New classical brackets for dissipative systems

TL;DR

This paper derives a new set of classical brackets for dissipative systems influenced by random forces, extending Peierls' method from conservative to dissipative dynamics, and shows their consistency with quantum commutation rules.

Contribution

It introduces a novel classical bracket formulation for dissipative systems with memory effects, generalizing Peierls' approach to include friction and stochastic forces.

Findings

Brackets satisfy the Jacobi identity.

Classical brackets match quantum commutation rules upon quantization.

Method applies to systems with linear friction and arbitrary memory functions.

Abstract

A set of brackets for classical dissipative systems, subject to external random forces, are derived. The method is inspired to the old procedure found by Peierls, for deriving the canonical brackets of conservative systems, starting from an action principle. It is found that an adaptation of Peierls' method is applicable also to dissipative systems, when the friction term can be described by a linear functional of the coordinates, as is the case in the classical Langevin equation, with an arbitrary memory function. The general expression for the brackets satisfied by the coordinates, as well as by the external random forces, at different times, is determined, and it turns out that they all satisfy the Jacobi identity. Upon quantization, these classical brackets are found to coincide with the commutation rules for the quantum Langevin equation, that have been obtained in the past, by appealing to microscopic conservative quantum models for the friction mechanism.