Generic Properties of Stochastic Entropy Production

Generic Properties of Stochastic Entropy Production

Simone Pigolotti    Izaak Neri    Édgar Roldán    Frank Jülicher Max Planck Institute for the Physics of Complex Systems, Nöthnitzerstraße 38, 01187 Dresden, Germany
Biological Complexity Unit, Okinawa Institute for Science and Technology and Graduate University, Onna, Okinawa 904-0495, Japan
Max Planck Institute of Molecular Cell Biology and Genetics, Pfotenhauerstraße 108, 01307 Dresden, Germany
Center for Systems Biology Dresden, Pfotenhauerstraße 108, 01307 Dresden, Germany

We derive an Itô stochastic differential equation for entropy production in nonequilibrium Langevin processes. Introducing a random-time transformation, entropy production obeys a one-dimensional drift-diffusion equation, independent of the underlying physical model. This transformation allows us to identify generic properties of entropy production. It also leads to an exact uncertainty equality relating the Fano factor of entropy production and the Fano factor of the random time, which we also generalize to non steady-state conditions.

05.70.Ln, 05.40.-a, 02.50.Le

The laws of thermodynamics can be extended to mesoscopic systems Sekimoto (2010); Bustamante et al. (2005); Maes (2000); Jarzynski (2008); Seifert (2012). For such systems, energy changes on the order of the thermal energy are relevant. Here, is the Boltzmann constant and the temperature. Therefore, thermodynamic observables associated with mesoscopic degrees of freedom are stochastic. A key example of such thermodynamics observables is the stochastic entropy production in nonequilibrium processes. Recent experimental advances in micromanipulation techniques permit the measurement of stochastic entropy production in the laboratory Martínez et al. (2017); Gomez-Solano et al. (2011); Speck et al. (2007); Gavrilov et al. (2017); Koski et al. (2015).

Certain statistical properties of stochastic entropy production are generic, i.e., they are independent of the physical details of a system. Examples of such generic properties are the celebrated fluctuation theorems, for reviews see Bustamante et al. (2005); Jarzynski (2008); Seifert (2012). Recently, it was shown that infima and passage probabilities of entropy production are also generic Neri et al. (2017). Other statistical properties of entropy production are system-dependent, such as the mean value Bo et al. (2016); Harada et al. (2016); Kawai et al. (2007); Parrondo et al. (2009), the variance Barato and Seifert (2015); Pietzonka et al. (2016), the first-passage times of entropy production Roldán et al. (2015); Saito and Dhar (2016); Gingrich et al. (2017) and the large deviation function Lebowitz and Spohn (1999); Mehl et al. (2008). Nevertheless, these properties are sometimes constrained by universal bounds Neri et al. (2017); Kawai et al. (2007); Barato and Seifert (2015); Gingrich et al. (2016); Pietzonka et al. (2016); Polettini et al. (2016); Garrahan (2017); Pietzonka et al. (2017); Gingrich et al. (2017). It remains unclear which statistical properties of stochastic entropy production are generic, and why.

In this Letter, we introduce a theoretical framework which addresses this question for nonequilibrium Langevin processes. We identify generic properties of entropy production by their independence of a stochastic variable which we call entropic time. We find that the evolution of steady-state entropy production as a function of is governed by a simple one-dimensional drift-diffusion process, independent of the underlying model. This allows us to identify a set of generic properties of entropy production and obtain exact results characterizing entropy production fluctuations.

We consider a mesoscopic system described by slow degrees of freedom . The system is in contact with a thermostat at temperature . The stochastic dynamics of the system can be described by the probability distribution to find the system in a configuration at time . This probability distribution satisfies the Smoluchowski equation


where the probability current is given by


Here we have introduced the force at time , , where is a potential and is a non-conservative force. We always imply no flux or periodic boundary conditions. The state-dependent mobility and diffusion tensors, and respectively, are symmetric and obey the Einstein relation . This system can also be represented by a Langevin equation with multiplicative noise as Lebowitz and Spohn (1999); Maes et al. (2008)


Here is a Gaussian white noise with mean and autocorrelation where denotes an ensemble average. Here and throughout the paper the noise terms are interpreted in the Itô sense. The tensor obeys and can be chosen as . In the Itô interpretation, the term is required for consistency with Eqs. (1) and (2) as it compensates a noise-induced drift Lau and Lubensky (2007). Examples of systems described by Eq. (31) that we consider in this paper are represented in Fig. 1: a colloidal particle driven by a constant force along a one-dimensional periodic potential (Fig. 1A); a colloidal particle in a two-dimensional non-conservative force field pointing in the direction (Fig. 1B); and a chiral active Brownian motion in two dimensions Bechinger et al. (2016) (Fig. 1C).

