Thermodynamics and stability of non-equilibrium steady states in open systems

TL;DR

This paper demonstrates how thermodynamic principles can be effectively used to construct physically meaningful Lyapunov functionals for analyzing the nonlinear stability of open systems, overcoming limitations of traditional methods.

Contribution

It introduces a new thermodynamically motivated approach to stability analysis of open systems, replacing artificial functionals with physically grounded ones.

Findings

Constructed a Lyapunov functional for heat conduction based on thermodynamics.

Showed the approach's potential applicability to complex thermomechanical systems.

Challenged the conventional view that thermodynamics is inapplicable to open systems.

Abstract

Thermodynamical arguments are known to be useful in the construction of physically motivated Lyapunov functionals for nonlinear stability analysis of spatially homogeneous equilibrium steady states in thermodynamically isolated systems. Unfortunately, the limitation to thermodynamically isolated systems is essential, and standard arguments are not applicable even for some very simple thermodynamically open systems. On the other hand, the nonlinear stability of thermodynamically open systems is usually investigated using the so-called energy method. Unfortunately, the designation "energy method" is clearly a misnomer. The mathematical quantity that is traditionally referred to as the "energy" is by no means linked to the energy in the physical sense of the word. Consequently, it would seem that genuine thermodynamical concepts are of no use in the nonlinear stability analysis of thermodynamically open systems. We show that this is not true. In particular, we propose a construction that in the case of simple heat conduction problem leads to a physically well-motivated Lyapunov functional, which effectively replaces the artificial Lyapunov functional used in the standard energy method. The proposed construction seems to be general enough to be applied in complex thermomechanical settings.