Statistical Mechanics - Physics at Oregon State University

1.3. STATES OF A SYSTEM. 9
which also is much smaller than one when N is very large. In other words, for
all values where the Gaussian approximation to the multiplicity function has a
large value it is close to the original function! In that sense we can replace the
multiplicity function by a Gaussian form for all values of x. The relative errors
are large when x is not small, but so is the value of g(N,x). In other words,
when the relative difference between the actual multiplicity function and the
Gaussian approximation is large, the values of the multiplicity function are very
small and the absolute value of the error is very small. This result is general for
all multiplicity functions. In the thermodynamic limit the deviations from the
maximum value are always of a Gaussian nature in the area where the values
are large. This is one of the reasons that statistical mechanics works!
It also reinforces the importance of the thermodynamic limit! It is
directly related to the law of large numbers in mathematics.
In order to keep the normalization correct we replace the factor N − 1 in the
exponent by N and will use later
Because we know that
g(N, x) ≈
2
πN 2N 1 −
e 2 x2N
g(N, x) = 2 N
x
(1.24)
(1.25)
and that the stepsize for x is 2
N (which includes a factor of two since M can only
change in units of two!), we can write
N 2
2
N
= g(N, x)∆x →
For the Gaussian approximation we find
1
−1
x
1
−1
2
πN 2N 1 −
e 2 x2
N
dx =
√ N
2
− √
2
N
2
∞
−∞
2
πN 2N e −t2
2
dt
N
which is the correct normalization indeed.
What is the physical meaning?
g(N, x)dx (1.26)
πN 2N e −t2
dt
= 2N 2
N
N
2 ≈
(1.27)
At this point we start to wonder what the multiplicity function means. Suppose
the parameter x in the previous example is an internal parameter of the
system. This would be the case in a system in zero magnetic field, where there
is no work involved when the magnetic moment changes. In thermodynamics
we have seen that the equilibrium state is the state of maximal entropy. Here
we see that the state with x = 0, which we expect to be the equilibrium state,