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Phases in the Diffusion of Gases via the Ehrenfest URN Modelx

Part of: Limit theorems Stochastic processes Combinatorics Combinatorial probability

Published online by Cambridge University Press:  14 July 2016

Srinivasan Balaji ,
Hosam Mahmoud  and
Zhang Tong
Srinivasan Balaji*
Affiliation:
The George Washington University
Hosam Mahmoud*
Affiliation:
The George Washington University
Zhang Tong*
Affiliation:
The George Washington University
*
∗Postal address: Department of Statistics, The George Washington University, Washington, DC 20052, USA.
∗Postal address: Department of Statistics, The George Washington University, Washington, DC 20052, USA.
∗Postal address: Department of Statistics, The George Washington University, Washington, DC 20052, USA.
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Abstract

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The Ehrenfest urn is a model for the diffusion of gases between two chambers. Classic research deals with this system as a Markovian model with a fixed number of balls, and derives the steady-state behavior as a binomial distribution (which can be approximated by a normal distribution). We study the gradual change for an urn containing n (a very large number) balls from the initial condition to the steady state. We look at the status of the urn after kn draws. We identify three phases of kn: the growing sublinear, the linear, and the superlinear. In the growing sublinear phase the amount of gas in each chamber is normally distributed, with parameters that are influenced by the initial conditions. In the linear phase a different normal distribution applies, in which the influence of the initial conditions is attenuated. The steady state is not a good approximation until a certain superlinear amount of time has elapsed. At the superlinear stage the mix is nearly perfect, with a nearly perfect symmetrical normal distribution in which the effect of the initial conditions is completely washed away. We give interpretations for how the results in different phases conjoin at the ‘seam lines’. In fact, these Gaussian phases are all manifestations of one master theorem. The results are obtained via martingale theory.

Keywords

Urn modelrandom structuremartingalecentral limit theoremdiffusion of gases

MSC classification

Secondary: 60C05: Combinatorial probability 60F05: Central limit and other weak theorems 05A05: Permutations, words, matrices 60G42: Martingales with discrete parameter
Type
Research Article
Information
Journal of Applied Probability , Volume 47 , Issue 3 , September 2010 , pp. 841 - 855
Copyright
Copyright © Applied Probability Trust 2010 

References

Antognini, A. B. (2005). On the speed of convergence of some urn designs for the balanced allocation of two treatments. Metrika 62, 309–322.Google Scholar
Bellman, R. and Harris, T. (1951). Recurrence times for the Ehrenfest model. Pacific J. Math. 1, 179–193.Google Scholar
Blom, G. (1989). Mean transition times for the Ehrenfest urn model. Adv. Appl. Prob. 21, 479–480.CrossRefGoogle Scholar
Diaconis, P. (1996). The cutoff phenomenon in finite Markov chains. Proc. Nat. Acad. Sci. USA 93, 1659–1664.Google Scholar
Ehrenfest, P. and Ehrenfest, T. (1907). Über zwei bekannte einwände gegen das Boltzmannsche H-theorem. Phys. Z. 8, 311–314.Google Scholar
Hall, P. and Heyde, C. C. (1980). Martingale Limit Theory and Its Applications. Academic Press, New York.Google Scholar
Karlin, S. and McGregor, J. (1965). Ehrenfest urn models. J. Appl. Prob. 2, 352–376.Google Scholar
Karr, A. F. (1993). Probability. Springer, New York.Google Scholar
Kolchin, V. F., Sevastyanov, B. A. and Chistyakov, V. P. (1976). Random Allocations. Moscow.Google Scholar
Mahmoud, H. M. (2008). Pólya Urn Models. CRC Press, Boca Raton, FL.Google Scholar
Mahmoud, H. (2010). Gaussian phases in generalized coupon collection. Unpublished manuscript.Google Scholar
Mikhailov, V. G. (1977). A Poisson limit theorem in the scheme of group disposal of particles. Theory Prob. Appl. 22, 152–156.Google Scholar
Mikhailov, V. G. (1980). Asymptotic normality of the number of empty cells in allocation of particles by groups. Theory Prob. Appl. 25, 82–90.CrossRefGoogle Scholar
Smythe, R. (2009). Phases in generalized coupon collection. Personal communication.Google Scholar
Vatutin, V. A. and Mikhailov, V. G. (1982). Limit theorems for the number of empty cells in an equiprobable scheme for the distribution of particles by groups. Theory Prob. Appl. 27, 734–743.Google Scholar