Brownian motion of classical spins : Anomalous dissipation and generalized Langevin equation

TL;DR

This paper investigates the relaxation dynamics of classical spins coupled to a heat bath, deriving a generalized Langevin equation with anomalous dissipation and analyzing the associated equilibrium distributions.

Contribution

It introduces a detailed analysis of classical spin relaxation using GLE with anomalous dissipation and addresses challenges in obtaining equilibrium distributions for non-Markovian systems.

Findings

Derived the Fokker-Planck equation for the GLE with anomalous dissipation.

Identified difficulties in obtaining equilibrium distributions in non-Markovian cases.

Proposed remedies to overcome these difficulties.

Abstract

In this work, we analyze the relaxation of a classical spin interacting with a heat bath, starting from the fully dynamical Hamiltonian description. An analogous problem in the framework of generalized Langevin equation (GLE) with anomalous dissipation is analyzed in details. The Fokker-Planck equation corresponding to GLE is derived and the concept of equilibrium probability distribution is analyzed. In this process we have identified few difficulties to obtain equilibrium distribution for the non-Markovian case and the remedy to overcome this difficulty is also discussed.