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

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.