Model of the ballot boxes of Ehrenfest

The model of the ballot boxes is a model Stochastique introduces in 1907 by the husbands Ehrenfest to illustrate some of the “paradoxes” appeared in the bases of the Mécanique incipient statistics. Little time indeed after Boltzmann had published his Théorème H, of virulent criticisms was formulated, in particular by Loschmidt, then by Zermelo, Boltzmann being shown to practice “doubtful mathematics”.

This model is sometimes also called the “ model of the dogs and the chips ”. The mathematician Mark Kac wrote in this connection that it was:

“… probably one of the most instructive models of all physics …”

The model of the ballot boxes

Definition of the stochastic model

One considers two ballot boxes has and B , like NR balls, numbered of 1 with NR . Initially, all the balls are in the ballot box has . The associated stochastic process is built in the following way:

At the moment $t_0 = 0$ , one draws randomly a number I ranging between 1 and NR , and one transfers the ball n° I from the ballot box has towards the ballot box B .

At the moment $t_1 = 1$ , one draws again randomly a number J ranging between 1 and NR .

If the ball n° J is in the ballot box has , one transfers it in the ballot box B .
If the ball n° J is in the ballot box B , one transfers it in the ballot box has .

And so on…

Dynamics of the model

In this model, one follows during time T (discrete) the full number of ball N (T) present in the ballot box has . One obtains a curve which initially leaves N (0) =N and starts by decreasing towards the median value N/2 , as one could expect it for “a good” thermodynamic system initially out of balance and releasing spontaneously towards balance.

But this decrease is irregular: there exist fluctuations around the median value N/2 , which can become sometimes very important (this is particularly visible when NR is small).

In particular, whatever the number of balls NR finished, there always exists recurrences in an initial state, for which all the balls return in the ballot box has after one finished duration. But, as average time between two consecutive recurrences believes very quickly with NR , these recurrences do not appear to us when NR is very large (typically in Physique statistics, NR is of about size of the Nombre of Avogadro).

Version “models dogs and chips”

In this version, the two ballot boxes are replaced by two dogs, and the NR balls by NR chips, jumping from one dog to another.

Recurrences & theorem of Kac (1947)

Recurrences in an initial state

There exist recurrences in an initial state, characterized by a countable succession of moments $\ {t_n \} _ {n=1, 2, \ finished dowries}$ for which all the balls return in the ballot box has , i.e. one a: $N (t_n) = NR$ (by convention, one poses $t_0 = 0$ ). One can then define a new countable continuation $\ tau_n = t_n - t_ {n-1}$ of the durations finished between two consecutive recurrences.

Theorem of Kac (1947)

It is possible to calculate the intermediate duration between two recurrences in an initial state consecutive:

\ langle \ \ tau \ \ rangle \ = \ \ lim_ {p \ to \ infty} \ \ frac {1} {p} \ \ sum_ {n=1} ^p \ \ tau_n

There is the following theorem - 1947:

\ langle \ \ tau \ \ rangle \ = \ 2^N

Moreover, one can show that the dispersion durations around their median value, characterized by the standard deviation $\ sigma$ , is of the even order of magnitude:

\ sigma \ = \ \ sqrt {\ \ lim_ {p \ to \ infty} \ \ frac {1} {(p - 1)} \ \ sum_ {n=1} ^p \ \ left \, \ tau_n \, - \, \ langle \ \ tau \ \ rangle \, \ right^2 \} \ \ sim \ \ langle \ \ tau \ \ rangle

See for example.

Examples of digital simulations

The great fluctuations relative around the average become less and less frequent for a given duration when the number NR of balls increases.

Exact solution

See for example: and:

Bond with a random walk

The model of the ballot boxes of Ehrenfest is formally similar to a random walk nonisotropic on the network $\ mathbb {Z}$ , whose limit continues converges towards the Brownian Movement of a particle elastically dependant .