### Statistical Mechanics Lecture 29 of 29

October 31, 2017 by Matteo Marsili

M. Marsili, ICTP

This lesson starts by the study of the internal degrees of freedom, examples are the electronic, the nuclear, the vibrational and the rotational etc.. The Hamiltonian will be then the sum of the Hamiltonians corresponding to each degree of freedom and the same for the specific heat. Then the lesson focused in the study of Ideal quantum gases in particular an ideal gas of bosons <span id="Ideal_Bose_and_Fermi_gases" class="mw-headline">Ideal Bose </span><span id="Ideal_Bose_and_Fermi_gases" class="mw-headline">gas and</span><span id="Ideal_Bose_and_Fermi_gases" class="mw-headline"> </span><span id="Ideal_Bose_and_Fermi_gases" class="mw-headline">ideal</span><span id="Ideal_Bose_and_Fermi_gases" class="mw-headline"> gas of fermions Fermi gases. </span>

The thermodynamics of the ideal Bose gas was studied with the help of the grand partition function
and the equation of state as well as the number of particles where calculated as a function of the thermal de Broglie wavelength and the Fugacity. It was shown that in order to have a positive number of particles in the lowest energy level the chemical potential (?) have to be smaller than the lowest energy level in this case ? is negative (?<0). Then both cases were considered the high and low temperature regime. It was demonstrated that in the high temperature regime the specific heat for the ideal Bose gas is larger than the Ideal gas converging to it from above for a very large temperature, on the other hand we know that at T=0 it has to go to zero thus a maximum in the specific heat was predicted. For very low temperature it was obtained that most of particles will occupy the ground state, this phenomenon is known as the Bose–Einstein condensate.

The critical temperature at which the Bose–Einstein condensate take place was calculated and the physical properties of the condensate studied, for example it was shown that this gas has infinite compressibility, the specific heat will goes to zero as T3/2 as the particles that are in the ground state does not contribute to the specific heat because they cannot transfer energy, they are in a zero energy state, and this is why the specific heat goes to zero.

As an example of this ideal Bose gas it was considered a system composed by photons. This system can be consider as a system of particles called photons occupying the energy levels of a harmonic oscillator and ?=0, the expressions for the calculation of the average number of particles and average energy for each possible frequency ? were obtained. The internal energy U is found to be U~T4, the equation of state is given by PV = U/3 and the entropy is S=4U/3T ~ T3 .

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