Statistical Mechanics - Physics at Oregon State University

10 CHAPTER 1. FOUNDATION OF STATISTICAL MECHANICS.
has the maximal multiplicity function. Therefore, the question can be asked if
there is any relation between a multiplicity function and the entropy. We will
return to this question in shortly.
How to use a multiplicity function?
Average values can also be calculated easily in a number of cases. If all
states of the system are equally probable, the expected value for a quantity f is
states < f >=
f(state)
states 1
(1.28)
If f only depends on the magnetization x in a given state, this reduces to
< f >= f(x)g(N, x)dx N
2 2−N
(1.29)
For the Ising model using the Gaussian approximation to the multiplicity
function we find
< x >= 0 (1.30)
< x 2 >= 1
N
(1.31)
which means that there is no net magnetization and that the fluctuations are
small. There is no net magnetization since for every state {si} there is a corresponding
state {s ′ i } with s′ j = −sj which has the opposite magnetization. One
has to apply a magnetic field in order to produce a magnetic state in our simple
Ising model. In realistic models to be discussed later there are interactions between
neighboring spins that lead to a net magnetic moment. The fluctuations
are small because the multiplicity function is very sharp as we have seen before.
1.4 Averages.
Once we have determined the eigenstates of a system, we have to describe in
which of these states the system actually is. The basic assumption in statistical
mechanics is:
All accessible quantum states are equally probable.
There is no a priori preference for any given state. One keyword in this statement
is accessible. This means that we only have to consider those states which have
properties consistent with the external constraints. For example, in a closed
system we know that the number of particles is fixed. If we start with 27
particles, we only consider quantum states with 27 particles.