# 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.