Statistical Mechanics - Physics at Oregon State University

12 CHAPTER 1. FOUNDATION OF STATISTICAL MECHANICS.
Finally, we can ask what happens in a truly isolated system. Is the time
evolution of such a system chaotic? This question is not easy to answer, since
we need to make a measurement in order to find out, and this measurement does
change the system! Another interesting theoretical question, but fortunately not
of practical importance. Nevertheless fun to think about.
What can we measure?
The quantities one can measure are always average quantities. It is impossible
to determine the positions as a function of time of molecules number two,
three, and seven in a gas of distinguishable molecules. We can measure quantities
like volume V and magnetization M. These are average quantities, and
they correspond to external handles on the system. The corresponding thermodynamic
variables are the intensive variables pressure p and magnetic field
H.
Everything independent.
Hence, we have to know how to calculate averages. There are three ways of
doing this. First, since all states are equally probable, we have that the average
value of a function f is given by
< f >ensemble=
states f(state)
states 1
(1.32)
In our simple Ising model this would mean a sum over 2 N states, if we assume
that N is specified. If we can only measure N and M, this expression is too
complicated. In this case f has to be a function of N and M only, otherwise it
would imply the ability to measure an additional quantity. Consequently, the
average reduces to
−N
< f >= 2 g(N, M)f(M) (1.33)
M
which only has N + 1 terms. We now have a sum over configurations of the
system, where a specific configuration is defined by the values of N and M. The
degeneracy of each configuration, or the number of states in each configuration,
is measured by the multiplicity function.
Averages taken this way are called ensemble averages. The formal justification
of this procedure in the old formulation of statistical mechanics is as
follows. Consider a set of L identical systems, in our case Ising systems with
N particles. The state of each system is chosen randomly from the accessible
states. If L ≫ 2 N , the number of systems with a configuration (N, M) is given
by Lg(N, M)2 −N and hence the average of a function f(M) over all systems is
given by