Langevin molecular dynamics derived from Ehrenfest dynamics
The author presents investigations in which the stochastic Langevin molecular dynamics for nuclei is derived from the Ehrenfest Hamiltonian system in a Kac-Zwanzig setting, with initial data for the electrons stochastically perturbed from the ground state and the ratio \(M\) of nuclei and electron mass tending to infinity.NEWLINENEWLINEThe paper is divided into five sections. In the introduction, the author gives preliminaries to ab initio molecular dynamics and also the motivation for this paper which is related to stochastic models proposed in [\textit{G. W. Ford} and \textit{M. Kac}, J. Stat. Phys. 46, No. 5--6, 803--810 (1987; Zbl 0682.60069); \textit{G. Ford, M. Kac} and \textit{P. Mazur}, J. Math. Phys. 6, 504--515 (1965; Zbl 0127.21605)] and by \textit{R. Zwanzig} [``Nonlinear generalized Langevin equations'', J. Statist. Phys. 9, No. 3, 215--220 (1973; \url{doi:10.1007/BF01008729})] who derived a Langevin equation for a heavy particle from a Hamiltonian system with the heavy particle coupled through a harmonic interaction potential to a heat bath particle system. In Section 2, the author showed that Zwanzig's model is closely related to the Ehrenfest Hamiltonian system and extends in Sections 4 and 5 ideas presented by Zwanzig [loc. cit.]. Section 3 is related to stability and a consistency argument used to answer the following question: why should the electron initial data be Gibbs distributed? In Section 5, proofs for two theorems given in Section 4 are presented. These theorems contribute to the central problem in statistical mechanics of showing that Hamiltonian dynamics of heavy particles, coupled to a heat bath of many lighter particles with random initial data, can be approximately described by Langevin's equation.NEWLINENEWLINEConsidering this problem at least four new issues are indicated.NEWLINENEWLINE(i) The slow nuclei dynamics compared to the fast electron dynamics is exploited to find a proper Langevin equation without using explicit heat bath frequencies.NEWLINENEWLINE(ii) The error analysis uses the residual in the Kolmogorov equation of the Langevin dynamics evaluated along the Ehrenfest dynamics instead of the explicit solution available for harmonic oscillators.NEWLINENEWLINE(iii) A long-term result uses exponential decay in time of the first variation of the observable with respect to perturbations in the momentum.NEWLINENEWLINE(iv) An assumption of sufficiently low temperature makes certain quadratic forcing terms in the Ehrenfest momentum equation negligible to resemble Zwanzig models.
scientific article; zbMATH DE number 6032266