Suppose that we have two states accessible to a system. Let the energy of the first be given by and the energy of the second be given by . It can be derived that the ratio of the probabilities of occupation of the two states is given by:
Any term of the form e -/τ is called a Boltzmann Factor.
You may wonder why we cannot simply write P(1) = e -/τ . The reason is that we are not guaranteed that the sum of the probabilities is equal to one yet, and therefore we can only talk of relative probabilities right now (see Quantum). In order to speak of the absolute probability we need to introduce a new concept.
We define the partition function as follows:
Notice that the partition function adds up all of the Boltzmann factors for a system. We can use it to make a crucial statement about absolute probability:
The equation should make sense to you. If the Boltzmann factor for a particular state were 2, and the partition function were 5, then we should expect our probability to by 0.4. Notice that P ranges from 0 to 1 as desired.
We can relate the partition function to the total energy of the system. Recall that we can determine the average value of a property by using the probability. Let us investigate < > , the average value of the energy of an occupied state. It turns out that < > is precisely what we mean when we say U . We can calculate it using the above equation for the absolute probability:
The final result seems strange but is mathematically tidier and is equivalent to the messier formulation.
Finally, and significantly, we can relate the Helmholtz free energy, sometimes just called the free energy, to Z as well. By a simple substitution, F = U - τσ , we find an important result: