Important Things to Note:
It is extremely important to be able to tell the difference between the canonical ensemble and the microcanonical ensemble.
In a nutshell:
The microcanonical ensemble keeps track of all of the possible states of a SINGLE energy. The partition function of the microcanonical ensemble is simply that: the amount of states in a SINGLE energy. It then uses simple probability to make statements about entropy, specific heat, etc.
The canonical ensemble models a small system in a heat bath. Now the “canonical partition function” is now completely different from the “partition function.” This time, you add up all the possible “total energy” configurations, (In addition, each “total energy” state in the sum is also multiplied by e^(E/kT)).
The key is seeing that the microcanonical ensemble is for a system of a FIXED energy, while in the canonical ensemble, your system has a distribution of energies (and you need to keep track of and sum together “total energy” states instead of just the configurations of states.
Understand EVERY WORD of these:
Canonical Ensemble Notes from Professor Mona Berciu at University of British Columbia
(Do NOT slack off here! Do NOT skim this! Every single equation is critically important.)
Helpful Secondary Notes (The first few pages are helpful, but this is not required reading)
Most important Problems & Solutions:
- Problems #1, 2, 3, 5 (questions come with solutions)
- Harmonic Oscillator Problem & Solution (Thanks to Ryan Richards)
- Problem from UAlbany & Solution (Thanks to Ryan Richards)
- #1,2,3 in solved problems by Ryan Richards
- Problem #2 (solution)
- Problem #2b (solution) (this is probably the most conceptually hard canonical problem, but if you can do this you’re good to go!)
- Fall 2015 Comprehensive Exam Problem #1
- Spring 2016 Comprehensive Exam Problem #2