Figure 1: Examples of nonequilibrium steady states. (A) Brownian particle driven by a constant non-conservative force in a periodic 1D sawtooth potential, , with the potential for and for . (B) 2D transport in a force field: and . (C) Chiral active Brownian motion described by 3 degrees of freedom: position coordinates , and orientation angle . In (B) and (C) and is an external non-conservative force.

We now discuss the stochastic thermodynamics of the process described by Eq. (3). In Itô’s calculus, the rate of change of the potential is given by Itô’s lemma Øksendal (2013):


where Tr denotes the trace and the dots denote tensor contractions. In stochastic thermodynamics, the first law can be expressed as , where is the work performed on the system and is the mesoscopic heat exchanged with the thermostat during a time interval  Sekimoto (2010). In Itô’s calculus, the rates of change of work and heat are given by Lebowitz and Spohn (1999)


The expressions (5) and (6) are the Itô versions of the stochastic work and mesoscopic heat originally defined by Sekimoto using the Stratonovich interpretation Sekimoto (1998, 2010).

We define the stochastic entropy production as the logarithm of the ratio of probabilities of forward and time-reversed stochastic trajectories Lebowitz and Spohn (1999); Maes (2000); Seifert (2005). This definition is equivalent to , where the first term can be interpreted as an exchange of entropy with the reservoir and the second term as a change of system entropy. Using Eq. (6) and Itô’s lemma, as in Eq. (4) (see foo ()), we obtain the following Itô stochastic differential equation for the entropy production rate

Figure 2: Illustration of the decomposition of stochastic entropy production. In nonequilibrium steady states, the stochastic entropy production (green) is given by the sum of the monotonously increasing entropic time (orange), and the martingale process (blue), see Eq. (11).

Here we define the entropic drift as


which on average equals the average rate of entropy production,  Maes et al. (2008); Seifert (2012). Entropy fluctuations are governed by the noise term  which is a one-dimensional Gaussian white noise with and . The Itô Eq. (7) is equivalent to the Langevin equation for entropy production in the Stratonovich interpretation given in Ref. Seifert (2005). For each trajectory generated by Eq. (3), Eq. (7) generates the corresponding entropy production. From Eq. (7) we can derive several generic properties of stochastic entropy production in nonequilibrium processes.

We first discuss properties of nonequilibrium steady states for which . We now calculate the time derivative of in steady state. Using Itô’s lemma, we obtain from Eq. (7)


which reveals that is a geometric Brownian motion with zero drift and time-dependent diffusion coefficient. The fact that has no drift implies that is a martingale process Øksendal (2013); Chetrite and Gupta (2011); Neri et al. (2017). Using the integral fluctuation theorem follows immediately from Eq. (9).

Figure 3: Generic properties of stochastic entropy production. Distributions of a) entropy production at fixed , (b) infimum of entropy production, c) supremum of entropy production before the infimum, d) number of crossings of entropy production, with . The symbols are obtained from numerical simulations of the three models sketched in Fig. 1 (blue squares, model A; red circles, model B; green diamonds, model C). The inset of a) shows numerically estimated distributions of for the three models at fixed for comparison. The solid orange curves are the theoretical expressions a) a Gaussian distribution with average and variance b) an exponential distribution with average c) Eq. (13); d) Eq. (14). The dashed line in c) is the theoretical distribution of minus the infimum for comparison. In all simulations, parameters are , , where is the identity matrix, and . In Model A we chose and . In model C we chose . Here and in the following figures, each point represents an average over simulations.

In steady state, Eq. (7) can be simplified by introducing the dimensionless entropic time


which is an example of a random time Øksendal (2013). Note that, in steady state, represents the expected rate of entropy production at a given point in phase space and thus represents the accumulated expected entropy production. In nonequilibrium situations with , the entropic time is monotonously increasing with . Integrating Eq. (7) we obtain


Equation (11) represents the decomposition of entropy production into a monotonously increasing process and a martingale that has zero mean, , as illustrated in Fig. 2. This decomposition is unique and is known as the Doob-Meyer decomposition Liptser and Shiryaev (2013).

We now discuss an important implication of Eqs. (7) and (10). Performing the random-time transformation in Eq. (7) we obtain a Langevin equation for steady state entropy production at entropic times Øksendal (2013)


where such that is Gaussian white noise with and . Equation (12) states that a temporal trajectory of entropy production of any nonequilibrium steady state can be mapped to a trajectory of a drift-diffusion process with constant drift and diffusion coefficient , where the mapping consists in a time-dependent, stochastic contraction or dilation of time. This implies that all properties of that are invariant under such transformation are generic.

