Statistical Mechanics - Physics at Oregon State University

6 CHAPTER 1. FOUNDATION OF STATISTICAL MECHANICS.
In order to show how to count states of a system it is very useful to have a
definite example at hand. Although we derive basic results in this model, the
final conclusions will be general. We will consider an Ising system (in one, two,
or three dimensions) with atomic moments µi of the form
µi = siµ (1.4)
with i = 1, · · · , N. The variables si can only take the values −1 or +1. A state
of the system is given by a specific series of values of the set of numbers {si}, for
example {−, −, +, +, +, · · ·}. Depending on our experimental abilities we can
define a number of quantities of interest for this system. Here we will restrict
ourselves to the total number of particles N and the total magnetic moment
M, with M =
si. The real magnetic moment is, of course, M = Mµ. One
i
could also include other quantities, like the number of domain walls (surfaces
separating volumes of spins which are all up from regions where all spins are
down). Or short range correlations could be included. Basically, anything that
can be measured in an experiment could be included.
How many states?
How many states are there for a system with a given value of N? Obviously,
2 N . How many states are there for this system with specified values of N and
M? The answer is g(N,M) and this function is called a multiplicity function.
Obviously, the 2 N mentioned before is an example of a simpler multiplicity
function g(N). If we define the number of spins up (with value +1) by N↑ and
the number of spins down by N↓ it is easy to show that
N
g(N, M) =
N↑
(1.5)
with M = N↑ − N↓. One way of deriving this is to choose N↑ elements out of a
total of N possibilities. For the first one we have N choices, for the second one
N − 1, and for the last one N − N↑ + 1. But the order in which we make the
choices is not important, and hence we need to divide by N↑!. We then use the
definition
N
N↑
= N(N − 1) · · · (N − N↑ + 1)
N↑!
= N!
N↑!N↓!
(1.6)
where we have used N = N↑ + N↓. The multiplicity function is properly normalized.
Using the binomial expansion
we find with x = y = 1 that
(x + y) N =
N
n=0
N
x
n
n y N−n
(1.7)