Microscopic quantum mechanical foundation of Fourier's law

Besides the growing interest in old concepts such as temperature and entropy at the nanoscale, theories of relaxation and transport have recently regained a lot of attention. With the electronic circuits and computer chips getting smaller and smaller, a fresh look on the equilibrium and nonequilibrium thermodynamics at small length scales far below the thermodynamic limit, i.e. on the theoretical understanding of original macroscopic processes, e.g. transport of energy, heat, charge, mass, magnetization, etc., should be appropriate. Only from the foundations of a theory its limits of applicability may be inferred. This review tries to give an overview on the background and recent developments in the field of nonequilibrium quantum thermodynamics, focusing on the transport of heat in small quantum systems.