Boltzmann distributions

We can think of the interaction of a molecule with the air in a classroom as analogous to the interaction of a small system in thermal contact with a heat reservoir . The air acts like a heat reservoir because its energy fluctuations due to any interactions with the molecule are far too small to affect any of its macroscopic parameters. Let us determine the probability of finding system in one particular microstate of energy when it is thermal equilibrium with the heat reservoir .

As usual, we assume fairly weak interaction between and , so that
the energies of these two systems
are additive. The energy of is not known at this
stage. In fact, only the *total* energy of the combined system
is known. Suppose that the
total energy lies in the range to
.
The overall energy is constant in time, since
is assumed to be an isolated system, so

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Let us now make use of the fact that system is far smaller than system .
It follows that
, so the slowly varying logarithm of
can be Taylor expanded about . Thus,

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The Boltzmann distribution often causes confusion. People who are used to the
principle of equal *a priori* probabilities, which says that all microstates
are equally probable, are understandably surprised when they come across the
Boltzmann distribution which says that high energy microstates are markedly less
probable then low energy states. However, there is no need for any
confusion. The principle of equal *a priori* probabilities applies to
the *whole* system, whereas the Boltzmann distribution only applies to
a small part of the system. The two results are perfectly consistent.
If the small system is in a microstate with a comparatively high energy then
the rest of the system (*i.e.*, the reservoir) has a slightly lower energy than
usual (since the overall energy is fixed). The number of accessible microstates
of the reservoir is a very strongly increasing function of its energy. It
follows that when the small system has a
high energy then significantly less states
than usual are accessible to the reservoir, and so the number of microstates
accessible
to the overall system is reduced, and, hence, the configuration is comparatively
unlikely. The strong increase in the number of accessible microstates of the
reservoir with increasing gives rise to the strong (*i.e.*, exponential) decrease
in the likelihood of a state of the small system with increasing .
The exponential factor
is called the *Boltzmann factor*.

The Boltzmann distribution gives the probability of finding the small system
in *one* particular state of energy . The probability
that has an energy in the small range between and
is just the sum of all the probabilities of the states which lie in this
range. However, since each of these states has approximately the same Boltzmann
factor this sum can be written

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