Statistical Mechanics - Physics at Oregon State University

1.5. THERMAL EQUILIBRIUM. 15
the magnetization of the subsystems. Since the total magnetization does not
change, the only independent variable is xA.
If we assume that the states of the subsystems are not correlated, we have
the following result for the multiplicity functions:
g(N, x) =
g(NA, xA)g(NB, xB) (1.39)
xA
with xA and xB related as described above.
Next, we introduce a short hand notation for the individual terms:
t(xA) = g(NA, xA)g(NB, xB) (1.40)
and notice that this function is also sharply peaked around its maximum value
when both NA and NB are large. For our model the multiplicity function is in
that case
Therefore
g(U) =
t(xA) = 2
1
2
π NANB
N 1 −
e
2
πN 2N 1 −
e 2 x2N 2 (x2
2
ANA+x B NB) = t0e
1 − 2 (x2
2
ANA+xB NB)
(1.41)
(1.42)
If the number of particles is very large, xA is approximately a continuous
variable and the number of states is maximal when the derivative is zero. It is
easier to work with the log of the product in t, which is also maximal when the
term is maximal. We find
log(t(xA)) = log(t0)− 1
2 (x2ANA+x 2 BNB) = log(t0)− 1
2 x2ANA− 1
(xN−xANA)
2NB
2 )
(1.43)
The derivatives are straightforward (thank you, logs)
∂ log(t)
= −xANA −
∂xA
NA
(xANA − xN)
NB
(1.44)
2 ∂ log(t)
∂xA 2
= −NA − N 2 A
NB
= − NNA
NB
(1.45)
The second derivative is negative, and hence we do have a maximum when the
first derivative is zero. This is the case when
or
0 = −xANB − (xANA − xN) (1.46)
xA = x = xB
(1.47)