Equilibrium and transport properties of quantum many-body systems

Résumé

Equilibrium and Transport Properties of Quantum Many-BodySystemsThis thesis is a study of equilibrium and dynamical properties ofmacroscopic quantum many-body problems. An important part of themanuscript concerns the study of heat and charge transportproperties of fermions on the lattice. This refers to thederivation, from first principles of quantum mechanics andthermodynamics, of the Ohm¿s law, first published in 1827 by G. S.Ohm, and of the heat equation, the well-known (classical) equationintroduced by J. Fourier in 1807. A complete derivation of the heatequation from quantum mechanics is still not achieved, but we provehere some preliminary results on this non-trivial issue. Bycontrast, the study of charge transport properties of fermionsystems on the lattice is largely developed in this thesis. Inparticular, we give a mathematical justification, for noninteractingfree-fermions, of recent experiments (in 2006 and 2012)which have shown that the classical Ohm's law remains valid asatomic scales are reached even at very low temperature. Equilibriumstate refers here to the notion of KMS state. We also set that theclassical KMS condition can be derived from the (quantum) KMScondition for the so-called Bose-Hubbard model.