Geometric Ergodicity of Two--dimensional Hamiltonian systems with a   Lennard--Jones--like Repulsive Potential
  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. 
Ben Cooke, David P. Herzog, Jonathan C. Mattingly, Scott A. McKinley,   and Scott C. Schmidler
math.PR math-ph math.MP