Introduction to Mathematical Physics/N body problems and statistical equilibrium/Exercises

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Exercice:

Paramagnetism. Consider a system constituted by N atoms located at nodes of a lattice. Let J_i be the total kinetic moment of atom number i in its fundamental state. It is known that to such a kinetic moment is associated a magnetic moment given by:


\mu_i=-g\mu_BJ_i

where \mu_B is the Bohr magneton and g is the Land\'e factor. J_i can have only semi integer values.

Assume that the hamiltonian describing the system of N atoms is:


H=\sum -\mu_i B_0

where B_0 is the external magnetic field. What sort of particles are the atoms in this systme, discernables or undiscernables? Find the partition function of the system.

Exercice:

Study the Ising model at two dimension. Is it possible to envisage a direct method to calculate Z ? Write a programm allowing to visualize the evolution of the spins with time, temperature being a parameter.

Exercice:

Consider a gas of independent fermions. Calculate the mean occupation number \bar N_\lambda of a state \lambda. The law you'll obtained is called Fermi distribution.

Exercice:

Consider a gas of independent bosons. Calculate the mean occupation number \bar N_\lambda of a state \lambda. The law you'll obtained is called Bose distribution.

Exercice:

Consider a semi--conductor metal. Free electrons of the metal are modelized by a gas of independent fermions. The states are assumed to be described by a sate density \rho(\epsilon), \epsilon being the nergy of a state. Give the expression of \rho(epsilon). Find the expression binding electron number \bar N to chemical potential. Give the expression of the potential when temperature is zero.