M. Marsili, ICTP
In this lesson continues with the study of the canonical ensemble for this type of ensembles the macrostate is specified by the number of particles N, the volume V and the temperature T. The probability distribution is then calculated by considering that each microstate as a system with a given energy and that the macrostate is composed by N systems and nR is the number of system with energy ER. Then the probability of finding a system with energy ER is PR=<nR>/N. To solve this problem first it was calculated the number nR* that maximized PR.
A trick to compute <nR> was to introduce the Generating functions, then <nR> can be calculated as a first derivative of this Generating function. After long and tedious mathematics it is obtained that <nR> = nR*. Then it was shown that PR fallows a Boltzmann distribution PR=Exp[-?ER]/Z where is called the partition function (Z = ? Exp[-?En]).
As a complementary tool you can also see some lessons on Statistical Mechanics given in the <span class="yt-user-name author">Stanford</span> University.
Lecture 5 | Modern Physics: Statistical Mechanics April 27, 2009 - Leonard Susskind discusses the basic physics of the diatomic molecule and why you don't have to worry about its structure at low temperature. Susskind later explores a black hole thermodynamics.