Sunday, January 27, 2008

Canon Not In 'D'

Yup, I'm talking about a canonical ensemble here; my Physical Chemistry professor, Dr. Fan Wai Yip, completely missed out this essential concept in my course last semester, which is essential to the concept of partition functions, statistical weight and the entropy of systems.

So what exactly is a canonical ensemble? Let us consider what it means by the word ensemble first:

Above we have a collection of atoms - any simple collection of microscopic objects can be referred to as an ensemble. Typically, we consider an ensemble to be a macroscopic system. So now, what is a canonical ensemble? Well, look at this:

Well, a canonical ensemble is simply a collection of ensembles as above! It is simply partitioning a huge, gigantic macroscopic system into smaller subsystems, each one still a macrosystem by itself. What this means is that each subsystem by itself, still obeys the statistical laws of statistical mechanics perfectly well, and therefore, by calculating the properties of one subsystem, one can simply multiply that property by the number of subsystems (if it is an extensive quantity) to obtain the overall property or quantity.

And the reverse is true: if you can calculate a quantity for the entire canonical ensemble, then it follows that you can divide by the number of such ensembles to obtain the quantity for the small subsystem by itself.

We shall now use this idea to derive a very general expression for the entropy of a system in terms of the probabilities of a macrosystem being in a certain energy state r. Let us consider a very general case where we have a huge macrosystem that can be partitioned into n identical subsystems. Each subsystem will then be in thermal contact with one another, interacting weakly, and each system will also have the same probabilities of being in certain energy states.

Let us then assume that each subsystem can exist in energy states 1, 2, 3, ..., r... and we allow the associated probability of each subsystem being found such an energy state be p(r). Then for n such subsystems, the number of systems to be found in the energy state r is simply:

With this, we then proceed to calculate the number of ways all of the subsystems can be partitioned within the huge system. What I mean to say is that we now proceed to determine the number of possible ways to distribute all of the subsystems into the various energy states 1, 2, 3, ..., r ... and we do this by determining the statistical weight of the entire canonical ensemble:

The corresponding entropy of the entire ensemble is then:And if we use Stirling's approximation we obtain: We then substitute in the very first relation we stated at the start of this derivation:

And the next step is to simply divide the entropy of this huge macrosystem by the number of subsystems in the ensemble, so that we obtain:

And there you have it, the entropy of a single macroscopic system entirely in terms of probabilities!

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