Ehrenfest model

The Ehrenfest model (also known as the Ehrenfest chain ) is a stochastic model that describes the exchange of substances between two containers separated by a membrane . The model was first by the Austrian physicist Paul Ehrenfest ( 1880 - 1933 ) proposed and is one of many contributions of physics to develop the mathematical theory of stochastic processes .

The model

With various substances it was observed that the distribution of the substance in such an experiment tends towards a state of equilibrium over time , but still remains exposed to uncontrollable, apparently random fluctuations even after it has been reached.

The following model tried to explain this fact:

At the beginning there are a finite number of particles in both containers together ; for example the individual molecules of the substance, which are initially in the left and analogously in the right container. In each time step exactly one of these particles is selected, which is uniformly distributed and which changes the container, so that and in each step it increases or decreases by exactly one. ${\displaystyle N}$ ${\displaystyle l_{0}\leq N}$ ${\displaystyle r_{0}=N-l_{0}\leq N}$ ${\displaystyle N}$ ${\displaystyle l}$ ${\displaystyle r}$

From a mathematical point of view, this random process is a Markov chain with state space and a transition matrix , given by ${\displaystyle (l_{n})_{n\in \mathbb {N} _{0}}}$ ${\displaystyle \{0,1,2,\ldots N\}}$ ${\displaystyle \Pi =\left(\pi _{a,b}\right)}$

${\displaystyle \pi _{a,b}={\begin{cases}{\frac {a}{N}},&{\text{falls }}b=a-1\,,\\{\frac {N-a}{N}},&{\text{falls }}b=a+1\,,\\0&{\text{sonst}}.\end{cases}}}$

Mathematical properties

• The Ehrenfest chain defined above has a uniquely determined stationary distribution : If the number of particles in the left (or right) container is binomially distributed with a parameter , i.e. is for , then has the same distribution. ${\displaystyle \ell _{n}}$ ${\displaystyle {\frac {1}{2}}}$ ${\displaystyle P(\ell _{n}=k)={\binom {N}{k}}{\frac {1}{2^{N}}}}$ ${\displaystyle k=0,1,\ldots N}$ ${\displaystyle \ell _{n+1}}$

• The chain does not converge against this distribution, however, because the chain is periodic (this can be seen from the fact that it always changes between even and odd numbers and is therefore equal to zero every second time). This can be avoided by switching to the aperiodic version of the chain and replacing the transition matrix for a fixed parameter with the matrix (where is the identity matrix ). Interpretation: with probability the number of particles in the containers remains unchanged, with probability it changes according to the procedure described above. This makes the chain aperiodic and converges against the stationary distribution, which does not change due to this modification. ${\displaystyle \ell _{n}}$ ${\displaystyle P(\ell _{n}=k)}$ ${\displaystyle \Pi }$ ${\displaystyle p\in {]0,1[}}$ ${\displaystyle {\hat {\Pi }}:=pI_{N+1}+(1-p)\Pi }$ ${\displaystyle I_{N+1}}$
  ${\displaystyle p}$ ${\displaystyle 1-p}$
  ${\displaystyle n\to \infty }$

example

Exemplary representation of the Ehrenfest model. At the beginning all 10 particles are still in the left container.

Transition graph (restricted to states 5 to 10) with the transition probabilities. The states represent the number of particles in the left container.

Two containers are given, which are separated from one another by a membrane. At the beginning of the experiment, the left container contains molecules and the right container is still empty. The membrane allows exactly one molecule to change container per unit of time. ${\displaystyle 10}$

Since the container on the right is still empty at the beginning, a molecule will fly from the container on the left into the container on the right in the first second. Then there are only molecules in the left container. Now there are two possibilities: Either one of the remaining molecules of the left container flies into the right area, or the molecule on the right flies back into the left area. Each molecule should have the same chance to change the container. Accordingly, the probability is  % that another molecule will fly from left to right. With molecules on the left, this probability is only  % and so on. ${\displaystyle 9}$ ${\displaystyle 9}$ ${\displaystyle 90}$ ${\displaystyle 8}$ ${\displaystyle 80}$

The transition graph contains the states to , which represent the number of molecules in the left container. The Markov chain starts in the state . If you complete the transition graph and create a matching transition matrix, you can determine the probability distributions for the number of molecules in the left container for each point in time. After time units there is a  possibility of physical equilibrium for the first time with a probability of %. ${\displaystyle 0}$ ${\displaystyle 10}$ ${\displaystyle 10}$ ${\displaystyle 5}$ ${\displaystyle 30{,}2}$

The stationary distribution can be calculated using the formula formulated above

${\displaystyle P(l_{n}=k)=P(l_{n+1}=k)={\binom {10}{k}}{\frac {1}{2^{10}}}}$ For ${\displaystyle k=0,1,\ldots ,10}$

determine. This results in the probability distribution

${\displaystyle (0.10,0.98,4.39,11.72,20.51,24.61,20.51,11.72,4.39,0.98,0.10)}$ %.

literature

• Hans-Otto Georgii: Stochastics: Introduction to probability theory and statistics. 4th edition. de Gruyter textbook, Berlin 2009, ISBN 978-3-11-021526-7 , p. 166f.