1207.3505v2.pdf (1.33 MB)
Velocity and energy distributions in microcanonical ensembles of hard spheres
journal contribution
posted on 2023-06-08, 22:13 authored by Enrico Scalas, Adrian T Gabriel, Edgar Martin, Guido GermanoIn 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
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Publication status
- Published
File Version
- Accepted version
Journal
Physical Review E (PRE)ISSN
1550-2376Publisher
American Physical SocietyExternal DOI
Issue
2Volume
92Page range
022140Department affiliated with
- Mathematics Publications
Full text available
- Yes
Peer reviewed?
- Yes
Legacy Posted Date
2015-08-28First Open Access (FOA) Date
2016-08-23First Compliant Deposit (FCD) Date
2016-08-23Usage metrics
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