The M-Wright function as a generalization of the Gaussian density for fractional diffusion processes

Abstract

This thesis is a study of equilibrium and dynamical properties of macroscopic quantum many-body problems. An important part of the manuscript concerns the study of heat and charge transport properties of fermions on the lattice. This refers to the derivation, from first principles of quantum mechanics and thermodynamics, of the Ohm’s law, first published in 1827 by G. S. Ohm, and of the heat equation, the well-known (classical) equation introduced by J. Fourier in 1807. A complete derivation of the heat equation from quantum mechanics is still not achieved, but we prove here some preliminary results on this non-trivial issue. By contrast, the study of charge transport properties of fermion systems on the lattice is largely developed in this thesis. In particular, we give a mathematical justification, for non-interacting free-fermions, of recent experiments (in 2006 and 2012) which have shown that the classical Ohm's law remains valid as atomic scales are reached even at very low temperature. Equilibrium state refers here to the notion of KMS state. We also set that the classical KMS condition can be derived from the (quantum) KMS condition for the so-called Bose-Hubbard model.