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 <N^{2}> - <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 (<?E^{2}> = <?E^{2}> + <?N^{2}>(?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.