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

At chapter chapproncorps, $$N$$ body problem was treated in the quantum mechanics framework. In this chapter, the same problem is tackled using statistical physics. The number $$N$$ 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  $$\frac{1}{N!}$$ (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 $$\mu$$. 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 . 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. In this chapter, we recall the most important properties of some of them.  