Thermodynamics of driven systems via the Kuramoto-Sivashinsky equation

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. The unified thermodynamic algorithm's implications for the KS equation and the transition between the KS equation and its metriplectic counterpart present a novel pathway to study deterministic dynamical systems with instability.