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Kinetic energy fluctuations and specific heat in generalized ensembles

Sergio Davis, Catalina Ruíz, Claudia Loyola, Carlos Femenías, Joaquín Peralta

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.

Kinetic energy fluctuations and specific heat in generalized ensembles

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.
Paper Structure (7 sections, 53 equations, 1 figure)

This paper contains 7 sections, 53 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Derivation of the main result
  3. Verification
  4. Canonical ensemble
  5. Superstatistical harmonic oscillator ensemble
  6. Uniform--energy ensemble
  7. Concluding remarks

Figures (1)

  • Figure 1: Relative variance of the kinetic energy as a function of the number of particles $N$ for a superstatistical ensemble of three--dimensional harmonic oscillators. Each microstate samples an inverse temperature $\beta$ from a gamma distribution with shape $\alpha=5$ and scale $\theta=1$ before drawing coordinates and momenta. Points correspond to Simulation data and the dashed line is the theoretical prediction from Eq. \ref{['eq:main']}.