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 (PRE), 92 (2). 022140. ISSN 1550-2376

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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.

Item Type: Article
Schools and Departments: School of Mathematical and Physical Sciences > Mathematics
Subjects: Q Science > QA Mathematics > QA0273 Probabilities. Mathematical statistics
Q Science > QC Physics > QC0170 Atomic physics. Constitution and properties of matter Including molecular physics, relativity, quantum theory, and solid state physics
Depositing User: Enrico Scalas
Date Deposited: 28 Aug 2015 14:51
Last Modified: 07 Mar 2017 05:48
URI: http://sro.sussex.ac.uk/id/eprint/56263

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