Velocity and energy distributions in microcanonical ensembles of hard spheres

Enrico Scalas, Adrian T. Gabriel, Edgar Martin, and Guido Germano
Phys. Rev. E 92, 022140 – Published 25 August 2015

Abstract

In a microcanonical ensemble (constant NVE, hard reflecting walls) and in a molecular dynamics ensemble (constant NVEPG, periodic boundary conditions) with a number N of smooth elastic hard spheres in a d-dimensional volume V having a total energy E, a total momentum P, and an overall center of mass position G, the individual velocity components, velocity moduli, and energies have transformed beta distributions with different arguments and shape parameters depending on d, N, E, the boundary conditions, and possible symmetries in the initial conditions. This can be shown marginalizing the joint distribution of individual energies, which is a symmetric Dirichlet distribution. In the thermodynamic limit the beta distributions converge to gamma distributions with different arguments and shape or scale parameters, corresponding respectively to the Gaussian, i.e., Maxwell-Boltzmann, Maxwell, and Boltzmann or Boltzmann-Gibbs distribution. These analytical results agree with molecular dynamics and Monte Carlo simulations with different numbers of hard disks or spheres and hard reflecting walls or periodic boundary conditions. The agreement is perfect with our Monte Carlo algorithm, which acts only on velocities independently of positions with the collision versor sampled uniformly on a unit half sphere in d dimensions, while slight deviations appear with our molecular dynamics simulations for the smallest values of N.

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  • Received 27 August 2013
  • Revised 16 July 2015

DOI:https://doi.org/10.1103/PhysRevE.92.022140

©2015 American Physical Society

Authors & Affiliations

Enrico Scalas1,2,*, Adrian T. Gabriel3,†, Edgar Martin3,‡, and Guido Germano4,5,§

  • 1School of Mathematical and Physical Sciences, University of Sussex, Falmer, Brighton BN1 9RH, United Kingdom
  • 2Basque Center for Applied Mathematics, Alameda de Mazarredo 14, 48009 Bilbao, Basque Country, Spain
  • 3Department of Chemistry and WZMW, Philipps-University Marburg, 35032 Marburg, Germany
  • 4Department of Computer Science, University College London, Gower Street, London WC1E 6BT, United Kingdom
  • 5Systemic Risk Centre, London School of Economics and Political Science, Houghton Street, London WC2A 2AE, United Kingdom

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Issue

Vol. 92, Iss. 2 — August 2015

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