Seifert derived an exact fluctuation relation for diffusion processes using
the concept of "stochastic system entropy". In this note we extend his
formalism to entropic transport. We introduce the notion of relative stochastic
entropy, or "relative surprisal", and use it to generalize Seifert's
system/medium decomposition of the total entropy. This result allows to apply
the concepts of stochastic thermodynamics to diffusion processes in confined
geometries, such as ion channels, cellular pores or nanoporous materials. It
can be seen as the equivalent for diffusion processes of Esposito and
Schaller's generalized fluctuation theorem for "Maxwell demon feedbacks".Comment: 3 pages, wording change

Smerlak, Matteo

English

arXiv

Seifert derived an exact fluctuation relation for diffusion processes using the concept of "stochastic system entropy". In this note we extend his formalism to entropic transport. We introduce the notion of relative stochastic entropy, or "relative surprisal", and use it to generalize Seifert's system/medium decomposition of the total entropy. This result allows to apply the concepts of stochastic thermodynamics to diffusion processes in confined geometries, such as ion channels, cellular pores or nanoporous materials. It can be seen as the equivalent for diffusion processes of Esposito and Schaller's generalized fluctuation theorem for "Maxwell demon feedbacks"

Smerlak, M.

MPG.PuRe

Stochastic thermodynamics of entropic transport

arXiv:1207.1026v4  [cond-mat.stat-mech]  29 Nov 2012Stochastic Thermodynamics of Entropic TransportMatteo Smerlak1, ∗1Max-Planck-Institut fu¨r Gravitationsphysik, Am Mu¨hlenberg 1, D-14476 Golm, Germany(Dated: March 8, 2013)Seifert derived an exact fluctuation relation for diffusion processes using the concept of “stochasticsystem entropy”. In this paper we extend his formalism to entropic transport. We introduce thenotion of relative stochastic entropy and use it to generalize Seifert’s system/medium decomposi-tion of the total entropy. This result allows to apply the concepts of stochastic thermodynamics todiffusion processes in confined geometries, such as ion channels, cellular pores or nanoporous mate-rials. It can be seen as the equivalent for diffusion processes of Esposito’s and Schaller’s generalizedfluctuation theorem for “Maxwell demon feedbacks”.PACS numbers: 05.40.-a, 05.70.LnKeywords: stochastic thermodynamics, fluctuation relation, entropic transport, relative entropyFrom a mathematical perspective, the fluctuation re-lations of non-equilibrium statistical mechanics [1] boildown to a couple of—almost-tautological—facts:• If ω is a random variable, P (ω) and P ′(ω) are twoprobability distributions on ω andR(ω) = lnP (ω)P ′(ω), (1)then the normalization of P ′ implies〈e−R〉 ≡∑ωP (ω) e−R(ω) = 1, (2)and therefore, by Jensen’s inequality,〈R〉 ≥ 0. (3)• In the particular case where P ′(ω) = P (ω†) forsome involution ( · )† (viz. ω†† = ω), then we havethe stronger relationProb(R(ω) = −r)= e−r Prob(R(ω) = r). (4)As stressed by Maes [2–4], the relevance of these lem-mas for non-equilibrium statistical mechanics becomesapparent relevant when (i) ω represents a path in the(suitably coarse-grained) state space of a mesoscopic sys-tem and (ii) ( · )† is the time-reversal operation. In thiscase, indeed, the quantity R(ω) often turns out to havea thermodynamic interpretation, in terms of the workreceived by the system, the heat dissipated or the en-tropy produced along ω. The identity (4) then becomesa Gallavotti-Cohen [5] type detailed fluctuation relation(DFR), the more general identity (2) becomes a Jarzyn-ski [6] type integral fluctuation relation (IFR), and theinequality (3) becomes the second law inequality. Thisunifying perspective on the non-equilibrium fluctuationrelations is developed in [7, 8].In the context of overdamped Brownian motion, whereω = (xt)0≤t≤T denotes the stochastic trajectory of a par-ticle driven by a protocol-dependent force F (x, λ) (in con-tact with a bath at inverse temperature β), this thermo-dynamic interpretation was clarified by Seifert in [9]: dueto the Markovian nature of Brownian motion, the irre-versibility function R(ω) splits into a “boundary” term∆s(ω) = − ln p(xT , T ) + ln p(x0, 0), (5)where p(x, t) is the solution of the corresponding Fokker-Planck equation with initial condition p(x, 0), and a“bulk” term1βQ(ω) = β∫ T0F (xt, λt) · dxt. (6)The interpretations of