A note on Harris' ergodic theorem, controllability and perturbations of harmonic networks

TL;DR

This paper introduces an alternative approach using control theory and Harris' ergodic theorem to prove exponential convergence to equilibrium for certain harmonic oscillator networks coupled to heat baths.

Contribution

It combines control theory with Harris' ergodic theorem to establish exponential mixing for networks of quasi-harmonic oscillators under specific conditions.

Findings

Exponential convergence to a unique stationary measure is proven.

Conditions include dissipativity of A, Kalman controllability, slow growth of F, and weak Hörmander condition.

Provides a new method for analyzing ergodicity in harmonic networks.

Abstract

We show that elements of control theory, together with an application of Harris' ergodic theorem, provide an alternate method for showing exponential convergence to a unique stationary measure for certain classes of networks of quasi-harmonic classical oscillators coupled to heat baths. With the system of oscillators expressed in the form $\mathrm{d} X_t = A X_t \,\mathrm{d} t + F(X_t) \,\mathrm{d} t + B \,\mathrm{d} W_t$ in $\mathbf{R}^d$, where $A$ encodes the harmonic part of the force and $-F$ corresponds to the gradient of the anharmonic part of the potential, the hypotheses under which we obtain exponential mixing are the following: $A$ is dissipative, the pair $(A,B)$ satisfies the Kalman condition, $F$ grows sufficiently slowly at infinity (depending on the dimension $d$), and the vector fields in the equation of motion satisfy the weak H\"ormander condition in at least one point of the phase space.