The Ehrenfest model of diffusion was proposed by Paul Ehrenfest

Paul Ehrenfest

Paul Ehrenfest was an Austrian and Dutch physicist, who made major contributions to the field of statistical mechanics and its relations with quantum mechanics, including the theory of phase transition and the Ehrenfest theorem.- Biography :Paul Ehrenfest was born and grew up in Vienna in a Jewish...

 to explain the second law of thermodynamics

Second law of thermodynamics

The second law of thermodynamics is an expression of the tendency that over time, differences in temperature, pressure, and chemical potential equilibrate in an isolated physical system. From the state of thermodynamic equilibrium, the law deduced the principle of the increase of entropy and...

. The model considers N particles in two containers. Particles independently change container at a rate λ. If X(t)=i is defined to be the number of particles in one container at time t, then it is a birth-death process

Birth-death process

The birth–death process is a special case of continuous-time Markov process where the states represent the current size of a population and where the transitions are limited to births and deaths...

 with transition rates

• for i=1,2,...,N
• for i=0,1,...,N–1

and equilibrium distribution .

It can be shown that if the initial system state is not equilibrium, then the entropy, given by


is monotonically increasing (H-theorem

H-theorem

In Classical Statistical Mechanics, the H-theorem, introduced by Ludwig Boltzmann in 1872, describes the increase in the entropy of an ideal gas in an irreversible process. H-theorem follows from considerations of Boltzmann's equation...

). This is a consequence of the convergence to the equilibrium distribution.