The Boltzmann distribution and the quantum-classical correspondence

TL;DR

This paper investigates whether quantum Boltzmann probabilities can be derived from the classical-quantum distribution similarity at thermal equilibrium, demonstrating that they approach the classical form at high temperatures in simple systems.

Contribution

It proposes a novel approach to derive quantum Boltzmann probabilities from quantum-classical distribution agreement, validated in simple models.

Findings

Quantum distributions approach Boltzmann probabilities at high temperature

Kullback-Leibler divergence between quantum and classical distributions tends to zero

Method demonstrated on particle in a box and harmonic oscillator

Abstract

In this paper we explore the following question: can the probabilities constituting the quantum Boltzmann distribution, $P^B_n \propto e^{-E_n/kT}$, be derived from a requirement that the quantum configuration-space distribution for a system in thermal equilibrium be very similar to the corresponding classical distribution? It is certainly to be expected that the quantum distribution in configuration space will approach the classical distribution as the temperature approaches infinity, and a well-known equation derived from the Boltzmann distribution shows that this is generically the case. Here we ask whether one can reason in the opposite direction, that is, from quantum-classical agreement to the Boltzmann probabilities. For two of the simple examples we consider---a particle in a one-dimensional box and a simple harmonic oscillator---this approach leads to probability distributions that provably approach the Boltzmann probabilities at high temperature, in the sense that the Kullback-Leibler divergence between the distributions approaches zero.