Physics 212: Statistical mechanics II, Fall 2006
Problem set 5
: due 11/9/06
1.
Suppose that one wants to calculate the specific heat of a system by Monte Carlo.
One
approach would be to calculate the free energy
F
or internal energy
U
at different temperatures,
then differentiate numerically. A better way is to use the following relation at fixed temperature
to relate the specific heat to fluctuations in
U
:
k
B
T
2
C
V
=
(
U

U
)
2
=
(Δ
U
)
2
.
(1)
In this notation,
U
=
Z

1
∑
i
U
(
i
)
e

βU
(
i
)
, where the sum is over the states of the system and
U
(
i
) is the energy of microstate
i
.
Prove the above relation. How must the typical fluctuation Δ
U
scale with system size? Assume
that
C
V
is proportional to the volume of the system, as is appropriate except possibly right at a
critical point.
What does this say about the relative size of fluctuations if we are away from a
critical point?
2. Wick’s theorem for the Gaussian model states that expectation values of products of
n
fields
factorize into products of 2body correlations. As an example, for
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 Fall '06
 MOORE
 mechanics, Energy, Statistical Mechanics, Heat, Ising

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