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Velocity and energy distributions in microcanonical ensembles of hard spheres

Scalas, E; Gabriel, AT; Martin, E; Germano, G; (2015) Velocity and energy distributions in microcanonical ensembles of hard spheres. Physical Review E , 92 , Article 022140. 10.1103/PhysRevE.92.022140. Green open access

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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.

Type: Article
Title: Velocity and energy distributions in microcanonical ensembles of hard spheres
Open access status: An open access version is available from UCL Discovery
DOI: 10.1103/PhysRevE.92.022140
Publisher version: http://dx.doi.org/10.1103/PhysRevE.92.022140
Language: English
Additional information: Copyright © 2015 American Physical Society.
UCL classification: UCL
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Engineering Science
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Engineering Science > Dept of Computer Science
URI: https://discovery.ucl.ac.uk/id/eprint/1407449
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