### Statistical Mechanics Lecture 19 of 29

July 4, 2012 by Matteo Marsili

M. Marsili, 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=kBlog[?(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=h3N this means that the volume occupied for one particle in phase space is h.<span class="l"> As a second example the harmonic oscillator</span> 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<span class="l">. Thus in general for a system of N particles we have </span>?(N,V,E)=?(E)/h3N.

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(Xn) of finding a particular microstate Xn with energy En is proportional to exp{-?En}.

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 4 | Modern Physics: Statistical Mechanics April 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.

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