A note on Harris' ergodic theorem, controllability and perturbations of 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 Hörmander condition in at least one point of the phase space.

References in corpus (3)

• Entropic fluctuations in thermally driven harmonic networks
• Non-Equilibrium Steady States for Networks of Oscillators
• Controllability implies mixing I. Convergence in the total variation metric

Cited by in corpus (5)

• Non-Equilibrium Steady States for Networks of Oscillators
• Quantitative Rates of Convergence to Non-Equilibrium Steady State for a Weakly Anharmonic Chain of Oscillators
• Spectral gap in mean-field $\mathcal{O}(n)$-model
• Cutoff thermalization for Ornstein-Uhlenbeck systems with small Lévy noise in the Wasserstein distance
• Exponential mixing of constrained random dynamical systems via controllability conditions