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Partition functions
It is clear that all important macroscopic quantities
associated with a system can be expressed in terms
of its partition function . Let us investigate how
the partition function is related to thermodynamical quantities.
Recall that is a function of both and (where is the single
external parameter). Hence,
, and we can write

(415) 
Consider a quasistatic change by which and change so slowly that
the system stays close to equilibrium, and, thus, remains
distributed according to the
Boltzmann distribution. If follows from Eqs. (399) and (409) that

(416) 
The last term can be rewritten

(417) 
giving

(418) 
The above equation shows that although the heat absorbed by the system
is not
an exact differential, it becomes one when multiplied by the temperature
parameter . This is essentially the second law of thermodynamics. In fact,
we know that

(419) 
Hence,

(420) 
This expression enables us to calculate the entropy of a system
from its partition function.
Suppose that we are dealing with a system consisting of
two systems and which only interact weakly with one another. Let each
state of be denoted by an index and have a corresponding energy .
Likewise, let each state of be denoted by an index and have a corresponding
energy . A state of the combined system is then denoted
by two indices and . Since and only interact weakly their
energies are additive, and the energy of state is

(421) 
By definition, the partition function of takes the form
Hence,

(423) 
giving

(424) 
where and are the partition functions of and , respectively.
It follows from Eq. (399) that the mean energies of , , and
are related by

(425) 
It also follows from Eq. (420) that the respective entropies of these systems
are related via

(426) 
Hence, the partition function tells us that the extensive thermodynamic
functions of two weakly interacting systems are simply additive.
It is clear that we can perform statistical thermodynamical calculations using
the partition function instead of the more direct approach in which we use the
density of states . The former
approach is advantageous because the partition function
is an unrestricted sum of Boltzmann factors over all
accessible states,
irrespective of their energy, whereas the
density of states is a restricted sum over all states whose energies
lie in some narrow range. In general, it is far easier to perform an unrestricted
sum than a restricted sum. Thus, it is generally easier to derive
statistical thermodynamical results using rather than , although
has a far more direct physical significance than .
Next: Ideal monatomic gases
Up: Applications of statistical thermodynamics
Previous: Mean values
Richard Fitzpatrick
20060202