### Statistical Mechanics Lecture 25 of 29

July 18, 2012 by Matteo Marsili

M. Marsili, 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 ER and number of particles NR 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 KBT log[Z] = PV where P is the pressure.

Then the parameter https://md.ictp.it/admin/media/https://md.ictp.it/admin/media/http://upload.wikimedia.org/wikipedia/en/math/7/7/6/77698ae92ac0435f8da1e266eeb528e3.png" alt="z\,"/> 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 <span class="yt-user-name author">Stanford</span> University.

Lecture 10 | Modern Physics: Statistical Mechanics June 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.

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