# High temperature convergence of the KMS boundary conditions: The Bose-Hubbard model on a finite graph

## TL;DR

This paper demonstrates that at high temperatures, the classical KMS condition can be derived from the quantum KMS condition for the Bose-Hubbard model on finite graphs, using semiclassical analysis and key inequalities.

## Contribution

It establishes a rigorous connection between quantum and classical KMS conditions in the high-temperature limit for the Bose-Hubbard model.

## Key findings

- Classical KMS condition derived from quantum condition at high temperature.
- Uses Golden-Thompson and Bogoliubov inequalities in proof.
- Provides semiclassical analysis for Bose-Hubbard system.

## Abstract

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.