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

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

2011-04-19

arxiv.org/abs/1104.3842v1

Mathematics

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20 Pages, 3 Figures

Scientific paper

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.

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