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Consider a
system in contact with a heat reservoir. The systems in the representative
ensemble are distributed over their accessible states in accordance with the
Boltzmann distribution.
Thus, the probability of occurrence of some state with energy
is given by

(395) 
The mean energy is written

(396) 
where the sum is taken over all states of the system,
irrespective of their energy. Note that

(397) 
where

(398) 
It follows that

(399) 
The quantity , which is defined as the sum of the Boltzmann factor over all
states, irrespective of their energy,
is called the partition function.
We have just demonstrated that it is fairly easy to work out the
mean energy of a system using its partition function. In fact, as we shall
discover, it is easy
to calculate virtually any piece of statistical information using the partition
function.
Let us evaluate the variance of the energy. We know
that

(400) 
(see Sect. 2).
Now, according to the Boltzmann distribution,

(401) 
However,

(402) 
Hence,

(403) 
We can also write

(404) 
where use has been made of Eq. (399). It follows from Eq. (400) that

(405) 
Thus, the variance of the energy can be worked out from the partition function
almost as easily as the mean energy. Since, by definition,
a variance can never be negative, it
follows that
, or, equivalently,
. Hence, the mean energy of a system
governed by the Boltzmann distribution always increases with temperature.
Suppose that the system is characterized by a single external parameter
(such as
its volume). The generalization to the case where there are several external
parameters is obvious. Consider a quasistatic change of the external parameter
from to . In this process, the energy of the system in state changes
by

(406) 
The macroscopic work
done by the system due to this parameter
change is

(407) 
In other words, the work done is minus the average change in internal
energy of the system, where the
average is calculated using the Boltzmann distribution. We can write

(408) 
which gives

(409) 
We also have the following general expression for the work done by the system

(410) 
where

(411) 
is the mean generalized force conjugate to (see Sect. 4). It follows that

(412) 
Suppose that the external parameter is the volume, so . It follows that

(413) 
and

(414) 
Since the partition function is a function of and (the energies
depend on ), it is clear that the above equation relates
the mean pressure to
(via
) and . In other words, the above expression
is the equation
of state. Hence, we can work out the pressure, and even the equation of state,
using the partition function.
Next: Partition functions
Up: Applications of statistical thermodynamics
Previous: Paramagnetism
Richard Fitzpatrick
20060202