On and beyond entropy production: the case of Markov jump processes (original) (raw)
Minimum entropy production principle from a dynamical fluctuation law
arXiv preprint math-ph/0612063, 2007
The minimum entropy production principle provides an approximative variational characterization of close-to-equilibrium stationary states, both for macroscopic systems and for stochastic models. Analyzing the fluctuations of the empirical distribution of occupation times for a class of Markov processes, we identify the entropy production as the large deviation rate function, up to leading order when expanding around a detailed balance dynamics. In that way, the minimum entropy production principle is recognized as a consequence of the structure of dynamical fluctuations, and its approximate character gets an explanation. We also discuss the subtlety emerging when applying the principle to systems whose degrees of freedom change sign under kinematical time-reversal.
Statistical Mechanics of Entropy Production: Gibbsian hypothesis and local fluctuations
2001
It is argued that a Gibbsian formula for the space-time distribution of microscopic trajectories of a nonequilibrium system provides a unifying framework for recent results on the fluctuations of the entropy production. The variable entropy production is naturally expressed as the time-reversal symmetry breaking part of the space-time action functional. Its mean is always positive. This is both supported by a Boltzmann type analysis by counting the change in phase space extension corresponding to the macrostate as by various examples of nonequilibrium models. As the Gibbsian set-up allows for non-Markovian dynamics, we also get a local fluctuation theorem for the entropy production in globally Markovian models. In order to study the response of the system to perturbations, we can apply the standard Gibbs formalism.
Nonequilibrium linear response for Markov dynamics, I: Jump processes and overdamped diffusions
Journal of Statistical Physics, 2009
Systems out of equilibrium, in stationary as well as in nonstationary regimes, display a linear response to energy impulses simply expressed as the sum of two specific temporal correlation functions. There is a natural interpretation of these quantities. The first term corresponds to the correlation between observable and excess entropy flux yielding a relation with energy dissipation like in equilibrium. The second term comes with a new meaning: it is the correlation between the observable and the excess in dynamical activity or reactivity, playing an important role in dynamical fluctuation theory out-of-equilibrium. It appears as a generalized escape rate in the occupation statistics. The resulting response formula holds for all observables and allows direct numerical or experimental evaluation, for example in the discussion of effective temperatures, as it only involves the statistical averaging of explicit quantities, e.g. without needing an expression for the nonequilibrium distribution. The physical interpretation and the mathematical derivation are independent of many details of the dynamics, but in this first part they are restricted to Markov jump processes and overdamped diffusions.
Entropy production in a non-Markovian environment
Physical Review E, 2015
Stochastic thermodynamics and the associated fluctuation relations provide means to extend the fundamental laws of thermodynamics to small scales and systems out of equilibrium. The fluctuating thermodynamic variables are usually treated in the context of isolated Hamiltonian evolution, or Markovian dynamics in open systems. In this work we introduce an explicitly non-Markovian model of dynamics of an open system, where the correlations between the system and the environment drive a subset of the environment outside of equilibrium. This allows us to identify the non-Markovian sources of entropy production. We show that the non-Markovian components lead to modifications in the standard fluctuation relations for entropy. As a concrete example, we explicitly derive such modified fluctuation relations for the case of an overheated single electron box.
Test of the fluctuation theorem for stochastic entropy production in a nonequilibrium steady state
Physical Review E, 2006
We derive a simple closed analytical expression for the total entropy production along a single stochastic trajectory of a Brownian particle diffusing on a periodic potential under an external constant force. By numerical simulations we compute the probability distribution functions of the entropy and satisfactorily test many of the predictions based on Seifert's integral fluctuation theorem. The results presented for this simple model clearly illustrate the practical features and implications derived from such a result of nonequilibrium statistical mechanics. PACS numbers: 05.70.Ln,05.40.-a.
