Nonlinear Stochastic Dynamics of Complex Systems, II: Potential of Entropic Force in Markov Systems with Nonequilibrium Steady State, Generalized Gibbs Function and Criticality

TL;DR

This paper explores the role of a generalized potential derived from stationary probabilities in nonequilibrium Markov systems, linking it to thermodynamics, criticality, and biological systems.

Contribution

It provides an axiomatic framework connecting stationary probabilities, entropic forces, and critical phenomena in nonequilibrium complex systems.

Findings

Identifies the generalized potential as a key thermodynamic quantity.

Derives a necessary and sufficient condition for criticality in stochastic systems.

Links criticality conditions to biological system behaviors.

Abstract

In this paper we revisit the notion of the "minus logarithm of stationary probability" as a generalized potential in nonequilibrium systems and attempt to illustrate its central role in an axiomatic approach to stochastic nonequilibrium thermodynamics of complex systems. It is demonstrated that this quantity arises naturally through both monotonicity results of Markov processes and as the rate function when a stochastic process approaches a deterministic limit. We then undertake a more detailed mathematical analysis of the consequences of this quantity, culminating in a necessary and sufficient condition for the criticality of stochastic systems. This condition is then discussed in the context of recent results about criticality in biological systems