### Statistical Mechanics Lecture 26 of 29

July 23, 2012 by Matteo Marsili

M. Marsili, ICTP

This lesson was dedicated to the study of the fluctuations of the number of particles in the Grand canonical ensemble and it was shown that <N2> - <N>2 = https://md.ictp.it/admin/media/https://md.ictp.it/admin/media/http://upload.wikimedia.org/wikipedia/en/math/7/7/6/77698ae92ac0435f8da1e266eeb528e3.png" alt="z\,"/> ?N/?https://md.ictp.it/admin/media/https://md.ictp.it/admin/media/http://upload.wikimedia.org/wikipedia/en/math/7/7/6/77698ae92ac0435f8da1e266eeb528e3.png" alt="z\," width="10" height="9"/> and that the relative fluctuation will be small of order 1/N, meaning that the description given in the Grand canonical ensemble is equivalent to the description of the canonical ensemble. Using thermodynamics it was shown that the maximum of relative fluctuations takes place when a phase transition occurs thus the canonical ensemble can not be used to describe phase transitions. The fluctuation of the energy are equal to the fluctuations of the energy at constant N plus the variation on the energy due to the fluctuation of N (<?E2> = <?E2> + <?N2>(?E/?N)2).

In the last part of the lesson it was started a new topic Quantum statistics with the introduction of new concepts like the ensemble in the quantum mechanical case and the Density matrix. The properties of the Density matrix where studied like the normalization and the evolution.

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