One such property is the distribution of entropy production at fixed values of , which must be a Gaussian with average and variance because of Eq. (12). This is indeed the case for all three model examples, see Fig. 3a. Note that the distribution of entropy production at fixed time are very different for the three models, as shown in the inset of Fig. 3a. Another generic property is the distribution of the global infimum of entropy production , previously derived using martingale theory Neri et al. (2017) and given by an exponential distribution with mean and (Fig. 3b). Also the supremum of entropy production before the infimum is generic and distributed according to


with . Its average value is . The number of times that entropy production crosses a given threshold value is also generic. An example is the number of times that entropy production crosses from to with . The distribution of is


Equation (13) and (14) are in excellent agreement with numerical simulations, as shown in Fig. 3c,d.

With equation (11), we can also compute the moments of . The first moment reads simply . The second moment is , see foo (). Combining these two results, the Fano factor of the entropy production can be expressed as


where denotes the variance.

Figure 4: Thermodynamic Fano factor equality. Long-time Fano factor of entropy production as a function of the external force . The symbols are obtained from numerical simulations of the models shown in Fig. 1A (blue), Fig. 1B (red), and Fig. 1C (green). The solid lines are the prediction of Eq. (15) and have been calculated by means of Eq. (16) (see foo ()). All the parameters of the numerical simulations except of the external force are the same as in Fig. 3.

The thermodynamic Fano factor equality given by Eq. (15) is an exact relation, valid for finite times, between the fluctuations of entropy production and the fluctuations of the entropic time . This equation provides further physical insight into the previously introduced finite-time uncertainty relation,  Pietzonka et al. (2017); Gingrich et al. (2017). The variance obeys the equality only if the entropic time satisfies , which holds e.g. near equilibrium. In this case, the distribution of entropy production is Gaussian. Another example for which is the chiral active Brownian motion shown in Fig. 1c.

For long times, the variance of the entropic time can be estimated by a Green-Kubo formula as an integral over a correlation function foo ()


Using Equations (15) and (16) we obtain explicit expressions for the Fano factor as a function of the driving force for our three models, see Fig. 4 for a comparison with numerical simulations.

Figure 5: Fano factor of stochastic entropy production out of steady state. The position of a Brownian particle is governed by the equation with and . The particle is initially at equilibrium with stiffness (see inset). a) Comparison between the exact value (orange line) of the long-time Fano factor of entropy production and the value obtained from numerical simulations (brown triangles) as a function of . The exact value is given by see Supplemental Material for details foo (). In simulations we measure , and and use Eq. (17). The horizontal purple line is set to for comparison. b) Behaviour of (solid line) and (dashed line) as a function of .

Our theory can also be applied to nonequilibrium processes out of steady state. From Eq. (7) we derive the general Fano factor equality




and is the entropic time for non-steady state processes. At steady state, , and Eq. (17) reduces to Eq. (15). Note that the argument of the integral in Eq. (18) is the correlation of the two drift terms in Eq. (7) at different times. In Fig. 5, we illustrate Eq. (17) for a particle confined in a harmonic trap, where the stiffness of the trap is instantaneously quenched from a value to a value . When , one has , so that the Fano factor of entropy production is larger than two according to Eq. (17). When instead , one has , and the Fano factor of entropy production is lower than two.

For nonequilibrium processes starting at thermal equilibrium and undergoing a defined protocol to a final state, one has , where is the work performed during the protocol and is the change of equilibrium free energy  associated with the final and initial states Jarzynski (1997); Crooks (1999); Bochkov and Kuzovlev (1977). Here denotes an equilibrium average over the Boltzmann distribution. For such protocols, Eq. (17) implies


Note that also obeys Jarzynski’s equality  Jarzynski (1997), which has the form of a cumulant generating function. Comparing it with Eq. (19), one can relate the term in parenthesis in (19) to a sum of cumulants of of order three and higher. This sum vanishes if the work distribution is Gaussian Jarzynski (1997).

We have shown that, in steady-state Langevin processes, entropy production is governed by a Langevin equation which only depends on the system’s details via the entropic drift . As a consequence all system-specific features of stochastic entropy production can be absorbed into a single stochastic quantity, the entropic time . Entropy productions of different systems at equal entropic time have the same statistics, and all properties independent of the entropic time are generic. Fluctuations of the entropic time uniquely determine the Fano factor of entropy production, providing physical insight for previously obtained bounds Barato and Seifert (2015); Gingrich et al. (2016); Pietzonka et al. (2016); Polettini et al. (2016); Garrahan (2017); Pietzonka et al. (2017); Gingrich et al. (2017).

