Physics – Condensed Matter – Statistical Mechanics
Scientific paper
2004-12-16
Eur. J. Phys. 26 (2005) 137
Physics
Condensed Matter
Statistical Mechanics
17 pages, 3 figures
Scientific paper
10.1088/0143-0807/26/1/014
The approach of an ideal gas to equilibrium is simulated through a generalization of the Ehrenfest ball-and-box model. In the present model, the interior of each box is discretized, {\it i.e.}, balls/particles live in cells whose occupation can be either multiple or single. Moreover, particles occasionally undergo random, but elastic, collisions between each other and against the container walls. I show, both analitically and numerically, that the number and energy of particles in a given box eventually evolve to an equilibrium distribution $W$ which, depending on cell occupations, is binomial or hypergeometric in the particle number and beta-like in the energy. Furthermore, the long-run probability density of particle velocities is Maxwellian, whereas the Boltzmann entropy $\ln W$ exactly reproduces the ideal-gas entropy. Besides its own interest, this exercise is also relevant for pedagogical purposes since it provides, although in a simple case, an explicit probabilistic foundation for the ergodic hypothesis and for the maximum-entropy principle of thermodynamics. For this reason, its discussion can profitably be included in a graduate course on statistical mechanics.
No associations
LandOfFree
The ideal gas as an urn model: derivation of the entropy formula does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with The ideal gas as an urn model: derivation of the entropy formula, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and The ideal gas as an urn model: derivation of the entropy formula will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-628311