2. Non-Equilibrium Statistical Mechanics

Overview :

[ Extract from : Platini, Phys. Rev. E 83, 011119 (2011) ]
In nature, typically every system is governed by well-known ’deterministic’ physical laws. However the microscopic details are usually unreachable and a full description of the system is impossible. The enormous number of degrees of freedom and the apparent ’chaotic’ motion of the microscopic elements leads to rather complicated phenomenon. To face such situations, the physics of statistical mechanics has been built over the last two centuries. Our best approximation is to assume that the interactions between the microscopic elements occur according to some probabilistic rules. It follows that the natural reformulation of many-body problems takes the form of stochastic processes. In addition, almost all systems are in interaction with the environment. It is only in particular limits (for defined time and spatial scales) that closed systems emerge. Generally, currents of particles, heat or magnetization are induced by the environment and testify to the non-equilibrium activity. If the presence of flux is a signature of a non-equilibrium steady state (NESS), the absence of macroscopic currents does not necessarily imply that the system is at equilibrium. By definition equilibrium is characterized, on a microscopic level, by the well-known ’detailed balance’ criterion. The latter ’balance’ is breaking for any non-equilibrium state and imposes non-vanishing currents of probability flowing between connected states.

If equilibrium systems are well described by the theory of ensembles, no global formalism exists for systems out of equilibrium. Recently, general relations for driven systems have attracted a lot of interest. One should mention the Kawasaki relation [*], the Jarzynski and Crooks relations [*] and ’fluctuation relations’ that include the fluctuation theorems of Evans-Searles [*] and Gallavotti-Cohen [*]. While analyzing NESS case-by-case, research works are traditionally focused on the probability distribution P of the micro-states.

This approach is usually relevant since almost all observables can be extracted from there. However, the analysis of currents or average production of entropy requires the knowledge of the transition rates which complete the characterization of the steady state. Recently, Zia and Schmittmann [*] suggested a general classification of NESS, where a complete description of the system is given by the distribution of the probabilities and probability currents {P , K}. This classification allows the identification of the transformations of the transition rates that leave the steady state invariant. Along these lines, a set of invariant quantities has been derived for a class of steady states driven by the boundaries dynamics [*]. From the probability currents a definition of the ’Euclidean distance’ from equilibrium has been proposed in [*]. A measure of the violation of ’detailed balance’ could define the ’distance’ from equilibrium and be quantified by the p-norm (||K ||_p ) of the matrix formed by the probability currents. For the periodic totally asymmetric simple exclusion process (TASEP), the exact expression of ||K||_p is given in [Platini, Phys. Rev. E 83, 011119 (2011) ]. We can show, in the thermodynamic limit, that the p-norm vanishes for any p ≠ 1. This result motivates the definition of the ’distance’ from equilibrium by ||K||_1 that is extensive for the periodic-TASEP, open-ASEP and open-ZRP. One should mention that the 1-norm was first defined in [*] and used to measure the violation of the ’detailed balanced’ criterion for different reaction-diffusion models.

* For a liste of citations please see the paper on ArXiv or PRE

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Measure of the violation of the detailed balance criterion: A possible definition of a “distance” from equilibrium

Phys. Rev. E 83, 011119 (2011) [6 pages]

Department of Physics and Virginia Bioinformatics Institute, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061-0435, USA

Received 1 July 2010; revised 30 September 2010; published 24 January 2011

Motivated by the classification of nonequilibrium steady states suggested by R. K. P. Zia and B. Schmittmann J. Stat. Mech. 2007 P07012 (), we propose to measure the violation of the detailed balance criterion by the p norm (||K*||p) of the matrix formed by the probability currents. Its asymptotic analysis, for the totally asymmetric simple exclusion process, motivates the definition of a “distance” from equilibrium K* obtained for p=1. In addition, we show that the latter quantity and the average activity 〈A*〉  are both related to the probability distribution of the entropy production. Finally, considering the open asymmetric simple exclusion process and open zero-range process, we show that the current of particles gives an exact measure of the violation of detailed balance.

Link to ArXiv paper

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