Canonical Ensemble - Deriving The Boltzmann Factor From Ensemble Theory

Deriving The Boltzmann Factor From Ensemble Theory

Let be the energy of the microstate and suppose there are members of the ensemble residing in this state. Further we assume the total number of members in the ensemble, and the total energy of all systems of the ensemble, are fixed, i.e.,

Since systems in the ensemble are indistinguishable with respect to a macrostate, for each set, the number of ways of shuffling systems is equal to

So for a given, there are rearrangements that specify the same state of the ensemble.

The most probable distribution is the one that maximizes . The probability for any other distribution to occur is extremely small in the limit . To determine this distribution, one should maximize with respect to the 's, under two constraints specified above. This can be done by using two Lagrange multipliers and . (The assumption that would be invoked in such calculation, which allows one to apply Stirling's approximation.) The result is

.

This distribution is called the canonical distribution. To determine and, it is useful to introduce the partition function as a sum over microscopic states

Comparing with thermodynamic formulae, it can be shown that, is related to the absolute temperature as, . Moreover the expression

is identified as the Helmholtz free energy . A derivation is given here. Consequently, from the partition function we can obtain the average thermodynamic quantities for the ensemble. For example, the average energy among members of the ensemble is

.

This relation can be used to determine . is determined from

.