Statistical optimization for passive scalar transport: maximum entropy production vs maximum Kolmogorov-Sinay entropy

TL;DR

This paper analytically compares maximum entropy production and Kolmogorov-Sinai entropy principles in passive scalar transport, revealing conditions under which their optimal points coincide and how system resolution influences this relationship.

Contribution

It provides a rigorous analytical link between maximum entropy production and Kolmogorov-Sinai entropy in a Markov model of passive scalar diffusion, highlighting their similarities and differences.

Findings

fmaxEP and fmaxKS have the same first-order Taylor expansion near equilibrium.

fmaxEP is largely independent of system resolution N.

fmaxKS depends strongly on N and tends to zero as N increases.

Abstract

We derive rigorous results on the link between the principle of maximum entropy production and the principle of maximum Kolmogorov-Sinai entropy using a Markov model of the passive scalar diffusion called the Zero Range Process. We show analytically that both the entropy production and the Kolmogorov-Sinai entropy seen as functions of f admit a unique maximum denoted fmaxEP and fmaxKS. The behavior of these two maxima is explored as a function of the system disequilibrium and the system resolution N. The main result of this article is that fmaxEP and fmaxKS have the same Taylor expansion at _rst order in the deviation of equilibrium. We find that fmaxEP hardly depends on N whereas fmaxKS depends strongly on N. In particular, for a fixed difference of potential between the reservoirs, fmaxEP (N) tends towards a non-zero value, while fmaxKS (N) tends to 0 when N goes to infinity. For values of N typical of that adopted by Paltridge and climatologists we show that fmaxEP and fmaxKS coincide even far from equilibrium. Finally, we show that one can find an optimal resolution N_ such that fmaxEP and fmaxKS coincide, at least up to a second order parameter proportional to the non-equilibrium uxes imposed to the boundaries.