Temporal Behavior of the Conditional and Gibbs' Entropies

TL;DR

This paper investigates how Gibbs' and conditional entropies evolve over time towards equilibrium in deterministic and stochastic systems, revealing diverse behaviors including monotonic and oscillatory approaches.

Contribution

It provides a comprehensive analysis of the temporal behavior of both entropies, including analytical examples illustrating various approach patterns to equilibrium.

Findings

Conditional entropy remains constant or increases monotonically to zero.

Gibbs' entropy exhibits diverse behaviors, including oscillations.

Analytical examples demonstrate all observed entropy dynamics.

Abstract

We study the temporal approach to equilibrium of the Gibbs' and conditional entropies for both invertible deterministic dynamics as well as non-invertible stochastic systems in the presence of white noise. The conditional entropy will either remain constant or monotonically increase to its maximum of zero. However, the Gibbs' entropy may have a variety of patterns of approach to its final value ranging from a monotone increase or decrease to an oscillatory approach. We have illustrated all of these behaviors using examples in which both entropy dynamics can be determined analytically.