# 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.