Communications in Mathematical Sciences

Volume 15 (2017)

Number 7

Geometric ergodicity of two-dimensional Hamiltonian systems with a Lennard–Jones-like repulsive potential

Pages: 1987 – 2025



Ben Cooke (Department of Mathematics, Duke University, Durham, North Carolina, U.S.A.)

David P. Herzog (Department of Mathematics, Iowa State University, Ames, Iowa, U.S.A.)

Jonathan C. Mattingly (Department of Mathematics, Center for Theoretical and Mathematical Science, Center for Nonlinear and Complex Systems, and Department of Statistical Sciences, Duke University, Durham, North Carolina, U.S.A.)

Scott A. McKinley (Department of Mathematics, Tulane University, New Orleans, Louisiana, U.S.A.)

Scott C. Schmidler (Department of Statistical Science, Department of Computer Science, Program in Computational Biology and Bioinformatics, Program in Structural Biology and Biophysics, Duke University, North Carolina, U.S.A.)


We establish ergodicity of the Langevin dynamics for a simple two-particle system involving a Lennard–Jones type potential. Moreover, we show that the dynamics is geometrically ergodic; that is, the system converges to stationarity exponentially fast. Methods from stochastic averaging are used to establish the existence of the appropriate Lyapunov function.


Langevin dynamics, Lennard–Jones potential, geometric ergodicity, Lyapunov function, stochastic averaging

2010 Mathematics Subject Classification

37A25, 60H10, 82C31

Received 21 April 2011

Accepted 1 July 2017

Published 16 October 2017