Thermodynamics of driven systems via the Kuramoto-Sivashinsky equation

TL;DR

This paper investigates the thermodynamic properties of the driven Kuramoto-Sivashinsky equation, analyzing how external energy sources influence entropy production and system evolution, and introduces a thermodynamic variant with monotonic entropy growth.

Contribution

It develops a thermodynamic variant of the Kuramoto-Sivashinsky equation that monotonically increases entropy and explores the effects of positive spectra on system dynamics.

Findings

Positive spectra prevent metriplectic description of KS

Thermodynamic variant of KS with monotonic entropy growth

Rescaling spectra affects system evolution and chaos transition

Abstract

We examine the differences between the driven turbulence described by the Kuramoto-Sivashinsky (KS) equation and the second law of thermodynamics. A general velocity and entropy density system is analyzed with the unified thermodynamic algorithm of metriplectic dynamics, and we show that the positive spectra of the KS equation due to an external energy source prevent its metriplectic description. A variant of the KS equation is produced that monotonically generates an entropy, but the only equilibria of this variant system are spatially constant. Numerical experiments are performed comparing the evolution of the KS equation and its thermodynamic variant. The entropy of this thermodynamic system is increased further by the driving effects of the KS equation, reconciling the generation of entropy with the energy source of the KS equation. Further numerical experiments restrict the positive spectra in the KS equation to determine the effect on the system time evolution. While rescaling the growth rates of instabilities reproduces similar behavior on a slower time scale, introduction of individual positive spectra reproduces the formation of equilibria, relative equilibria, and a transition to chaos.