# The Boltzmann distribution and the quantum-classical correspondence. (arXiv:1710.06051v5 [quant-ph] UPDATED)

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.