Statistical Mechanics - Physics at Oregon State University

16 CHAPTER 1. FOUNDATION OF STATISTICAL MECHANICS.
The last statement is intuitively correct. In equilibrium, any part of a magnet
has the same partial magnetization as the whole magnet.
Next, we approximate the form of the terms t(xA) in the case that both
NA/gg1 and NB/gg1. In that case we can write
t(xA) ≈ t(x)e − N A N
2N B (xA−x) 2
(1.48)
which has the right value and curvature at the maximum. For large numbers of
particles this form has a small relative error when the value is large, and a large
relative error when the value is very small (and hence does not matter). This is
again one of those cases where we rely on the thermodynamic limit. Statistical
mechanics can be applied to small systems, but in such a case one cannot make
a connection with thermodynamics.
The value of the term is reduced by a factor e for a fluctuation
2NB
δxA =
(1.49)
NAN
For a given ratio of NA and NB this becomes very small in the limit N → ∞.
Again, if system A is small, the relative fluctuations in energy are large and one
has to be careful in applying standard results of thermodynamics.
We are now able to do the summation over all terms:
or
g(N, x) ≈ t(x)
g(N, x) =
t(xA) ≈ t(x)
xA
xA
e − N A N
2N B (xA−x) 2
NA
dxA
2 e− NAN 2N (xA−x)
B 2
= t(x) 1
πNANB
2 N
(1.50)
(1.51)
Next, we use are familiar friends the logarithms again to get
log(g(N, x)) ≈ log(t(x)) + log( 1
πNANB
) (1.52)
2 N
The first term on the right hand side contains terms proportional to N or NA,
while the second term contains only the logarithm of such numbers. Therefore,
in the thermodynamic limit we can ignore the last terms and we find that the
logarithm of the sum (or of g) is equal to the logarithm of the largest term!
This might seem a surprising result, but it occurs in other places too. In a
mathematical sense, it is again connected with the law of large numbers.
1.6 Entropy and temperature.
The statements in the previous section can be generalized for arbitrary systems.
If two systems A and B, with multiplicity functions gA(N, U) and gB(N, U)