Derivation of Langevin equation of a Brownian particle in a harmonic oscillator bath with space-dependent damping

A. Bhattacharyay

Published 2015-03-01Version 1

In ref.[1] an alternative approach to the equilibrium of a Brownian particle in an inhomogeneous space has been proposed. The sources of inhomogeneity, in general, in such cases, are the space dependent damping and the presence of a conservative force. Based on the phenomenological treatment at the level of Langevin dynamics, it was shown that, for equilibrium of such systems one cannot take the stochastic force strength to be a straightforward generalization of the Einstein relation as $\sqrt{2\Gamma(x)k_BT}$, rather, one has to do a generalization of the form $\Gamma(x)\sqrt{2k_BT/<\Gamma(x)>}$. Here we present a microscopic derivation of this generalized stochastic force strength considering a Brownian particle interacting with a bath of harmonic oscillators.