Symmetry-Based Balance Equation for Local Entropy Density in a Dissipative Multibaker Chain System

Abstract

In this study, the balance equation for local entropy density defined on each partition is obtained by the decomposition of the time-evolution operator for local entropy density, on the level of the master equation, by using symmetric and antisymmetric properties for the inversion of partition, density pairs and a given drift velocity. The resultant equation includes the following terms: convection, diffusion, entropy flow due to a thermostat and entropy production. The averaging of the four terms recover the corresponding terms in a balance equation for the macroscopic entropy density of irreversible thermodynamics for a thermostated system. Moreover, an empirical law of order estimation is introduced to explain the limiting behavior of the averaged quantities in the macroscopic limit for the bulk system. The law makes it possible to separate some minor contributions from the major four terms and, for example, to explain the positive entropy production rate in a nonequilibrium state for volume-preserving systems, even if the state is far from steady state. They are numerically confirmed on an invertible, dissipative multibaker chain system, named a circuit model. These properties are independent of partitioning.Keywords: symmetry-based decomposition; local entropy density; coarse graining; dissipative system; positive entropy production; law of order estimation; multibaker map

A law of order estimation and leading-order terms for a family of averaged quantities on a multibaker chain system

“…with Table 1 in Ref. 19, we are easily convinced that the quantity of the present study is an extension in the sense that the symmetric partĈ(Y ) (Y ≡ (ρ…”

“…site on a multibaker chain, has been decomposed into three terms: entropy flow, entropy flow due to a thermostat, and nonnegative entropy production 1 , and its similar or extended formalism is introduced by Vollmer, Breymann, Mátyás, et al [2][3][4][14][15][16]18 . Although the entropy balance equation, based on the information theory, is confirmed to agree with the phenomenological one in the macroscopic limit, each term does not necessarily agree with a corresponding macroscopic one: the estimated entropy flow due to a thermostat is contaminated by a flux term and the contributions of convection and diffusion cannot be distinguished 19 . The conventional decomposition inherently suffers from the need of an appropriate physical principle when finding a coarse-grained form corresponding to an arbitrary macroscopic quantity.…”

“…Ishida 19 introduced a family of local quantities defined on each partition. They are found by the decomposition of the equations for local probability density and local entropy density, based on symmetric properties.…”

“…where ∆ i,j is expected to reflect the distance between i-th and j-th partitions. This is a component of local probability density change 19 .…”

“…Hereafter we shall confine our discussion to a triadic multibaker chain, introduced by Vollmer et al [2][3][4][5] and Ishida 6,19 .…”