Kinetic energy fluctuations and specific heat in generalized ensembles

TL;DR

This paper presents a generalized formula linking kinetic energy fluctuations to specific heat across various ensembles, validated through simulations and applicable to finite, non-equilibrium systems.

Contribution

It derives an exact, versatile relation between kinetic energy fluctuations and specific heat valid for arbitrary steady-state ensembles and system sizes.

Findings

Validated the generalized formula with Monte Carlo simulations.

Showed relevance to systems with negative heat capacity and ensemble inequivalence.

Applicable to non-equilibrium phase transitions in finite systems.

Abstract

We derive an exact generalization of the well-known Lebowitz--Percus--Verlet (LPV) formula that relates the kinetic energy fluctuations of an isolated system to its specific heat. Our general formula, obtained by the application of expectation identities, is valid for arbitrary steady--state ensembles and system sizes, expressing the relative variance of the kinetic energy in terms of the variance of total energy and the microcanonical specific heat. The usual microcanonical LPV formula can be readily recovered as a particular case where energy fluctuations vanish. We test the validity of the generalized formula by performing Monte Carlo simulations of a superstatistical system of harmonic oscillators, as well as by exact calculation of energy variances in a uniform--energy ensemble, discussing its relevance to systems exhibiting negative heat capacity and ensemble inequivalence, as encountered in finite nuclei and self--gravitating models. Our results may provide useful in the study of non-equilibrium phase transitions in finite systems.