Brownian Motion of Solitons in the $\Phi^4$ Model

TL;DR

This paper develops a theoretical framework to analyze the Brownian motion of solitons in the $\

Contribution

It introduces a method to derive the correlation function of the random force on solitons, accounting for zero modes and phonon contributions in the $\

Findings

Correlation function consistent with zero mode constraints

Low frequency phonons dominate when non-physical operators are used

Optical phonons can be included with physical operators

Abstract

We derive an expression for the correlation function of the random force on a soliton which is consistent with the constraints needed to integrate out the zero modes which appear due to the broken translational symmetry of the soliton solution. It is shown that when the constraint does not commute with the operator which defines the correlation function, i.e. when the operator is not physical, only low frequency phonons contributions may be considered. On the contrary, when the correlation function of the random force on the soliton is constructed with physical operators one may also include in a correct manner the contributions from the optical phonons.