### Statistical Mechanics Lecture 23 of 29

July 13, 2012 by Antonello Scardicchio

A. Scardicchio, ICTP

This lesson started with a proof that in classical statistical mechanics, the variance of a variable on which the Hamiltonian depends quadratically is proportional to kT. In particular, this result implies that the specific heat is a constant along the whole energy range. This important classical result breaks down when quantum behavior starts to dominate.

In order to understand better what happens, this lesson started the study of a system composed of N harmonic oscillators fixed in space. This differs from the ideal gas in the fact that the Hamiltonian includes a potential energy and in that the particles are distinguishable since they
are assumed to be in fixed points in space.

The partition function of this system and, hence, all of its thermodynamics can be found straightforwardly following the procedure studied in previous lessons. In particular the entropy is found to be negative for T below some value (hbar*?/E). More importantly, it diverges to minus infinity for T -> 0. This implies that we must be doing some wrong when counting the microstates.

The solution is in the fact that as the system cools, quantum effects start to dominate. The partition function for a quantum system of N harmonic oscillators can also be computed and the thermodynamics deduced from it doesn't have any of the problems above. The entropy is shown to tend to zero as T -> 0 and also the specific heat. On the other hand, for very large T the result reduced to the classical case.

In the last part of the lecture, an alternative way of deriving the Virial theorem from statistical mechanics was shown.

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 8 | Modern Physics: Statistical Mechanics May 19, 2009 - Leonard Susskind lectures on a new class of systems, magnetic systems. He goes on to talk about mean field approximations of molecules in multidimensional lattice systems.

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