Ehrenfest's Urn Model
Paul (1880-1933) and Tatyana (born Afanasjeva, 1905-1984) Ehrenfest came
up with another example to illustrate how to reconcile thermodynamics and
irreversibility with the underlying reversible laws of classical mechanics. Paul
Ehrenfest, born in 1880, was a student of Boltzmann's. The following descrip-
tion of Ehrenfest's urn model was taken out of the wonderful book by J.R.
Introduction to Chaos in Nonequilibrium Statistical Mechanics
Consider two urns,
, and a bag.
balls, labeled 1
· · ·
, are dis-
tributed between the urns.
The bag contains
pieces of paper that carry
· · ·
. At each time step, somebody draws a number out of the
bag at random, moves the corresponding ball to the other urn, and then puts
the piece of paper back in the bag. This is repeated a large number of times.
Consider the (absolute) difference
between the number
of balls in urn
and in urn
as a function of time.
If you can, write a computer programme to simulate this experiment!
Derive an exact equation for the time evolution of the expected value of
) assuming transition probabilities to move a ball from urn
Make an approximation for the transition probabilities based on macro-
scopic variables (your
) and derive a closed evolution equation for
the macroscopic variable of the expected
). How would you characterize the
) for short and long times?
How can you relate the behaviour of the distribution of the balls in the urns
for this model to the resolution of the Zermelo and the Loschmidt paradoxes?
What would be the
for this system with its associated irreversible
Notice that the chance that a ball will move from the fuller
urn to the emptier urn is always greater than the chance that the opposite
will happen, therefore the system has the tendency to approach an equilibrium,
where the balls are distributed equally between the urns. Fluctuations will also
occur, even very large ones.