A. Schadicchio, 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 onĀ Statistical Mechanics given in theĀ

University.Lecture 8 | Modern Physics: Statistical MechanicsMay 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.