Ehrenfest model

The Ehrenfest model (or dog–flea model) of diffusion was proposed by Tatiana and Paul Ehrenfest in 1907 to explain the second law of thermodynamics. 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 with transition rates

q

i

,

i

−

1

=

i

λ

{\displaystyle q_{i,i-1}=i\,\lambda }

for i = 1, 2, ..., N

q

i

,

i

+

1

=

(

N

−

i

)

λ

{\displaystyle q_{i,i+1}=(N-i\,)\lambda }

for i = 0, 1, ..., N – 1

and equilibrium distribution

π

i

=

2

−

N

(

N

i

)

{\displaystyle \pi _{i}=2^{-N}{\tbinom {N}{i}}}

.

Mark Kac proved in 1947 that if the initial system state is not equilibrium, then the entropy, given by

H

(

t

)

=

−

∑

i

P

(

X

(

t

)

=

i

)

log

⁡

(

P

(

X

(

t

)

=

i

)

π

i

)

,

{\displaystyle H(t)=-\sum _{i}P(X(t)=i)\log \left({\frac {P(X(t)=i)}{\pi _{i}}}\right),}

is monotonically increasing (H-theorem). This is a consequence of the convergence to the equilibrium distribution.

Interpretation of results

Consider that at the beginning all the particles are in one of the containers. It is expected that over time the number of particles in this container will approach

N

/

2

{\displaystyle N/2}

and stabilize near that state (containers will have approximately the same number of particles). However from mathematical point of view, going back to the initial state is possible (even almost sure). From mean recurrence theorem follows that even the expected time to going back to the initial state is finite, and it is

2

N

{\displaystyle 2^{N}}

.