Quantum Systems Obeying a Generalized Exclusion - Inclusion Principle

The aim of this work is to describe, at the quantum level, a many body system obeying to a generalized Exclusion-Inclusion Principle (EIP) originated by collective effects, the dynamics, in mean field approximation, being ruled by a nonlinear Schrödinger equation. The EIP takes its origin from a nonlinear kinetic equation where the nonlinearities describe interactions of different physical nature. The method starts from the study of the kinetic behavior of many particle system which results to be nonlinear because of the interaction among the particles and introduces an effective nonlinear potential U EIP which permits us to simulate the true interactions governing the dynamics of the system. The power of the method is tested in the case of spatially homogeneous classical N-particles system. Its kinetic in the momenta space is described by a Markoffian process. A judicious generalization of the particle current, permits us to obtain at the equilibrium, a statistical distribution interpolating in a continuous way the well known quantum statistics (Fermi-Dirac or Bose-Einstein). Systems with statistical behavior interpolating between the Bose-Einstein and the Fermi-Dirac were introduced fifty years ago. Up to now many have been the attempts to generalize quantum statistics. As we will discuss in the first chapter I many of these generalized statistics can be obtained by means of EIP by an appropriate generalization of the many particle current expression. The result of this approach is a nonlinear Schrödinger equation with a complex nonlinearity. Only the imaginary part of this nonlinearity is fixed by EIP, while the real one, is obtained by the requirement that the system is canonical. This permits us to include together within EIP other interactions of different physical origin. Systems obeying to EIP can be used to describe a wide range of physical situations. In condensed matter we can find many topics where EIP concepts can be applied like, for instance, in superconductivity to describe the formation of vortex-like exci-tations or in superfluidity to describe the formation of Bose-Einstein condensates. The plane of the work is the following. In the first chapter, after an introduction, we explain the origin of EIP. In the same chapter we summarize the fundamental mathematical tools used in the thesis: variational principle, Lagrangian and Hamil-tonian formalism for systems with infinity degrees of freedom, symmetries and conservation quantities. In chapter II, EIP is obtained from an appropriate deformation of the quantum current. From all the possible deformations of …