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On an urn problem of Paul and Tatiana Ehrenfest

Published online by Cambridge University Press:  24 October 2008

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

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 = (mi)/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 iI, kI and n = 0,1, 2,….

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1979

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References

REFERENCES

(1)Bellman, R. and Harris, T.Recurrence times for the Ehrenfest model. Pacific J. Math. 1 (1951), 179193.Google Scholar
(2)Boltzmann, L.Lectures on gas theory. University of California Press, Berkeley, 1964. English translation of L. Boltzmann: Vorlesungen über Gastheorie (Leipzig, J. A. Barth, 1896, 1898).Google Scholar
(3)Ehrenfest, P. and Ehrenfest, T.Über zwei bekannte Einwände gegen das Boltz-mannsche H-Theorem. Physik. Zeit. 8 (1907), 311314.Google Scholar
(4)Ehrenfest, P. and Ehrenfest, T. Begriffliche Grundlagen der statistischen Auffassung in der Mechanik. Enzyklopädie der Mathematischen Wissenschaften, vol. iv, 32. Mechanik (Leipzig, B. G. Teubner, 19071914), pp. 390.Google Scholar
(5)Friedman, B.A simple urn model. Comm. Pure Appl. Math. 2 (1949), 5970.Google Scholar
(6)Kac, M.Random walk and the theory of Brownian motion. American Mathematical Monthly 54 (1947), 369391. Reprinted: Selected papers on noise and stochastic processes, ed. N. Wax (New York, Dover, 1954), pp. 295–317.Google Scholar
(7)Kemble, E. C.The fundamental principles of quantum mechanics with elementary applications (New York, McGraw Hill, 1937). Reprinted: New York, Dover, 1958.Google Scholar
(8)Kemperman, J. H. B.The passage problem for a stationary Markov chain (University of Chicago Press, 1961).Google Scholar
(9)Rózsa, P.Megjegyzések egy sztochasztikus matrix spektrálfelbontáshoz. Magyar Tud. Akad. Mat. Fiz. Oszt. Közl. 7 (1957), 199206.Google Scholar
(10)Schlapp, R.The Stark effect of the fine-structure of hydrogen. Proc. Roy. Soc. London, Ser. A 119 (1928), 313334.Google Scholar
(11)Schrödinoer, E.Quantisierung als Eigenwertproblem. III. Annalen der Physik 80 (1926), 437490. Reprinted: E. Schrödinger: Abhandlungen zur Wellenmechanik, 2nd ed. (Leipzig, J. A. Barth, 1928), pp. 85–138.Google Scholar
(12)Schrödinger, E. and Kohlrausch, F.Das Ehrenfestsche Model der H-Kurve. Physikal-ische Zeitschrift 27 (1926), 306313.Google Scholar
(13)Siegert, A. J. F.Note on the Ehrenfest problem. Los Alamos Scientific Laboratory, MDDC-1406 (LADC-438).Google Scholar
(14)Siegert, A. J. F.On the approach to statistical equilibrium. Phys. Rev. 76 (1949), 17081714.CrossRefGoogle Scholar
(15)Sylvester, J. J.Théorème sur les déterminants. Nouvelles Annales de Mathématiques 13 (1854), 305. Reprinted: The collected mathematical works of James Joseph Sylvester, vol. II (Cambridge University Press, 1908), p. 28.Google Scholar
(16)Vincze, I. Über das Ehrenfestsche Modell der Warmeübertragung. Arch. Math. 15 (1964), 394400.Google Scholar
(17)Wang, M. C. and Uhlenbeck, G. E.On the theory of Brownian motion. II. Rev. Mod. Phys. 17 (1945), 323342. Reprinted: Selected papers on noise and stochastic processes, ed. N. Wax (New York, Dover, 1954) pp. 113–132.Google Scholar