Communications in Mathematical Sciences

Volume 16 (2018)

Number 8

Exponential relaxation of the Nosé–Hoover thermostat under Brownian heating

Pages: 2231 – 2260



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


We study a stochastic perturbation of the Nosé–Hoover equation (called the Nosé–Hoover equation under Brownian heating) and show that the dynamics converges at a geometric rate to the augmented Gibbs measure in a weighted total variation distance. The joint marginal distribution of the position and momentum of the particles in turn converges exponentially fast in a similar sense to the canonical Boltzmann–Gibbs distribution. The result applies to a general number of particles interacting through a wide class of potential functions, including the usual polynomial type as well as the singular Lennard–Jones variety.


Langevin dynamics, Nosé–Hoover equation, Lennard–Jones potential, geometric ergodicity, molecular dynamics simulation, random sampling

2010 Mathematics Subject Classification

37A25, 60H10, 60J22, 65C05, 82C31

Received 24 April 2018

Accepted 22 August 2018

Published 18 April 2019