We study a system of N interacting fermions at positive temperature in a confining potential. In the regime where the intensity of the interaction scales as 1/N and with an effective semi-classical parameter = N −1/d where d is the space dimension, we prove the convergence to the corresponding Thomas-Fermi model at positive temperature.

We consider a system of N interacting fermions in R 3 confined by an external potential and in the presence of a homogeneous magnetic field. The intensity of the interaction has the mean-field scaling 1/N . With a semi-classical parameter ∼ N −1/3 , we prove convergence in the large N limit to the appropriate Magnetic Thomas-Fermi type model with various strength scalings of the magnetic field.

A stereographic projection is used to map the bosonic nonlinear sigma model with torsion from two-dimensional Euclidean space onto a sphere-S2 embedded in three dimensions. The one-loop β-function of the torsionless σ-model is determined using operator regularization to handle ultraviolet divergences. Only by excluding the lowest eigenstate of the rotation operator on the sphere can the usual β-function be recovered; inclusion of this eigenstate leads to severe infrared divergences. Both the ultraviolet and infrared divergences can be regulated by working in n, rather than two, dimensions, in which case the contribution of the lowest mode cancels exactly against the contribution of all other modes, resulting in a vanishing β-function.

We provide upper and lower bounds on the lowest free energy of a classical system at given one-particle density $$\rho (x)$$ ρ ( x ) . We study both the canonical and grand-canonical cases, assuming the particles interact with a pair potential which decays fast enough at infinity.