M. Masili, 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 n_{R} is the number of system with energy E_{R}. Then the probability of finding a system with energy E_{R} is P_{R}=<n_{R}>/**N. **To solve this problem first it was calculated the number n_{R}^{*} that maximized P_{R}.

A trick to compute <n_{R}> was to introduce the **Generating functions,** then <n_{R}> can be calculated as a first derivative of this **Generating function**. After long and tedious mathematics it is obtained that <n_{R}> = n_{R}^{*}. Then it was shown that P_{R} fallows a **Boltzmann distribution** P_{R}=Exp[-?E_{R}]/Z where is called the **partition function** (*Z = ? *Exp[*-?E _{n}*]).

As a complementary tool you can also see some lessons on Statistical Mechanics given in the

University.Lecture 5 | Modern Physics: Statistical MechanicsApril 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.