Maximum Kolmogorov-Sinai entropy vs minimum mixing time in Markov chains

TL;DR

This paper proposes using Kolmogorov-Sinai entropy as a faster way to estimate the mixing time of Markov chains, linking entropy maximization to minimal mixing time, with implications for physics and computer science.

Contribution

It establishes a novel link between Kolmogorov-Sinai entropy maximization and the minimization of mixing time in Markov chains, offering a more efficient estimation method.

Findings

KSE maximization correlates with minimal mixing time.

New method provides faster approximation of mixing time.

Implications for selecting stationary states in physics.

Abstract

Many modern techniques employed in physics, such a computation of path integrals, rely on random walks on graphs that can be represented as Markov chains. Traditionally, estimates of running times of such sampling algorithms are computed using the number of steps in the chain needed to reach the stationary distribution. This quantity is generally defined as mixing time and is often difficult to compute. In this paper, we suggest an alternative estimate based on the Kolmogorov-Sinai entropy, by establishing a link between the maximization of KSE and the minimization of the mixing time. Since KSE are easier to compute in general than mixing time, this link provides a new faster method to approximate the minimum mixing time that could be interesting in computer sciences and statistical physics. Beyond this, our finding will also be of interest to the out-of-equilibrium community, by providing a new rational to select stationary states in out-of-equilibrium physics: it seems reasonable that in a physical system with two simultaneous equiprobable possible dynamics, the final stationary state will be closer to the stationary state corresponding to the fastest dynamics (smallest mixing time).Through the empirical link found in this letter, this state will correspond to a state of maximal Kolmogorov-Sinai entropy. If this is true, this would provide a more satisfying rule for selecting stationary states in complex systems such as climate than the maximization of the entropy production.