Previously, we derived the state probability distribution for the canonical ensemble. The quantity , given by
was the probability that a closed system occupies the state with associated energy . The variable is as yet unendowed with physical meaning, but we can tell that, since it appears in an exponent with an energy term, it must have units of reciprocal energy.
For sake of ease, we give the sum on the denominator of its own special letter:
This sum is given the stately name of the partition function. Unwieldy as it appears, it turns out that any thermodynamic property of the system you’d care to ask out can be written in terms of the partition function.
Note that to evaluate we need to know how the energy levels of the system are discretised. This requires that we understand the system’s microphysics. So for now we’ll have to talk in general terms. But if we have the set , the partition function is the means by which we translate this microphysical knowledge to macrophysical knowledge.
As an example, consider the mean internal energy of the system. It is related to the set of probabilities by
Given the expression for above, we can rewrite this as
The reciprocal of can be taken out of the sum:
By inspection we see this is equal to
This is in fact equivalent to
applying the chain rule ‘in reverse’. So if we know the partition function, we know the internal energy of the system!
Here’s a further example. Recall the expression for the Gibbs entropy:
Again, we use the expression for :
So if we know the partition function, we know the Gibbs entropy of the system.
This stuff is still pretty abstract. In a few posts’ time, we’ll actually look at a toy system to see how everything we’ve done so far comes together. But there’s a little more work to do yet.
In the next post, we’ll consider small changes to the Gibbs entropy, and in doing so, construct the first law of thermodynamics for a closed system.