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Velocity and energy distributions in microcanonical ensembles of hard spheres

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journal contribution
posted on 2023-06-08, 22:13 authored by Enrico Scalas, Adrian T Gabriel, Edgar Martin, Guido Germano
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.

Funding

ES/K002309/1; ESRC

The growth of firms and countries: distributional properties and economic determinants—Finitary and non-finitary probabilistic models in economics.; Ministero dell'Istruzione, dell'Universita' e della Ricerca

History

Publication status

  • Published

File Version

  • Accepted version

Journal

Physical Review E (PRE)

ISSN

1550-2376

Publisher

American Physical Society

Issue

2

Volume

92

Page range

022140

Department affiliated with

  • Mathematics Publications

Full text available

  • Yes

Peer reviewed?

  • Yes

Legacy Posted Date

2015-08-28

First Open Access (FOA) Date

2016-08-23

First Compliant Deposit (FCD) Date

2016-08-23

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