# Classical Langevin dynamics derived from quantum mechanics

## TL;DR

This paper extends Zwanzig's classical Langevin dynamics derivation to quantum systems, showing that quantum Langevin molecular dynamics can more accurately approximate quantum observables than Hamiltonian systems, especially at large mass ratios.

## Contribution

It introduces a quantum generalization of Langevin dynamics derived from first principles, improving the approximation of quantum observables over traditional Hamiltonian models.

## Key findings

- Quantum Langevin dynamics better approximates quantum observables.
- The model applies for any temperature and large mass ratios.
- A rank one friction matrix is used in the model.

## Abstract

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 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.