Langevin molecular dynamics derived from Ehrenfest dynamics

TL;DR

This paper derives Langevin molecular dynamics for nuclei from Ehrenfest dynamics in a quantum classical setting, showing the approximation accuracy and stability properties, with implications for understanding molecular systems at low temperatures.

Contribution

It provides a rigorous derivation of Langevin dynamics from Ehrenfest systems, including error estimates and stability analysis, connecting quantum and classical molecular dynamics.

Findings

Langevin dynamics approximates Ehrenfest nuclei dynamics with error o(M^{-1/2})

Initial electron distribution as Gibbs density ensures stability and convergence

Diffusion and friction coefficients satisfy Einstein's fluctuation-dissipation relation

Abstract

Stochastic Langevin molecular dynamics for nuclei is derived from the Ehrenfest Hamiltonian system (also called quantum classical molecular dynamics) in a Kac-Zwanzig setting, with the initial data for the electrons stochastically perturbed from the ground state and the ratio, $M$, of nuclei and electron mass tending to infinity. The Ehrenfest nuclei dynamics is approximated by the Langevin dynamics with accuracy $o(M^{-1/2})$ on bounded time intervals and by $o(1)$ on unbounded time intervals, which makes the small $\mathcal{O}(M^{-1/2})$ friction and $o(M^{-1/2})$ diffusion terms visible. The initial electron probability distribution is a Gibbs density at low temperture, derived by a stability and consistency argument: starting with any equilibrium measure of the Ehrenfest Hamiltonian system, the initial electron distribution is sampled from the equilibrium measure conditioned on the nuclei positions, which after long time leads to the nuclei positions in a Gibbs distribution (i.e. asymptotic stability); by consistency the original equilibrium measure is then a Gibbs measure.The diffusion and friction coefficients in the Langevin equation satisfy the Einstein's fluctuation-dissipation relation.