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

Exercice:

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

${\displaystyle \mu _{i}=-g\mu _{B}J_{i}}$

where ${\displaystyle \mu _{B}}$ is the Bohr magneton and ${\displaystyle g}$ is the Land\'e factor. ${\displaystyle J_{i}}$ can have only semi integer values.

Assume that the hamiltonian describing the system of ${\displaystyle N}$ atoms is:

${\displaystyle H=\sum -\mu _{i}B_{0}}$

where ${\displaystyle B_{0}}$ 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 ${\displaystyle Z}$ ? 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 ${\displaystyle {\bar {N}}_{\lambda }}$ of a state ${\displaystyle \lambda }$. The law you'll obtained is called Fermi distribution.

Exercice:

Consider a gas of independent bosons. Calculate the mean occupation number ${\displaystyle {\bar {N}}_{\lambda }}$ of a state ${\displaystyle \lambda }$. 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 ${\displaystyle \rho (\epsilon )}$, ${\displaystyle \epsilon }$ being the nergy of a state. Give the expression of ${\displaystyle \rho (epsilon)}$. Find the expression binding electron number ${\displaystyle {\bar {N}}}$ to chemical potential. Give the expression of the potential when temperature is zero.