We have demonstrated our results for coupled overdamped Langevin equations but expect our results to hold more generally for continuous processes, as is the case for the infimum of entropy production Neri et al. (2017). Using the Doob-Meyer decomposition of entropy production, our definition of entropic time can also be generalized to underdamped systems Celani et al. (2012); Ge (2014) and jump processes Gaspard (2004). Our results can be experimentally tested for example with optical tweezers Gomez-Solano et al. (2011); Speck et al. (2007); Martínez et al. (2017); Krishnamurthy (2016); Argun (2016), feedback traps Gavrilov et al. (2017), single-electron transistors Koski et al. (2015) and light-activated phototactic microparticles Lozano (2016).

We thank AC Barato for stimulating discussions and A Mazzino and A Vulpiani for suggesting the example of model B.


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Appendix A Supplemental Material

This document provides additional information for the manuscript “Generic Properties of Stochastic Entropy Production”. It is organized as follows. Section S1 sketches the derivation of the Itô stochastic differential equation for the entropy production. Section S2 presents a derivation of the Fano-factor equality for steady-state processes. Sections S3A, S3B, and S3C detail the calculations of the Fano factor of entropy production for the three steady-state models discussed in the Main Text. Section S4 presents the derivation of the Fano-factor equality out of steady state. Section S5 describes details on the non-equilibrium process shown in Fig. 5 of the Main text.

Appendix B S1.Ito stochastic differential equation for entropy production

In this section, we sketch the derivation of the evolution for the stochastic entropy production (Eq. (7) in the Main Text). We recall that the rate of total entropy production change can be decomposed into the rates of system-entropy change and heat change Seifert (2005)


with the system entropy. We express the rate of heat change as


where we have used the Einstein relation . We use Itô’s lemma Øksendal (2013) to find the following expressions for the rate of system-entropy change

The Fokker-Planck equation can be rewritten as


After substituting equation (23) into the expression for the rate of system-entropy change, Eq. (B), and adding the rate of heat change, Eq. (21), we obtain the following compact expression for the rate of entropy production change


with , , and the probability currents .

Appendix C S2.Fano Factor of entropy production

In this section we derive the Fano-factor equality for stochastic entropy production at finite times


We show that (25) holds for steady state processes satisfying the Langevin Eq. (3) in the Main Text. The Langevin equation for entropy production (24) implies that the steady-state stochastic entropy production can be expressed as


where , and we recall that . Taking the average of Eq. (26), we find


From Eq. (26) we find for the second moment of stochastic entropy production

where the contribution has been obtained using Ito’s isometry Øksendal (2013). Subtracting from both sides, further dividing by and using Eq. (27) we obtain


We show below that


for stationary processes. Therefore the Fano-factor equality (15) for stochastic entropy production follows from Eq. (29). To show (30) we first note that for the relation (30) is a direct consequence of the rules of Itô calculus. This is because the noise in the future is uncorrelated with the trajectory in the past. The case of requires a careful analysis, since we have to average over the noise conditioned on at a future time.

To compute this conditioned average, we apply Doob’s h-transform Doob (1957) (see also Satya (2015); Chetrite and Touchette (2015)). In short, Doob’s h-transform maps a stochastic process with noise variables conditioned on a future event to a stochastic process with unconditioned noise variables, but with an additional drift term. For example, consider the Langevin equation


with and . We calculate averages conditioned on the future constraint , with . In other words, when taking averages we only consider the trajectories generated by (31) for which , and disregard the other ones. In general, averages involving the noise variables become biased by this condition, i.e., . Introducing the -function, , where is the solution of the corresponding Fokker-Planck equation with initial condition . Doob’s h-transform generates a Langevin equation for a process which reads Doob (1957); Satya (2015); Chetrite and Touchette (2015)


where and is a white noise with zero mean, i.e., . Doob showed that (32) generates an ensemble of trajectories identical to the ensemble of trajectories generated by the stochastic differential equation (31) and conditioned on the event in the future Doob (1957); Satya (2015); Chetrite and Touchette (2015). Comparing Eq. (31) with Eq. (32), reveals that replacing the noise in Eq. (31) with the noise process defined by


allows to use standard noise averages when calculating averages conditioned on the future event at time . Note that in (33) we have used .

Using the Doob -transform, we compute now the average of Eq. (30) for


where is the -function for the future condition . Note that we have used , which follows from the fact that is not fluctuating and is a white noise which is uncorrelated with at the same time. We also used . We proceed by writing the average explicitly in terms of the distribution of the system states and at times and :

Using , we integrate by parts:


In the second step, we have used that no boundary term arise for periodic or no flux boundary conditions. Indeed either the flux or the difference in probability must vanish for this boundary conditions. In steady state because .

Appendix D S3.Fano Factor of the entropic time

The Fano factor of the entropic time at finite times can be expressed as


that in the limit reduces to the Green-Kubo-like expression