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Reversibility and entropy production of inhomogeneous Markov chains

Part of: Markov processes Time-dependent statistical mechanics (dynamic and nonequilibrium)

Published online by Cambridge University Press:  14 July 2016

Hao Ge ,
Da-Quan Jiang  and
Min Qian
Hao Ge*
Affiliation:
Peking University
Da-Quan Jiang*
Affiliation:
Peking University
Min Qian*
Affiliation:
Peking University
*
∗Postal address: LMAM, School of Mathematical Sciences, Peking University, Beijing, 100871, P. R. China.
∗Postal address: LMAM, School of Mathematical Sciences, Peking University, Beijing, 100871, P. R. China.
∗Postal address: LMAM, School of Mathematical Sciences, Peking University, Beijing, 100871, P. R. China.
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Abstract

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In this paper we introduce the concepts of instantaneous reversibility and instantaneous entropy production rate for inhomogeneous Markov chains with denumerable state spaces. The following statements are proved to be equivalent: the inhomogeneous Markov chain is instantaneously reversible; it is in detailed balance; its entropy production rate vanishes. In particular, for a time-periodic birth-death chain, which can be regarded as a simple version of a physical model (Brownian motors), we prove that its rotation number is 0 when it is instantaneously reversible or periodically reversible. Hence, in our model of Markov chains, the directed transport phenomenon of Brownian motors can occur only in nonequilibrium and irreversible systems.

Keywords

Brownian motorinhomogeneous Markov chaininstantaneous reversibilityinstantaneous entropy productionrotation number

MSC classification

Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 82C31: Stochastic methods (Fokker-Planck, Langevin, etc.) 82C70: Transport processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 43 , Issue 4 , December 2006 , pp. 1028 - 1043
Copyright
© Applied Probability Trust 2006 

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