Irreversible circulation of fluctuation and entropy production

Abstract

Physical and chemical stochastic processes described by the master equation are investigated. The system-size expansion, called the Ω-expansion, transforms the master equation to the corresponding Fokker-Planck equation. In this paper, we examine the entropy production for both the master equation and the corresponding Fokker-Planck equation. For the master equation, the exact expression of the entropy production was recently derived by Gaspard using Kolmogorov-Sinai entropy (J. Stat. Phys. 117 (2004), 599 [Errata; 126 (2006), 1109]). Although Gaspard's expression is derived from a stochastic consideration, it should be noted that it coincides with the thermodynamical expression. For the corresponding Fokker-Planck equation, by using the detailed imbalance relation, which appears in the process of deriving the fluctuation theorem through the Onsager-Machlup theory, the entropy production is expressed in terms of the irreversible circulation of fluctuation, which was proposed by Tomita and Tomita (Prog. Theor. Phys. 51 (1974), 1731 [Errata; 53 (1975), 1546b]). However, this expression for the corresponding Fokker-Planck equation differs from that of the entropy production for the master equation. This discrepancy is due to the difference between the master equation and the corresponding Fokker-Planck equation, namely the former treats discrete events, but the latter equation is an approximation of the former one. In fact, in the latter equation, the original discrete events are smoothed out. To overcome this difficulty, we propose a hypothetical path weight principle. By using this principle, the modified expression of the entropy production for the corresponding Fokker-Planck equation coincides with that of the master equation (i.e., the thermodynamical expression) for a simple chemical reaction system and a diffusion system.