Statistical Mechanics - Physics at Oregon State University

14 CHAPTER 1. FOUNDATION OF STATISTICAL MECHANICS.
almost identical to the subsystem at time t. In a spatial average, a change in a
given subsystem is directly related to a change in the neighboring systems. The
time average only approaches the ensemble average when we measure very long,
or T → ∞. The spatial average only reduces to the ensemble average if we can
divide the system in an infinite number of subsystems, which reduces the relative
effects of the correlation. This requires N → ∞. In the ensemble average
we have to take L → ∞, but that is not a problem for a theorist. The limit that
the system becomes infinitely large is called the thermodynamic limit and is
essential in a number of derivations in thermodynamics. Keep in mind, though,
that we still need an even larger outside world to justify the basic assumption!
1.5 Thermal equilibrium.
Isolated systems are not that interesting. It is much more fun to take two
systems and bring them in contact. Suppose we have two systems, A and B,
which are in thermal contact. This means that only energy can flow back and
forth between them. The number of particles in each system is fixed. Also,
each system does not perform work on the other. For example, there are no
changes in volume. Energy can only exchange because of the interactions of
the two systems across their common boundary. Conservation of energy tells us
that the total energy U of the combined system is constant. The big question
is: what determines the energy flow between A and B and what is the
condition for thermal equilibrium, e.g. no net energy flow. We invoke our basic
assumption and note that all accessible states of the total system are equally
probable. A configuration of the total system is specified by the distribution of
energy over A and B, and hence by the value of UA ( since UB = U − UA and
U is constant ). The most probable configuration, or most probable value of
UA , corresponds to the configuration which has the largest number of states
available. The energy UA of this most probably configuration will be well defined
in the thermodynamic limit, since in that case multiplicity functions become
very sharp on a relative scale.
Explanation by example.
As an example, consider the Ising model in the presence of a magnetic induction
B. Since the number of particles does not vary, we will drop the reference
to N in the multiplicity functions in this paragraph. It is easy to show that
U = −
siµ · B = −Mµ · B = −xNµ · B (1.38)
i
The energy of subsystem A is UA = −xANAµ · B and of subsystem B is UB =
−xBNBµ · B. Because energy is conserved we have xN = xANA + xBNB
and therefore the total relative magnetization x is the average of the values of