title: Velocity and energy distributions in microcanonical ensembles of hard spheres.
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.
authors.affiliation: School of Mathematical and Physical Sciences, University of Sussex, Falmer, Brighton BN1 9RH, United Kingdom.|Basque Center for Applied Mathematics, Alameda de Mazarredo 14, 48009 Bilbao, Basque Country, Spain.
authors.forename: Enrico
authors.initials: E
authors.lastname: Scalas
authors.affiliation: Department of Chemistry and WZMW, Philipps-University Marburg, 35032 Marburg, Germany.
authors.forename: Adrian T
authors.initials: AT
authors.lastname: Gabriel
authors.affiliation: Department of Chemistry and WZMW, Philipps-University Marburg, 35032 Marburg, Germany.
authors.forename: Edgar
authors.initials: E
authors.lastname: Martin
authors.affiliation: Department of Computer Science, University College London, Gower Street, London WC1E 6BT, United Kingdom.|Systemic Risk Centre, London School of Economics and Political Science, Houghton Street, London WC2A 2AE, United Kingdom.
authors.forename: Guido
authors.initials: G
authors.lastname: Germano