Linear Hamiltonian Systems under Microscopic Random Influence

TL;DR

This paper proves that finite-dimensional linear Hamiltonian systems with a single dissipative and noisy coordinate converge to a unique Gibbsian equilibrium distribution, regardless of initial conditions and most Hamiltonians.

Contribution

It establishes the convergence to a Gibbs measure for linear Hamiltonian systems with microscopic noise and dissipation in almost all cases, addressing a longstanding issue about invariant measures.

Findings

Existence of a unique limiting Gibbs measure under dissipation and noise

Convergence holds for almost all Hamiltonians and initial conditions

The limiting temperature depends on dissipation and noise variance

Abstract

It is known that a linear hamiltonian system has too many invariant measures, thus the problem of convergence to Gibbs measure has no sense. We consider linear hamiltonian systems of arbitrary finite dimension and prove that, under the condition that one distinguished coordinate is subjected to dissipation and white noise, then, for almost any hamiltonians and almost any initial conditions, there exists the unique limiting distribution. Moreover, this distribution is Gibbsian with the temperature depending on the dissipation and of the variance of the white noise.