In this note we study the asymptotic limit of large variance in a stochastically perturbed thermostat model, the Nos\'{e}-Hoover-Langevin device. We show that in this limit, the model reduces to a Langevin equation with one-dimensional Wiener process, and that the perturbation is in the direction of the conjugate momentum vector. Numerical experiments with a double well potential corroborate the asymptotic analysis.

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Springer
doi.org/10.1007/s10955-011-0203-1
Journal of Statistical Physics
Adaptive Multisymplectic Box Schemes for Hamiltonian Wave Equations
Computational Dynamics

Frank, J., & Gottwald, G. A. (2011). The Langevin Limit of the Nosé-Hoover-Langevin Thermostat. Journal of Statistical Physics, 143, 715–724. doi:10.1007/s10955-011-0203-1