Quantum Brownian motion and its conflict with the second law

The Brownian motion of a harmonically bound quantum particle and coupled to a harmonic quantum bath is exactly solvable. At low enough temperatures the stationary state is non-Gibbsian due to an entanglement with the bath. This happens when a cloud of bath modes around the particle is formed. Equilibrium thermodynamics for particle plus bath together, does not imply standard thermodynamics for the particle itself at low T . Various formulations of the second law are then invalid. First, the Clausius inequality can be violated. Second, when the width of the confining potential is suddenly changed, there occurs a relaxation to equilibrium during which the rate of entropy production is partly negative. Third, for non-adiabatic changes of system parameters the rate of energy dissipation can be negative, and, out of equilibrium, cyclic processes are possible which extract work from the bath. Conditions are put forward under which perpetuum mobile of the second kind, having several work extraction cycles, enter the realm of condensed matter physics. Introduction. There are not two fundamental theories of nature, quantum mechanics and thermodynamics. There is only one: quantum mechanics, while thermodynamics must emerge from it. The universal character of equilibrium thermodynamics led to the general expectation that in one way or another, thermodynamics will apply to the full quantum domain [1]. Few people have taken the painful road to check this emergence, yet this is what we have set out to do. Here we discuss the results for quantum Brownian motion that have been presented [2, 3] and were discussed in the scientific literature [4]. Brownian motion has numerous applications in condensed matter physics [5, 6, 7, 8], atomic physics [5], quantum optics and chemistry [9]. Some realizations involve weak coupling with the thermal bath [9]. However, there are well-known experimental situations, which are essentially far from the weak-coupling regime. Here standard thermodynamics may not apply. The main example of this is the case of weak links between superconductive regions, the so-called Josephson junctions, in their overdamped regime [10, 11], where the relevant ranges of parameters were achieved already twenty years ago. Even in quantum optics, which has often been satisfactorily described by weakcoupling theories [9], there are recent experiments showing the necessity for moderate and strong coupling approaches [12]. The Hamiltonian. We consider an ‘ideal gas’ of non-interacting harmonic oscillators coupled to a bath. For the total Hamiltonian Htot = H +HB +HI we thus assume [6]