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On an urn problem of Paul and Tatiana Ehrenfest

Published online by Cambridge University Press:  24 October 2008

Lajos Takács
Lajos Takács
Affiliation:
Case Western Reserve University, Cleveland, Ohio
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Extract

A short solution is given for the urn problem proposed by Paul and Tatiana Ehrenfest in 1907.

In 1907 P. and T. Ehrenfest(3) proposed an urn model for the resolution of the apparent discrepancy between irreversibility and recurrence in Boltzmann's theory of gases (2). In this model it is assumed that m balls numbered 1, 2, …, m are distributed in two boxes. We perform a series of trials. In each trial we choose a number at random among 1, 2, …, m in such a way that each number has probability 1/m. If we choose j, then we transfer the ball numbered j from one box to the other. Denote by ξn the number of balls in the first box at the end of the nth trial. Initially there are ξ0 balls in the first box. If the trials are independent, then the sequence {ξn;n = 0, 1, 2,…} forms a homogeneous Markov chain with state space I = {0, 1, 2,…, m} and transition probabilities pi,i+1 = (m − i)/m for i = 0, l,…, m − 1, Pi,i−1 = i/m for i = 1, 2,…, m, and pi,k = 0 otherwise. The problem is to determine the transition probabilities

for i∈I, k∈I and n = 0,1, 2,….

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 86 , Issue 1 , July 1979 , pp. 127 - 130
Copyright
Copyright © Cambridge Philosophical Society 1979

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