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Velocity and energy distributions in microcanonical ensembles of hard spheres

Scalas, Enrico, Gabriel, Adrian T., Martin, Edgar and Germano, Guido (2015) Velocity and energy distributions in microcanonical ensembles of hard spheres. Physical Review E, 92. ISSN 2470-0045

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Identification Number: 10.1103/PhysRevE.92.022140


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.

Item Type: Article
Official URL:
Additional Information: © 2015 The Authors
Divisions: Systemic Risk Centre
Subjects: Q Science > QA Mathematics
Date Deposited: 13 Oct 2015 11:42
Last Modified: 16 May 2024 02:05
Projects: ES/K002309/1, PRIN 2009 H8WPX5
Funders: Economic and Social Research Council, Italian MIUR

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