Geometric Ergodicity of Two--dimensional Hamiltonian systems with a Lennard--Jones--like Repulsive Potential

2011/04/19 by Ben Cooke, David P. Herzog, Cooke, Ben +7
Mathematics · Physics and Astronomy · #Markov Chains and Monte Carlo Methods #Quantum chaos and dynamical systems #Statistical Mechanics and Entropy

paper · pdf · doi:10.48550/arxiv.1104.3842

Abstract

In this paper we establish the ergodicity of Langevin dynamics for simple two-particle system involving a Lennard-Jones type potential. To the best of our knowledge, this is the first such result for a system operating under this type of potential. Moreover we show that the dynamics are \it geometrically ergodic (have a spectral gap) and converge at a geometric rate. Methods from stochastic averaging are used to establish the existence of a Lyapunov function. The existence of a Lyapunov function in this setting seems resistant to more traditional approaches. This is a corrected version of the article.

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