Statistical Mechanics in Collective Coordinates

TL;DR

This paper explores transforming particle-based statistical mechanics into a field-based framework using collective coordinates, deriving key equations and approximations relevant to many-body systems.

Contribution

It provides explicit formulas for the functional Fourier transform of the Jacobian and derives the Fokker-Planck equation in collective coordinates, connecting microscopic and field descriptions.

Findings

Derived the functional Fourier transform of the Jacobian.

Formulated the Fokker-Planck equation in collective coordinates.

Discussed approximations leading to Debye-Huckel and Percus-Yevick theories.

Abstract

We study the transformation of the statistical mechanics of N particles to the statistical mechanics of fields, that are the collective coordinates, describing the system. We give an explicit expression for the functional Fourier transform of the Jacobian of the transformation from particle to collective coordinate and derive the Fokker-Planck equation in terms of the collective coordinates. Simple approximations, leading to Debye-Huckel theory and to the hard sphere Percus-Yevick equation are discussed.