  The Kubo-Martin-Schwinger condition is a widely studied fundamental property
in quantum statistical mechanics which characterises the thermal equilibrium
states of quantum systems. In the seventies, G. Gallavotti and E. Verboven,
proposed an analogue to the KMS condition for classical mechanical systems and
highlighted its relationship with the Kirkwood-Salzburg equations and with the
Gibbs equilibrium measures. In the present article, we prove that in a certain
limiting regime of high temperature the classical KMS condition can be derived
from the quantum condition in the simple case of the Bose-Hubbard dynamical
system on a finite graph. The main ingredients of the proof are Golden-Thompson
inequality, Bogoliubov inequality and semiclassical analysis.
