# Ensemble dependence of fluctuations and the canonical/micro-canonical equivalence of ensembles

## TL;DR

This paper investigates the relationship between microcanonical and canonical ensembles in continuous systems, establishing convergence of Gibbs measures through local limit theorems and applying results to fluctuations of observables.

## Contribution

It introduces a rigorous proof of ensemble equivalence using local limit theorems and derives a formula for fluctuations in microcanonical ensembles.

## Key findings

- Proves convergence of Gibbs measures for continuous systems.
- Derives a formula for fluctuations of observables like kinetic energy.
- Establishes conditions for ensemble equivalence in the thermodynamic limit.

## Abstract

We study the equivalence of microcanonical and canonical ensembles in continuous systems, in the sense of the convergence of the corresponding Gibbs measures. This is obtained by proving a local central limit theorem and a local large deviations principle. As an application we prove a formula due to Lebowitz-Percus-Verlet. It gives mean square fluctuations of an extensive observable, like the kinetic energy, in a classical micro canonical ensemble at fixed energy.