sum\limits_{n}\hat{O}_{n}{\rm e}^{inx}\hskip 0.71114pt$, one obtains easily the equalities:

The result for the Fourier transforms (4.11) and (4.12) follows immediately:

The modified FDT (4.5) or (4.13) are clearly satisfied. Taking $\,a=b$, one may define the factors

Note that $\,X^{n}(0)=1=\hat{X}^{n}(\infty)\,$ but that these factors are not necessarily positive. This is also true for the “effective temperatures”

In particular, $\,T^{n}_{\rm eff}(\omega)\,$ is positive only in the region where $\,\omega^{2}>(n^{2}G^{2}-\beta^{-2}n^{4})\,$ and in this region it decreases with $\,\omega^{2}\,$ approaching for $\,\omega^{2}\to\infty\,$ the value $\,\beta^{-1}$.

The above calculations show that the equilibrium relation (1.2) between the response and correlation functions is strongly violated in the Langevin equation on a circle with a constant drift unless the drift vanishes. On the other hand, the drift may be removed altogether by passing into the frame moving with constant velocity $\,\nu_{{}_{0}}=G$, see Eq. (B.1). This is captured by our MFDT (1.3). Indeed, the solutions of the advection equation (1.4) are

so that the Lagrangian-frame response and correlation functions are

and the MFDT (1.3) takes the form of the equilibrium FDT holding for $\,G=0$.

References