Microscopic Quantum Mechanical Foundation of Fourier's Law

Abstract

Besides the growing interest in old concepts like 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 should be appropriate 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. Only from the foundations of a theory its limits of applicability may be inferred. This review tries to give an overview of the background and recent developments in the field of nonequilibrium quantum thermodynamics focusing on the transport of heat in small quantum systems.

Entropy production and thermal conductivity of a dilute gas

“…one will find that the new result (22) is the same as the kinetic theory result (2). In fact, ε is a little bit less than cT , as discussed before Eq.…”

Energy flux operator, current conservation and the formal Fourier's law

“…[29,36,37,38], In order to increase generality, Ref. [36] further suggests a 'symmetrized local flux' that has the same form as j (2) s→s+1 .…”

“…We discuss next some special cases: (i) A = −B, corresponding to the antiferromagnetic phase. Here j [29,38]. (iii) The XY model, A = B and C = 0.…”

“…This is the discrete Fourier's law [29,38]. We can identify the microscopic-local thermal conductivity as κ s = a As an example we consider a three-spin system.…”

Geometry of system-bath coupling and gauge fields in bosonic ladders: Manipulating currents and driving phase transitions

“…A typical example is that of a chain of particles connected, at its extremities, to two different baths, thus imposing a flow within the system. This scenario has been studied both theoretically and experimentally for various kinds of physical realizations including spin systems, photonic lattices, ions, metals, semiconductors and Josephson junctions arrays [7,8]. Experiments with ion traps promise to be an ideal set-up to simulate boundary driven systems [9][10][11].…”