these terms are, respectively, thestochastic system entropy—we prefer the term “surprisal”[10]—variation, and (β times) the dissipated heat alongω. In other words, R(ω) is nothing but the total entropyvariationR(ω) = ∆stot(ω) = ∆s(ω) + βQ(ω). (7)The identities (2), and (4) in the case of steady states,provide fluctuation relations for the total entropy produc-tion. This interpretation forms the backbone of stochasticthermodynamics [8, 11].The purpose of this note is to point out that Seifert’sinterpretation (7) must be generalized to become appli-cable for entropic transport, that is, to diffusion processeswhere the density (or degeneracy) of states Ω(x)—hencethe hence the free equilibrium state p∗(x) ∝ Ω(x)—isstate-dependent. An example of such situation is the “en-tropic barrier” studied by Jacobs [12] and Zwanzig [13]:a narrow channel whose sectional area A(z) = πR2(z)1 The stochastic integral should be interpreted in the Stratonovitchsense.2varies along the main axis z. This entropic barrier mod-els various physically, chemically and biologically relevanttransport processes, including tracer diffusion accross ionchannels, biological membranes, zeolites and nano-porousmaterials; see [14] for a review. Provided the transversedimensions are small enough for the corresponding diffu-sion times to be negligible, and assuming that |R′(z)| . 1,the Fokker-Planck equation for the longitudinal motion(in the absence of forcing) is the Fick-Jacobs equation[12, 13]∂tp(z, t) = ∂z(D(z)A(z)∂z(p(z, t)A(z))). (8)where D(z) is an effective state-dependent diffusivity.The density of states of this projected diffusion processis Ω(z) = A(z), and the equilibrium state takes the formp∗(z) ∝ A(z).2 More generally, entropic transport phe-nomena are modeled by general Fokker-Planck equationof the form∂tp(x, t) = ∇ ·(µ(x)F (x, λ)p(x, t) +D(x)∇(p(x, t)Ω(x))),(9)with the local Einstein relation µ(x) = βD(x).It is easy to see that Seifert’s intepretation (7) doesnot hold for these state-dependent diffusion processes.Consider for simplicity the forcing-free case, F = 0 in(9). Because the conventional (“Gibbs”) entropy is max-imized by uniform distributions, and p∗(x) 6= const., itsuffices to take p(x0, 0) = const to find〈∆stot〉 = 〈∆s〉 = ∆(−∫dx p(x, t) ln p(x, t))< 0,(10)in direct violation of (3). This shows that R(ω) 6=∆stot(ω) in this case, i.e. that the irreversibility of en-tropic transport is not measured by the combination ofthe “surprisal” − ln p(xt, t) and the dissipated heat Q(ω),as in Seifert’s relation (7). In Bayesian terms, this is be-cause the non-trivial density of states Ω(x) is a relevantprior, which must be taken into account in the very def-inition of the surprisal.Taking our cues from Kullback and Leibler’s definitionof relative entropy [16],DKL[p|Ω] = −∫dx p(x, t) lnp(x, t)Ω(x), (11)which is well-known to satisfy D˙KL[p|Ω] ≥ 0 for any solu-tion p(x, t) of the Fokker-Planck equation (9) with F = 0,2 Such non-standard Fokker-Planck equations, with space-dependent density of states, arise in other contexts as well, in-cluding general relativity [15]. In this case, the role of thespatially-varying density of states Ω(x) is played by the inverseof the comoving lapse function.let us define the relative system entropy variation or rel-ative surprisal variation by∆srel(ω, t|Ω) ≡ − lnp(xT , T )Ω(xT )+ lnp(x0, 0)Ω(x0). (12)Here as before, p(x, t) is the solution of the Fokker-Planckequation with initial condition p(x, 0). The Bayesianinterpretation of the relative surprisal srel(ω, t|Ω) ≡− ln p(xt, t)/Ω(xt) is the following: srel(ω, t|Ω) is the sur-prise experienced by an observer finding the system instate xt at time t when she expected to find it in theforcing-free equilibrium state p∗(x) ∝ Ω(x).Now the main point: for a diffusion process describedby the Fokker-Planck equation (9), consider the proba-bility P [ω] of a path ω = (xt)0≤t≤T under the drivingλ = (λt)0≤t≤T , and the probability P′[ω] ≡ DP †[ω†]of the time-reversed path ω† ≡ (xT−t)0≤t≤T under thetime-reversed protocol λ† ≡ (λT−t)0≤t≤T . Then the irre-versibility function R(ω) defined by (1) decomposes asR(ω) = ∆srel(ω|Ω) + βQ(ω). (13)To show this, first isolate the initial and distributionsp(·, 0) and p(·, T ),R(ω) = lnp(y0, 0)p(y1, T )+ lnP [ω|x0 = y0]P †[ω†|xT = y1](14)and, second, use the Girsanov formula as in [17],lnP [ω|x0 = y0]P †[ω†|xT = y1]=∫ T0µ(xt)F (xt, λt)−D(xt)∇Ω−1(xt)D(xt)Ω−1(xt)· dxt. (15)With the