M. Masili, ICTP

This lesson starts by recalling the relevant variables of the **microcanonical **and the **canonical ensembles **as an introduction for the **Grand canonical ensemble** where the temperature (T) the volume (V) and a quantity introduced here called **chemical potential** (?) remains constant and the energy (E) and number of particles (N) are allowed to fluctuate. The aim of this lesson is to find the analog for this kind of systems of the **Helmholtz free energy** and the **partition function** defined for the previous ensembles. The Grand canonical probability distribution of finding the system on a state with a given energy E_{R }and number of particles N_{R} was obtained using two methods. This probability distribution was shown to be the connection to thermodynamics, because it was used as a hint to calculate the **partition function** deriving the following relation *K _{B}T log[Z] = PV *where P is the pressure.

Then the parameter called the **fugacity** was introduced and the **partition function **was rewritten as a function of it. Some examples were then considered the first was the case of non interacting indistinguishable particles (the **Ideal gas**) for these case it was obtained the **equation of state** for the average number of particles. The second example considered was a gas of oscillators in which the particles are distinguishable. For this system it was found that the pressure is approximately zero, physically this was expected because the position of the particles does not depend on the external forces but just the oscillations.

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

University.Lecture 10 | Modern Physics: Statistical MechanicsJune 1, 2009 - Leonard Susskind presents the final lecture of Statistical Mechanics 10. In this lecture, he cover such topics as inflation, adiabatic transformation and thermal dynamic systems.