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

**Exercice:**

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

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

Assume that the hamiltonian describing the system of atoms is:

where 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 ? 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 of a state . The law you'll obtained is called Fermi distribution.

**Exercice:**

Consider a gas of independent bosons. Calculate the mean occupation number of a state . 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 ,
being the nergy of a state. Give the expression of .
Find the expression binding electron number to chemical
potential. Give the expression of the potential when temperature is zero.