Conditions for the validity of the quantum Langevin equation

TL;DR

This paper investigates the conditions under which the quantum Langevin equation derived from microscopic models remains valid, focusing on frequency proximity and slow variable assumptions, tested against an exactly solvable model.

Contribution

It identifies and tests specific conditions for the validity of the quantum Langevin equation, highlighting their limitations through comparison with an exactly solvable model.

Findings

Frequency proximity condition is necessary for validity.

Slow variable assumption has limitations in certain regimes.

Comparison with an exactly solvable model clarifies the applicability of these conditions.

Abstract

From microscopic models, a Langevin equation can in general be derived only as an approximation. Two possible conditions to validate this approximation are studied. One is, for a linear Langevin equation, that the frequency of the Fourier transform should be close to the natural frequency of the system. The other is by the assumption of `slow' variables. We test this method by comparison with an exactly soluble model, and point out its limitations. We base our discussion on two approaches. The first is a direct, elementary treatment of Senitzky. The second is via a generalized Langevn equation as an intermediate step.