Assignment
Boltzmann's Dilemma
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.
Dorfman
Introduction to Chaos in Nonequilibrium Statistical Mechanics
.
Consider two urns,
I
and
II
, and a bag.
N
balls, labeled 1
,
2
· · ·
,N
, are dis-
tributed between the urns.
The bag contains
N
pieces of paper that carry
numbers 1
,
2
· · ·
,N
. 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
D
(
t
) =
bardbl
N
I
(
t
)
−
N
II
(
t
)
bardbl
between the number
of balls in urn
I
and in urn
II
as a function of time.
(0.)
If you can, write a computer programme to simulate this experiment!
(1.)
Derive an exact equation for the time evolution of the expected value of
D
(
t
) assuming transition probabilities to move a ball from urn
I
to urn
II
, and
vice versa.
(2.)
Make an approximation for the transition probabilities based on macro-
scopic variables (your
Stoßzahlansatz
) and derive a closed evolution equation for
the macroscopic variable of the expected
D
(
t
). How would you characterize the
behaviour of
D
(
t
) for short and long times?
(3.)
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
"entropy"
for this system with its associated irreversible
time evolution?
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.
1