Flow Decomposition Reveals Dynamical Structure of Markov Process

TL;DR

This paper introduces a novel decomposition method for Markov processes that separates the process into stationary, symmetric, and anti-symmetric parts, offering new insights into their dynamics and entropy properties.

Contribution

It presents a complete decomposition of Markov processes into three independent components, providing a new perspective for analysis and construction of such processes.

Findings

Decomposition captures both steady state and dynamics.

A unique relative entropy definition separates detailed-balance effects.

Gini entropy production is unaffected by non-detailed balance parts.

Abstract

Markov process is widely applied in almost all aspects of literature, especially important for understanding non-equilibrium processes. We introduce a decomposition to general Markov process in this paper. This decomposition decomposes the process into 3 independent parts: stationary distribution, symmetric detailed-balance part and anti-symmetric breaking detailed-balance part. This complete decomposition captures the steady state as well as the dynamics of the process, providing an elegant perspective for construction or analyzing problems. In light of the decomposition, a unique definition of relative entropy is found to formally separate the effect of detailed-balance part and breaking detailed-balance part. We find that the relative Gini entropy production introduced in the paper is not affected by the non-detailed balance part of the process. This property do not holds for other entropy definition in general discrete case.