Non-Markovian quantum Brownian motion: a non-Hamiltonian approach

TL;DR

This paper extends classical Brownian motion theory to include non-Markovian effects, develops a non-Hamiltonian quantum approach, and derives new non-Markovian quantum master equations applicable to various environments and temperature regimes.

Contribution

It introduces a non-Hamiltonian, non-Markovian framework for classical and quantum Brownian motion, including derivation of new master equations without relying on specific Hamiltonian models.

Findings

Non-Markovian effects cause differentiable trajectories and break energy equipartition at short times.

Derived non-Markovian quantum master equations for various environments and temperature ranges.

Extended classical non-Markovian equations to quantum systems using a dynamical-quantization method.

Abstract

We generalize the classical theory of Brownian motion so as to reckon with non-Markovian effects on both Klein-Kramers and Smoluchowski equations. For a free particle and a harmonic oscillator, it is shown that such non-Markovian effects account for the differentiability of the Brownian trajectories as well as the breakdown of the energy equipartition of statistical mechanics at short times in some physical situations. This non-Markovian approach is also extended to look at anomalous diffusion. Next, we bring in the dynamical-quantization method for investigating open quantum systems, which does consist in quantizing the classical Brownian motion starting directly from our non-Markovian Klein-Kramers and Smoluchowski equations, without alluding to any model Hamiltonian. Accordingly, quantizing our non-Markovian Klein-Kramers in phase space gives rise to a non-Markovian quantum master equation in configuration space, whereas quantizing our non-Markovian Smoluchowski equation in configuration space leads to a non-Markovian quantum Smoluchowski equation in phase space. In addition, it is worth noticing that non-Markovian quantum Brownian motion takes place in presence of a generic environment (e.g. a non-thermal quantum fluid). As far as the special case of a heat bath comprising of quantum harmonic oscillators is concerned, a non-Markovian Caldeira-Leggett master equation and a thermal quantum Smoluchowski equation are derived and extended to bosonic and fermionic heat baths valid for all temperatures.