M. Masili, ICTP

This lesson starts by recalling the **Liouville's theorem** that was described in the previous lecture. Then the **microcanonical ensemble** was introduced this ensemble is used to describe **isolated systems n** , systems were the macrostate is characterize by a given energy. In the **microcanonical ensemble** the number of particles N, the volume V, and the energy E remains constant. In order to compute thermodynamics quantities we have to calculate the number of **microstates** ?(N,V,E)=?(E)/?_{o} that correspond to one **macrostates**, where ?(N,V,E) is the total volume on phase space occupied by the macrostate and ?_{o} the volume of one microstate, then the **entropy** S was defined as S=k_{B}log[?(N,V,E)].

The number of microstates ?(N,V,E) for the **Ideal classical gas** was then obtained as an example as well as the entropy S. The volume of one microstate in phase space was obtained and it was shown to be equal to ?_{o}=h^{3N} this means that the volume occupied for one particle in phase space is h. As a second example the **harmonic oscillator** was studied and ?(N,V,E) calculated, again in this case the volume occupied for one particle in phase space is h. These results are related with the **Uncertainty principle**. Thus in general for a system of N particles we have ?(N,V,E)=?(E)/h^{3N}.

The second part of the lesson was dedicated to study system in **Thermal equilibrium**. These type of systems are considered to be in equilibrium with a **thermal reservoir** or **thermal bath**, they can interchange heat but the temperature remains constant, the **thermal bath** is considered to be much larger than the system of interest in order that its properties does not change due to the presence of it. All systems of these types are described by the **Canonical ensemble **. It was shown that the probability P(*X*_{n}) of finding a particular microstate *X*_{n} with energy *E _{n}* is proportional to exp{

*-?E*}.

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

University.Lecture 4 | Modern Physics: Statistical MechanicsApril 20, 2009 - Leonard Susskind explains how to calculate and define pressure, explores the formulas some of applications of Helm-Holtz free energy, and discusses the importance of the partition function.