Phases in the Mixing of Gases via the Ehrenfest Urn Model

The Ehrenfest urn is a model for the mixing of gases in 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 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 either 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 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.” The Gaussian results are obtained via martingale theory. 1 The Ehrenfest urn as a model for gas mixing The Ehrenfest urn was first proposed as a model for the mixing of nonreacting gases [4]. We deal here with the speed of this mixing across time phases. The model is for two chambers (say A and B) containing gases (possibly the same). The two chambers are connected through a pipe controlled by a valve. The ∗Department of Statistics, The George Washington University, Washington, D.C. 20052, U.S.A. valve is opened at time 0 and the mixing proceeds over epochs of time, which we can take as the unity. In each time unit (mixing step) one molecule of gas randomly chosen from the population of molecules in both chambers jumps from its chamber to the other one. This continual switching of sides affects a gradual mixing; inducing change in the amount of gas in each chamber. It is of interest to know the amount of gas (number of molecules) in chamber A after a certain period of time. This physical model of gas mixing can be visualized in terms of a scheme of drawing balls from an urn. We can think of the molecules in chamber A as balls of a certain color (say white) and those in chamber B as balls of an antithetical color (say red). The gas model with n molecules can then be viewed as n balls of two colors all residing in one urn, which evolves in the following manner. At each discrete point in time, we pick a ball at random from the urn. We paint that ball with the opposite color and put it back in the urn. In this equivalent model, the interest is to know the number of white balls (the amount of gas in Chamber A) after a certain period of time. The 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; see [1], [2] and [6]; for an overview see [8]. In a physical system the number of gas molecules is very large, we shall take it to be n, and is apportioned as bαnc ∼ αn in chamber A and n − bαnc ∼ (1 − α)n in Chamber B, for some α ∈ (0, 1). One is interested in knowing the behavior of the gases after a certain finite interval of time. So, the question is How many white balls are in the urn after k = kn draws, for functions kn of various growth rates? 9 Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 2 Scope We shall identify three phases of kn: (a) The sublinear phase, when kn = o(n); (b) The linear phase, when kn ∼ λnn, for some λn > 0 of a magnitude bounded from above and below; (c) The superlinear phase, when n = o(kn). We shall prove the following general trends. Trivially, at the very low end of the sublinear phase, when kn = O(1), as n → ∞, there is not much change in the content of the two chambers, only a finite perturbation on the initial conditions can be felt. However, when enough time has elapsed, that is when kn grows sublinearly to ∞, one sees normal behavior in the amount of gas in each chamber, even for a fairly slowly growing function kn. We call the phase when kn grows sublinearly to ∞, the growing sublinear phase. Functions that are asymptotically as small as 1 20 ln lnn, for example, are sufficient to give a normally distributed mix in each chamber. For the sublinear phase, the initial conditions persist, and the asymptotic normal result in this case contains the initial condition α. Theorem 1 Let Wkn be the number of white balls in the Ehrenfest urn (molecules in Chamber A) after kn draws (gas mixing steps) from an urn with n balls, of which initially the number of white balls is bαnc, where kn is in the growing sublinear phase. Then, Wkn − n ( 1 2 + ( α− 1 2 )( n−2 n )kn) √ kn D −→ N ( 0, 4α(1− α) ) . Normality continues to hold in the linear and superlinear phases. However, in each phase we get a different normal distribution. The mean and scale factors are essentially different. In the linear phase a different normal distribution (in the usual style of central limit theorems) is in effect, and the parameters of the distribution depend on both the initial condition α and the coefficient of linearity. However, the influence of the initial conditions is attenuated as we get deeper in the linear phase. Theorem 2 Let Wkn be the number of white balls in the Ehrenfest urn (molecules in Chamber A) after kn draws (gas mixing steps) from an urn with n balls, of which initially the number of white balls is bαnc, when kn is in the linear phase, where kn ∼ λnn, for some λn such that 0 < S1 ≤ λn ≤ S2 < ∞. Then, Wkn − (( α− 12 ) e−2λn + 12 ) n √ e4λn−1−4λn(2α−1)2 4e4λn n D −→ N (0, 1). As one might expect, after a very long period of time, as in the superlinear case, the mixing is nearly complete, and the result is a central limit theorem in which the effect of any initial conditions is washed away. Theorem 3 Let Wkn be the number of white balls in the Ehrenfest urn (molecules in Chamber A) after kn draws (gas mixing steps) from an urn with n balls, of which initially the number of white balls is bαnc, where kn is in the superlinear phase. Then, Wkn − 12n √ n D −→ N ( 0, 1 4 ) . To put our results in perspective note that these Gaussian laws in different phases consider the diffusion of gas when the chambers contain a large number of particles, tending to infinity. Some earlier research considers the transience in Ehrenfest models for a fixed number of particles, where the case is a finite Markov chain possessing a stationary distribution. Discussion of the variation distance from the stationary distribution is given in [3], where the diffusion of a fixed number of particles is considered over an extended number of draws. A cutoff phenomenon is reported in [3]: The total variation distance stays high (close to 1) up until a logarithmic number of draws when it drops sharply to low values near 0. 10 Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.