Einstein relation βD(x) = µ(x), this giveslnDP [ω|x0 = y0]DP [ω†|xT = y1]= βQ(ω)−∫ T0∇Ω−1(xt)Ω−1(xt)· dxt, (16)and evaluating the second integral explicitly, we arrive atlnDP [ω|x0 = y0]DP †[ω†|xT = y1]= βQ(ω) + lnΩ(y1)Ω(y0). (17)Combining (17) with (14) gives the desired result (13).Accordingly, the IFR for entropic transport can bewritten〈e−∆srel(ω|Ω)+βQ(ω)〉 = 1. (18)In the special case of steady states, p(·, 0) = p(·, T ), withsteady forcing λT−t = λt—in which case P† = P—wehave the corresponding DFRProb(∆srel(ω|Ω)− βQ(ω) = −r)= e−r Prob(∆srel(ω|Ω)− βQ(ω) = r). (19)3The second law inequality, in turn, readsDKL[p( · , T )|Ω]−DKL[p( · , 0)|Ω] ≥ β〈Q〉. (20)These relations extend the framework of stochastic ther-modynamics to entropic transport.We close by observing that our results are somewhatcomplementary to those of [18]. In that paper, Espositoand Schaller study, within the framework of Markov jumpprocesses, the stochastic thermodynamics of “Maxwelldemon feedbacks”. These are defined as systems satis-fying a modified detailed balance condition of the formlnWmm′Wm′m= −β((ǫ− ǫm)−µ(Nm−Nm′))+ fmm′ , (21)where m is a state, ǫm its energy, Nm its number of par-ticles, µ the chemical potential, Wmm′ the m→ m′ tran-sition rate, and fmm′ a “feedback parameter” such thatfmm = 0 and fmm′ = −fm′m, measuring the departurefor the standard detailed balance condition. Comparingwith (17), we see that the feedback parameters play thesame role in Esposito’s and Schaller’s setting as a non-trivial density of states Ω(x) in ours. This is intuitivelyclear: a gradient∇Ω(x) 6= 0 effectively acts as a “Maxwelldemon” for the diffusing particles, biasing their Brownianmotion towards regions of high Ω(x). As emphasized in[18], the term lnΩ(xT )/Ω(x0) in (17) can be interpretedas the integral of an “information current”.The author thanks C. Jarzynski for bibliographical ad-vice, R. Chetrite, M. Esposito and G. Schaller for use-ful comments a previous version of this manuscript, thephysics department of Ecole normale supe´rieure de Lyon(France) for an invitation to give a seminar on this topic,and the Inter-University Center for Astrophysics and As-tronomy (India), where part of this work was performed,for hospitality.∗ Electronic address: smerlak@aei.mpg.de[1] C. Jarzynski, “Equalities and inequalities: Irreversibilityand the second law of thermodynamics at thenanoscale,” Annu. Rev. Condens. Matter Phys. 2 (2011)329–351.[2] C. Maes, “The fluctuation theorem as a Gibbsproperty,” J. Stat. Phys. 95 (1999) 367.[3] C. Maes, F. Redig, and A. Van Moffaert, “On thedefinition of entropy production, via examples,” J.Math. Phys. 41 (2000) 1528.[4] C. Maes and K. Netocny, “Time-Reversal and Entropy,”J. Stat. Phys. 110 no. 1/2, (2003) 269.[5] G. Gallavotti and E. Cohen, “Dynamical Ensembles inNonequilibrium Statistical Mechanics,”Phys. Rev. Lett. 74 no. 14, (Apr., 1995) 2694–2697.[6] C. Jarzynski, “Nonequilibrium Equality for Free EnergyDifferences,”Phys. Rev. Lett. 78 no. 14, (Apr., 1997) 2690–2693.[7] R. Chetrite and K. Gawedzki, “Fluctuation relations fordiffusion processes,” Commun. Math. Phys. 282 (2007)469–518.[8] U. Seifert, “Stochastic thermodynamics, fluctuationtheorems, and molecular machines,” Rep. Prog. Phys.75 (2012) 126001.[9] U. Seifert, “Entropy Production along a StochasticTrajectory and an Integral Fluctuation Theorem,”Phys. Rev. Lett. 95 no. 4, (July, 2005) 4.[10] M. Tribus, Thermostatics and thermodynamics. D. VanNostrand Company, 1961.[11] U. Seifert, “Stochastic thermodynamics: principles andperspectives,” Eur. Phys. J. B 64 (2008) 423–431.[12] M. Jacobs, Diffusion processes. Springer, New York,1967.[13] R. Zwanzig, “Diffusion past an entropy barrier,”J. Phys. Chem. 96 no. 10, (May, 1992) 3926–3930.[14] P. S. Burada, P. H a¨nggi, F. Marchesoni, G. Schmid,and P. Talkner, “Diffusion in Confined Geometries,”ChemPhysChem 10 (2009) 45–54.[15] M. Smerlak, “Diffusion in curved spacetimes,” New. J.Phys. 14 (2012) 023019.[16] S. Kullback, Information Theory and Statistics. Wiley,1959.[17] J. L. Lebowitz and H. Spohn, “AGallavotti-Cohen-Type Symmetry in the LargeDeviation Functional for Stochastic Dynamics,” J. Stat.Phys. 95 no. 1/2, (1999) 333.[18] M. Esposito and G. Schaller, “Stochasticthermodynamics for ”Maxwell demon” feedbacks,”EPL 99 no. 3, (Aug., 2012) 30003.

Stochastic thermodynamics of entropic transport

Abstract

Similar works

Full text

Available Versions

MPG.PuRe

MPG.PuRe