The Ehrenfest fleas: from model to theory
The authors propose a generalized Ehrenfest urn model (i.e., \(n\) individuals distributed on \(d\) categories) and develop an abstract theory based on some axioms for the conditional probability \(\text{Pr}\{n(j)| n\}\), where the occupation vector \(n= (n_1,\dots, n_d)\in N^n_d\), \(\sum^d_{j=1} n_j= n\), and \(n(j)\) is the occupation vector resulted from \(n\) increasing or decreasing by \(1\) at the \(j\)th category (\(n(j)\in N^{n+1}_d\) or \(N^{n-1}_d\)). In fact, the exchangeability and invariance (for increasing or decreasing) are assumed in these axioms. Under some additional simple conditions the generalized model yields transition probability and equilibrium distribution. It then can lead to the unified statistics of Bose-Einstein, Fermi-Dirac and Maxwell-Boltzmann and can also follow many applications in population genetics, economics etc. This paper strongly means that the time evolution or explication of many natural phenomena can be given in terms of conditional probabilities, which play an essential role in science.
scientific article; zbMATH DE number 2081517