Exact Markovian kinetic equation for a quantum Brownian oscillator

TL;DR

This paper derives an exact Markovian kinetic equation for a quantum Brownian oscillator, revealing how irreversibility and stochasticity emerge from the dynamics in generalized function spaces.

Contribution

It introduces a novel derivation of the Markovian kinetic equation using subdynamics, highlighting differences between integrable and non-integrable systems.

Findings

Invariant subspaces follow uncoupled, renormalized particle dynamics in integrable systems.

Non-integrable systems exhibit broken-time symmetry in their invariant subspaces.

Irreversibility and stochasticity are shown to be inherent properties of the dynamics in generalized function spaces.

Abstract

We derive an exact Markovian kinetic equation for an oscillator linearly coupled to a heat bath, describing quantum Brownian motion. Our work is based on the subdynamics formulation developed by Prigogine and collaborators. The space of distribution functions is decomposed into independent subspaces that remain invariant under Liouville dynamics. For integrable systems in Poincar\'e's sense the invariant subspaces follow the dynamics of uncoupled, renormalized particles. In contrast for non-integrable systems, the invariant subspaces follow a dynamics with broken-time symmetry, involving generalized functions. This result indicates that irreversibility and stochasticity are exact properties of dynamics in generalized function spaces. We comment on the relation between our Markovian kinetic equation and the Hu-Paz-Zhang equation.