Stochastic Dynamics II: Finite Random Dynamical Systems, Linear Representation, and Entropy Production

TL;DR

This paper analyzes finite state random dynamical systems and their Markov chains, focusing on entropy measures and entropy production, providing new insights into their stochastic properties and time-reversal asymmetry.

Contribution

It introduces a linear representation of deterministic maps in RDS, relates entropy production to cycle distributions, and bounds entropy production rate using Kullback-Leibler divergence.

Findings

Entropy production rate expressed via cycle distributions.

Bound on entropy production rate using KL divergence.

Relation between deterministic map probabilities and time-reversal duals.

Abstract

We study finite state random dynamical systems (RDS) and their induced Markov chains (MC) as stochastic models for complex dynamics. The linear representation of deterministic maps in RDS are matrix-valued random variables whose expectations correspond to the transition matrix of the MC. The instantaneous Gibbs entropy, Shannon-Khinchin entropy per step, and the entropy production rate of the MC are discussed. These three concepts as key anchor points in stochastic dynamics, characterize respectively the uncertainties of the system at instant time $t$, the randomness generated per step, and the dynamical asymmetry with respect to time reversal. The entropy production rate, expressed in terms of the cycle distributions, has found an expression in terms of the probability of the deterministic maps with the single attractor in the maximum entropy RDS. For finite RDS with invertible transformations, the non-negative entropy production rate of its MC is bounded above by the Kullback-Leibler divergence of the probability of the deterministic maps with respect to its time-reversal dual probability.