Stochastic thermodynamics of quantum maps with and without equilibrium

We study stochastic thermodynamics for a quantum system of interest whose dynamics are described by a completely positive trace-preserving (CPTP) map as a result of its interaction with a thermal bath. We define CPTP maps with equilibrium as CPTP maps with an invariant state such that the entropy production due to the action of the map on the invariant state vanishes. Thermal maps are a subgroup of CPTP maps with equilibrium. In general, for CPTP maps, the thermodynamic quantities, such as the entropy production or work performed on the system, depend on the combined state of the system plus its environment. We show that these quantities can be written in terms of system properties for maps with equilibrium. The relations that we obtain are valid for arbitrary coupling strengths between the system and the thermal bath. The fluctuations of thermodynamic quantities are considered in the framework of a two-point measurement scheme. We derive the entropy production fluctuation theorem for general maps and a fluctuation relation for the stochastic work on a system that starts in the Gibbs state. Some simplifications for the probability distributions in the case of maps with equilibrium are presented. We illustrate our results by considering spin 1/2 systems under thermal maps, non-thermal maps with equilibrium, maps with non-equilibrium steady states and concatenations of them. Finally, we consider a particular limit in which the concatenation of maps generates a continuous time evolution in Lindblad form for the system of interest, and we show that the concept of maps with and without equilibrium translates into Lindblad equations with and without quantum detailed balance, respectively. The consequences for the thermodynamic quantities in this limit are discussed.Comment: 17 pages, 4 figures; new section added, typos correcte

The smallest absorption refrigerator: the thermodynamics of a system with quantum local detailed balance

We study the thermodynamics of a quantum system interacting with different baths in the repeated interaction framework. In an appropriate limit, the evolution takes the Lindblad form and the corresponding thermodynamic quantities are determined by the state of the full system plus baths. We identify conditions under which the thermodynamics of the open system can be described only by system properties and find a quantum local detailed balance condition with respect to an equilibrium state that may not be a Gibbs state. The three-qubit refrigerator introduced in [N. Linden, S. Popescu and P. Skrzypczyk, Phys. Rev. Lett.${\bf 105}$, 130401 (2010)] is an example of such a system. From a repeated interaction microscopic model we derive the Lindblad equation that describes its dynamics and discuss its thermodynamic properties for arbitrary values of the internal coupling between the qubits. We find that external power (proportional to the internal coupling strength) is required to bring the system to its steady state, but once there, it works autonomously as discussed in [N. Linden, S. Popescu and P. Skrzypczyk, Phys. Rev. Lett. ${\bf 105}$, 130401 (2010)].Comment: 11 pages, 2 figure

Nonequilibrium Quantum Phase Transitions in the XY model: comparison of unitary time evolution and reduced density matrix approaches

We study nonequilibrium quantum phase transitions in XY spin 1/2 chain using the $C^*$ algebra. We show that the well-known quantum phase transition at magnetic field $h = 1$ persists also in the nonequilibrium setting as long as one of the reservoirs is set to absolute zero temperature. In addition, we find nonequilibrium phase transitions associated to imaginary part of the correlation matrix for any two different temperatures of the reservoirs at $h = 1$ and $h = h_{\rm c} \equiv|1-\gamma^2|$, where $\gamma$ is the anisotropy and $h$ the magnetic field strength. In particular, two nonequilibrium quantum phase transitions coexist at $h=1$. In addition we also study the quantum mutual information in all regimes and find a logarithmic correction of the area law in the nonequilibrium steady state independent of the system parameters. We use these nonequilibrium phase transitions to test the utility of two models of reduced density operator, namely Lindblad mesoreservoir and modified Redfield equation. We show that the nonequilibrium quantum phase transition at $h = 1$ related to the divergence of magnetic susceptibility is recovered in the mesoreservoir approach, whereas it is not recovered using the Redfield master equation formalism. However none of the reduced density operator approaches could recover all the transitions observed by the $C^*$ algebra. We also study thermalization properties of the mesoreservoir approach.Comment: 25 pages, 10 figure

Fractality of the non-equilibrium stationary states of open volume-preserving systems: II. Galton boards

Galton boards are models of deterministic diffusion in a uniform external field, akin to driven periodic Lorentz gases, here considered in the absence of dissipation mechanism. Assuming a cylindrical geometry with axis along the direction of the external field, the two-dimensional board becomes a model for one-dimensional mass transport along the direction of the external field. This is a purely diffusive process which admits fractal non-equilibrium stationary states under flux boundary conditions. Analytical results are obtained for the statistics of multi-baker maps modeling such a non-uniform diffusion process. A correspondence is established between the local phase-space statistics and their macroscopic counter-parts. The fractality of the invariant state is shown to be responsible for the positiveness of the entropy production rate.Comment: Second of two papers, 17 double column pages, 10 figure

Multiple scattering of elastic waves by pinned dislocation segments in a continuum

The coherent propagation of elastic waves in a solid filled with a random distribution of pinned dislocation segments is studied to all orders in perturbation theory. It is shown that, within the independent scattering approximation, the perturbation series that generates the mass operator is a geometric series that can thus be formally summed. A divergent quantity is shown to be renormalizable to zero at low frequencies. At higher frequencies said quantity can be expressed in terms of a cut-off with dimensions of length, related to the dislocation length, and physical quantities can be computed in terms of two parameters, to be determined by experiment. The approach used in this problem is compared and contrasted with the scattering of de Broglie waves by delta-function potentials as described by the Schr\"odinger equation

Non-equilibrium Lorentz gas on a curved space

The periodic Lorentz gas with external field and iso-kinetic thermostat is equivalent, by conformal transformation, to a billiard with expanding phase-space and slightly distorted scatterers, for which the trajectories are straight lines. A further time rescaling allows to keep the speed constant in that new geometry. In the hyperbolic regime, the stationary state of this billiard is characterized by a phase-space contraction rate, equal to that of the iso-kinetic Lorentz gas. In contrast to the iso-kinetic Lorentz gas where phase-space contraction occurs in the bulk, the phase-space contraction rate here takes place at the periodic boundaries