Entropy production and coarse graining in Markov processes
Journal of Statistical Mechanics: Theory and Experiment, 2010
We study the large time fluctuations of entropy production in Markov processes. In particular, we consider the effect of a coarse-graining procedure which decimates fast states with respect to a given time threshold. Our results provide strong evidence that entropy production is not directly affected by this decimation, provided that it does not entirely remove loops carrying a net probability current. After the study of some examples of random walks on simple graphs, we apply our analysis to a network model for the kinesin cycle, which is an important biomolecular motor. A tentative general theory of these facts, based on Schnakenberg's network theory, is proposed. To our friend and colleague Massimo Falcioni, on his 60th birthday I. INTRODUCTION The coarse-graining procedure is a fundamental ingredient of the statistical description of physical systems [1-3]. By coarse-graining we mean a procedure which reduces the number of observables to simplify the physical description. For instance, it is used to describe the behaviour of the physically relevant quantities, or slow variables, which depends on the coupling among all variables characterizing the system of interest, including the so-called fast variables. The archetype of such a procedure is the treatment of Brownian colloidal particles, immersed in a fluid, in terms of the Langevin equation. In this sense any model meant to represent a real phenomenon may be thought of as a coarsegrained, i.e. reduced, description. The purpose of a model is, indeed, to advance our understanding of the object under investigation, by highlighting its interesting features and discarding the irrelevant ones. In turn, the roles of relevant and irrelevant characteristics depend on the purpose of the analysis to be performed. Furthermore, it isn't always obvious which quantities should be listed as interesting, and which ones should be neglected, especially if a new problem is to be tackled [1-5]. Therefore, it is critical to understand how specific physical observables depend on the coarse-graining procedure. Examples of coarse-grained descriptions at different resolution levels include the steps meant to connect the microscopic descriptions of systems of physical interest to the macroscopic ones, for instance the passage from the deterministic Γ-space description (positions and momenta of the N particles) to the stochastic µ-space description (position and momentum of one particle), up to macroscopic descriptions such as hydrodynamics, Fourier law, Navier-Stokes equations, etc. Other methods use the coarse-graining procedure in order to reduce the number of variables, e.g. by a decimation method which suppresses the fast variables, or perform a spatial coarse-graining, as in the renormalization group approach. In these methods the coarse-graining is parametrized by some threshold, here denoted as coarse-graining level (CGL). This paper is devoted to the investigation of the impact of variations of CGL on the entropy production of non-equilibrium systems. In the last decades, the introduction of the so-called Fluctuation Relations (FR) for deterministic dynamics, by Evans, Cohen, Morriss, Gallavotti, Jarzynski and other authors brought about important developements in the physics of far from equilibrium systems [6-8]. In the specific context of Markov processes, here discussed, Lebowitz and Spohn [9] showed that the "entropy production" per unit time, measured on a time-interval t, W t say, is described by a large deviation theory whose Cramer function, C, enjoys the following symmetry property: C(W t) − C(−W t) = −W t .
Fluctuations of the total entropy production in stochastic systems
EPL (Europhysics Letters), 2008
Fluctuations of the total entropy are experimentally investigated in two stochastic systems in a non-equilibrium steady state : an electric circuit with an imposed mean current and a harmonic oscillator driven out of equilibrium by a periodic torque. In these two linear systems, we study the total entropy production which is the entropy created to maintain the system in the non-equilibrium steady state. Fluctuation theorem holds for the total entropy production in the two experimental systems, both for all observation times and for all fluctuation magnitudes.
On the definition of entropy production, via examples
Journal of Mathematical Physics, 1999
We present a definition of entropy production rate for classes of deterministic and stochastic dynamics. The point of departure is a Gibbsian representation of the steady state path space measure for which ''the density'' is determined with respect to the time-reversed process. The Gibbs formalism is used as a unifying algorithm capable of incorporating basic properties of entropy production in nonequilibrium systems. Our definition is motivated by recent work on the Gallavotti-Cohen ͑lo-cal͒ fluctuation theorem and it is illustrated via a number of examples.
Big entropy fluctuations in statistical equilibrium: The macroscopic kinetics
Journal of Experimental and Theoretical Physics, 2001
Large entropy fluctuations in an equilibrium steady state of classical mechanics were studied in extensive numerical experiments on a simple 2-freedom strongly chaotic Hamiltonian model described by the modified Arnold cat map. The rise and fall of a large separated fluctuation was shown to be described by the (regular and stable) "macroscopic" kinetics both fast (ballistic) and slow (diffusive). We abandoned a vague problem of "appropriate" initial conditions by observing (in a long run) spontaneous birth and death of arbitrarily big fluctuations for any initial state of our dynamical model. Statistics of the infinite chain of fluctuations, reminiscent to the Poincaré recurrences, was shown to be Poissonian. A simple empirical relation for the mean period between the fluctuations (Poincaré "cycle") has been found and confirmed in numerical experiments. A new representation of the entropy via the variance of only a few trajectories ("particles") is proposed which greatly facilitates the computation, being at the same time fairly accurate for big fluctuations. The relation of our results to a long standing debates over statistical "irreversibility" and the "time arrow" is briefly discussed too.
Physical Review E, 2012
We present a general framework for systems which are prepared in a non-stationary nonequilibrium state in the absence of any perturbation, and which are then further driven through the application of a time-dependent perturbation. We distinguish two different situations depending on the way the non-equilibrium state is prepared, either it is created by some driving; or it results from a relaxation following some initial non-stationary conditions. Our approach is based on a recent generalization of the Hatano-Sasa relation for non-stationary probability distributions.