Equivalence of Irreversible Entropy Production in Driven Systems: An Elementary Chaotic Map Approach

Abstract

A multibaker map is generalized in order to mimic the thermostating algorithm of transport models. Elementary calculations yield the irreversible entropy production caused by coarse graining of the phase-space density. For different systems, either in steady states (periodic or flux boundaries) or subjected to absorbing boundaries, the specific irreversible entropy production is shown to be u 2 ͞D, where u denotes the local streaming velocity (current per density) and D is the diffusion coefficient. [S0031-9007(97)04219-1] PACS numbers: 05.70.Ln, 05.45. + b, 51.10. + y The connection between nonequilibrium statistical physics and the underlying chaotic dynamics has become a subject of vivid interest [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. The central questions are how the microscopic reversible dynamics can appear as an irreversible process on the macroscopic level, and how the macroscopic transport coefficients are related to microscopic characteristics of the chaotic dynamics. A careful analysis of the rate of irreversible entropy production is at the heart of this problem [16 -20], but the relation between complementary approaches has been poorly understood. Here, we present a consistent derivation of the irreversible entropy production for three main approaches to describe transport in driven systems producing currents. They model either nonequilibrium steady states (A) or the relaxation towards steady states (B):(A1) In the thermostating algorithm a special force is introduced to avoid an uncontrolled growth of the kinetic energy of particles moving in external fields [2][3][4][5][6][7][8][9]. The force mimics the presence of a thermostat and makes the particle dynamics dissipative on average, although it preserves time reversibility. The systems investigated up to now were assumed to be periodic of large spatial extension, and hence to be closed. The long time dynamics exhibits permanent chaos on an underlying chaotic attractor. Transport coefficients and the irreversible entropy production are connected with the average phase-space contraction rates ͑ p͒ on the attractor.(A2) By applying flux boundary conditions to an open Hamiltonian system it was shown that the steady-state density follows Fick's law [10], and the irreversible entropy production has been calculated [19]. In this case the current running through the system is due to the boundary condition only, and the phase-space contraction rate is zero,s ͑ f͒ 0.(B) The escape-rate formalism of transport processes is based on the investigation of open systems of large spatial extensions subjected to absorbing boundary conditions [11 -13]. In such cases the particle dynamics is chaotic in the sense of transient chaos, and there exists an underlying nonattracting chaotic set (a chaotic saddle) in the phase space. In the regime of linear response, at least, relaxation is closely related to steady-state transport. The transport coefficients of Hamiltonian systems are related [1,[11][12][13] to the chaotic saddle's escape rate k. In a recent ...

Entropy Production, Fractals, and Relaxation to Equilibrium

“…Significant progress has been made recently in the theory for entropy production in nonequilibrium, Hamiltonian systems, using ideas from dynamical systems theory. Gaspard [1], Tél, Vollmer, and Breymann [2][3][4][5], Gilbert and Dorfman [6], as well as Tasaki and Gaspard [7,8], have described coarse graining procedures applied to the Gibbs entropy, defined in the phase space of systems maintained in nonequilibrium steady states. These procedures lead to results in accord with the predictions of the thermodynamics of irreversible processes.…”

Entropy production in a persistent random walk

“…A closely related, independent approach to the problem of entropy production in non-equilibrium steady states is provided by Tél, Vollmer, and Breymann (TVB) in a series of papers [1,19,2,20], also devoted to diffusion in multibaker models. These authors considered the entropy production in measure preserving maps as well as in dissipative maps that do not preserve the Lebesgue measure and model systems with Gaussian thermostats.…”

“…A generalization of the methods of Gaspard [7,8] and TVB [1,19,2,20] was proposed by the present authors in a recent paper [9], where it was shown that a coarse-grained form of the Gibbs entropy, which can be expressed in terms of the measures and volumes of the coarse graining sets partitioning the phase, leads to an entropy production formula similar to Gaspard's and applicable to more general volume-preserving as well as dissipative models, such as those considered by TVB, as well as multi-baker maps with energy flow considered by Tasaki and Gaspard [15].…”

Untitled

“…In the following we revisit this argument in the light of recent developments [13,17] dealing with steady states instead of empty asymptotic states. We work out the irreversible entropy production for an isothermal multibaker map with reversible microscopic dynamics subjected to absorbing boundary conditions.…”

“…Multibaker maps model particle transport in spatially extended systems by a chain of mutually interrelated baker maps [13,14,17,18,19,20,21,22,23,24]. They consist of N identical cells of width a and height 1 (the phase-space) in the (x, p) plane.…”