Path Integral Solutions to the Distributions of Statistical Mechanics

TL;DR

This paper derives path-integral solutions for classical and quantum statistical distributions, providing a unified framework and explicit kernels for harmonic oscillator and free particle cases, facilitating perturbative expansions.

Contribution

It introduces a unified path-integral approach to derive distributions in classical and quantum mechanics, demonstrating the equivalence of two methods and providing explicit kernels for key systems.

Findings

Derived the phase space kernels using two methods and showed their perturbative equivalence.

Provided explicit Wigner and Liouville kernels for harmonic oscillator and free particle.

Established a foundation for perturbative expansion of the Wigner kernel for arbitrary potentials.

Abstract

We present the path-integral solutions to the distributions in classical (Gibbs) and quantum (Wigner) statistical mechanics. The kernel of the distributions are derived in two ways - one by time slicing and defining the appropriate short-time interval phase space matrix element and second by making use of the kernel in the path-integral approach to quantum mechanics. We show that the two approaches are perturbatively identical. We also present another computation for the Wigner kernel, which is also the Liouville kernel, for the harmonic oscillator and free particle. These kernels may be used as the starting point in the perturbative expansion of the Wigner kernel for any potential. With the kernel solved, we essentially solve also the distributions in classical and quantum statistical mechanics.