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- W1975804092 abstract "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 ξ n the number of balls in the first box at the end of the n th trial. Initially there are ξ 0 balls in the first box. If the trials are independent, then the sequence {ξ n ; n = 0, 1, 2,…} forms a homogeneous Markov chain with state space I = {0, 1, 2,…, m } and transition probabilities p i , i +1 = ( m − i )/ m for i = 0, l,…, m − 1, P i , i −1 = i/m for i = 1, 2,…, m , and p i,k = 0 otherwise. The problem is to determine the transition probabilities for i ∈ I , k ∈ I and n = 0,1, 2,…." @default.
- W1975804092 created "2016-06-24" @default.
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- W1975804092 date "1979-07-01" @default.
- W1975804092 modified "2023-09-29" @default.
- W1975804092 title "On an urn problem of Paul and Tatiana Ehrenfest" @default.
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- W1975804092 doi "https://doi.org/10.1017/s0305004100000694" @default.
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