M. Masili, ICTP

This lecture began the study of **statistical mechanics**. It was recalled that thermodynamics, as studied so far, dealt with the macrostates of statistical system. A particular macrostate is defined by a set of macroscopic quantities such as the energy *E*, the volume *V* and the number of particles *N*.

Statistical mechanics on the other hand has to do with the relation betwen the macrostates the **microstates** compatible with them. The latter are defined by the position and momenta of the particles in the classical case or by the wavefunctions of the particles in the quantum case. In a macroscopic system, to each macrostate correspond a great number of microstates. The number of microstates *?* that correspond to a given macrostate is a function of *E*, *V* and *N* and if that function is known, all thermodynamics can be computed. So, the basic problem of statistical mechanics is to determine *?*.

A key assumption in this program is that any microstate that is compatible with a given macrostate has an equal probability of ocurrence. In order to derive thermodynamics relations one then uses the equilibrium condition on the system. This allows to identify the entropy of a system with the logarithm of the number of microstates up to a constant factor. This identification:

*S* = *k* log *?*

is the **Boltzmann equation** and provides a microscopic interpretation of entropy.

A statistical interpretation may be now abscribed to the **Third Law of Thermodynamics**. Namely, there is only one microstate state compatible with the energy being the absolute minimum minimal, thus the entropy of this state is zero.

In the last part of the lecture, the study of the **ideal monoatomic gas** was commenced. Using some approximations, the number of states of an ideal gas was calculated and from there the equation of state for an ideal gas was derived.

Written by R. K. Pathria,Paul D. Beale, pages from 9-172 and from 195-218 are the one related with this course.

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

University.Lecture 1 | Modern Physics: Statistical MechanicsMarch 30, 2009 - Leonard Susskind discusses the study of statistical analysis as calculating the probability of things subject to the constraints of a conserved quantity. Susskind introduces energy, entropy, temperature, and phase states as they relate directly to statistical mechanics.