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## Classical Langevin dynamics derived from quantum mechanics

 1 Chair of Mathematics for Uncertainty Quantification, RWTH Aachen University, Germany 2 Institutionen för Matematik, Kungl. Tekniska Högskolan, 100 44 Stockholm, Sweden

* Corresponding author: Anders Szepessy (szepessy@kth.se)

Received  June 2019 Revised  January 2020 Published  April 2020

Fund Project: The second author is supported by the Swedish Research Council grants 2014-04776 and 2019-03725

The classical work by Zwanzig [J. Stat. Phys. 9 (1973) 215-220] derived Langevin dynamics from a Hamiltonian system of a heavy particle coupled to a heat bath. This work extends Zwanzig's model to a quantum system and formulates a more general coupling between a particle system and a heat bath. The main result proves, for a particular heat bath model, that ab initio Langevin molecular dynamics, with a certain rank one friction matrix determined by the coupling, approximates for any temperature canonical quantum observables, based on the system coordinates, more accurately than any Hamiltonian system in these coordinates, for large mass ratio between the system and the heat bath nuclei.

Citation: Håkon Hoel, Anders Szepessy. Classical Langevin dynamics derived from quantum mechanics. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020135
##### References:

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##### References:
(Left column) unit-volume scaled histogram for the final time position and momentum of the heat bath dynamics for a series of $m$-values and (right column) corresponding histograms for the Langevin dynamics
(Left column) heat bath dynamics autocorrelation function $\mathbb E[X^1(t)X^1(10)]$ for a series of $m$-values and (right column) corresponding Langevin dynamics autocorrelation functions $\mathbb E[X^1_L(t)X^1_L(10)]$
(Left column) heat bath dynamics autocorrelation function $\mathbb E[P^1(t)P^1(10)]$ for a series of $m$-values and (right column) corresponding Langevin dynamics autocorrelation functions $\mathbb E[P^1_L(t)P^1_L(10)]$
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