Strict Lyapunov functions and energy decay in Hamiltonian chains with degenerate damping

TL;DR

This paper develops a strict Lyapunov function for a damped Hamiltonian chain of rotators, providing insights into energy decay and dissipation rates relevant to nonequilibrium statistical mechanics.

Contribution

It introduces an explicit, universal Lyapunov function for Hamiltonian chains with degenerate damping, enabling analysis of energy decay in nonlinear systems.

Findings

Lower bound for energy dissipation rate established

Explicit Lyapunov function constructed and analyzed

Method applicable to chains of oscillators

Abstract

We consider a Hamiltonian chain of rotators (in general nonlinear) in which the first rotator is damped. Being motivated by problems of nonequilibrium statistical mechanics of crystals, we construct a strict Lyapunov function that allows us to find a lower bound for the total energy dissipation rate when the energy and time are large. Our construction is explicit and its analysis is rather straightforward. We rely on a method going back to Matrosov, Malisoff and Mazenc, which we review in our paper. The method is rather universal and we show that it is applicable to a chain of oscillators as well.