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

At chapter chapproncorps, ${\displaystyle N}$ body problem was treated in the quantum mechanics framework. In this chapter, the same problem is tackled using statistical physics. The number ${\displaystyle N}$ of body in interaction is assumed here to be large, of the order of Avogadro number. Such types of N body problems can be classified as follows:

• Particles are undiscernable. System is then typically a gas. In a classical approach, partition function has to be described using a corrective factor ${\displaystyle {\frac {1}{N!}}}$ (see section secdistclassi). Partition function can be factorized in two cases: particles are independent (one speaks about perfect gases)\index{perfect gas} Interactions are taken into account, but in the frame of a mean field approximation\index{mean field}. This allows to considerer particles as if they were independent (see van der Waals model at section secvanderwaals) In a quantum mechanics approach, Pauli principle can be included in the most natural way. The suitable description is the grand-canonical description: number of particles is supposed to fluctuate around a mean value. The Lagrange multiplier associated to the particles number variable is the chemical potential ${\displaystyle \mu }$. Several physical systems can be described by quantum perfect gases (that is a gas where interactions between particles are neglected): a fermions gas can modelize a semi--conductor. A boson gas can modelize helium and described its properties at low temperature. If bosons are photons (their chemical potential is zero), the black body radiation can be described.
• Particles are discernable. This is typically the case of particles on a lattice. Such systems are used to describe for instance magnetic properties of solids. Taking into account the interactions between particles, phase transitions like paramagnetic--ferromagnetic transition can be described [1]. Adsorption phenomenom can be modelized by a set of independent particles in equilibrium with a particles reservoir (grand-canonical description). Those models are described in detail in [ph:physt:Diu89]. In this chapter, we recall the most important properties of some of them.
1. A mean field approximation allows to factorize the partition function. Paramagnetic--ferromagnetic transition is a second order transition: the two phases can coexist, on the contrary to liquid vapour transition that is called first order transition.