M. Masili, ICTP

This lesson begins with a discussion of the **Gibbs paradox**. It appears as a violation of the Second Law of Thermodynamics (the entropy of a closed system is shown to decrease) when the indistinguishability of the particles of a gas is not taken into accont in the definition of entropy. Another way of seeing this is noticing that the entropy is not an extensive quantity if identical particles are not considered indistinguishable. Once this is corrected, the Gibbs paradox dissapears.

The lesson proceeds to give a more rigorous treatment of statistical mechanics linking macrostates and microstates through the concept of **statistical ensemble**. A statistical ensemble is a collection of (infinitely) many points in the *6N*-dimensional phase space of the system where *N* is, as usual, the number of particles. Each of these points correspond to a copy of the statistical system and the density of points in a region of phase space corresponds to the probability that the system is in that region. A reduction of the statistical mechanics of one system to the study of ensembles needs an assumption of ergodicity, i.e., the assumption that averaging a property of the system over a time yields the same result than averaging the same property over many copies of the system.

Under these assumptions, the system under study can be modelled as a fluid that moves in phase space. These fluid must satisfy a **continuity equation** since ensembles are not created, nor destroyed. This continuity equation supplemented with the Hamiltonian equations of motion can be used to prove that **the fluid of ensembles is actually incompressible**! That is, the density of the fluid around an infinitesimal volume remains constant as the volume moves through the fluid or, what is the same thing, the amount of fluid entering a portion of phase space is equal to the amount of fluid that leaves it (*incompressible*). This is **Liouville's theorem**, a fundamental result of statistical mechanics.

Why is it fundamental? Well, it guarantees that the density of microstates around an infinitesimal volume of phase space is a constant of motion. This simplifies enormously the task of finding the distribution of states! Because, no matter how many particles the system might have, the constants of motion are few. With a couple more plausible assumptions it can be shown that actually, the density of states must be only a function of the energy.

Statistical Mechanics (Text Book) 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 3 | Modern Physics: Statistical MechanicsApril 13, 2009 - Leonard Susskind reviews the Lagrange multiplier, explains Boltzmann distribution and Helm-Holtz free energy before oulining into the theory of fluctuations.