CONTENTS I. Introduction A. Length scales B. Weak localization, closed paths and the backscatter cone. C. Anderson localization D. Correlation of different diffusons II. Macroscopics: the diffusion approximation A. Transmission through a slab and Ohm's law B. Diffusion propagator for slabs III. Mesoscopics: the radiative transfer equation A. Specific intensity B. Slab geometry 1. Isotropic scattering 2. Anisotropic scattering and Rayleigh scattering 3. The transport mean free path and the absorption length C. Injection depth and the improved diffusion approximation IV. Microscopics: wave equations, t matrix, and cross sections A. Schrödinger and scalar wave equations B. The t matrix and resonant point scatterers C. The t matrix as a series of returns D. Point scatterer in three dimensions 1. Second-order Born approximation 2. Regularization of the return Green's function 3. Resonances 4. Comparison with Mie scattering for scalar waves E. Cross sections and the albedo V. Green's functions in disordered systems A. Diagrammatic expansion of the self-energy B. Self-consistency VI. Transport in infinite media: isotropic scattering A. Ladder approximation to the Bethe-Salpeter equation B. Diffusion from the stationary ladder equation C. Diffusion coefficient and the speed of transport VII. Transport in a semi-infinite medium A. Plane wave incident on a semi-infinite medium B. Air-glass-medium interface C. Solutions of the Schwarzschild-Milne equation VIII. Transport through a slab A. Diffuse transmission B. Electrical conductance and contact resistance IX. The enhanced backscatter cone A. Milne kernel at nonzero transverse momentum B. Shape of the backscatter cone C. Decay at large angles D. Behavior at small angles X. Exact solution of the Schwarzschild-Milne equation A. The homogeneous Milne equation B. The inhomogeneous Milne equation C. Enhanced backscatter cone D. Exact solution for internal reflections in diffusive media E. Exact solution for very anisotropic scattering XI. Large index mismatch A. Diffuse reflected intensity B. Limit intensity and injection depth C. Comparison with the improved diffusion approximation D. Backscatter cone XII. Semiballistic transport XIII. Imaging of objects immersed in opaque media A. Spheres B. Cylinders XIV. Interference of diffusons: Hikami vertices A. Calculation of the Hikami four-point vertex B. Six-point vertex: H 6 50 C. Beyond the second-order Born approximation 51 D. Corrections to the conductivity 51 XV. Short range, long range, and conductance correlations: C 1 , C 2 , and C 3 51 A. Angular resolved transmission: the speckle pattern 52 B. Total transmission correlation: C 2 52 C. Conductance correlation: C 3 53 XVI. Calculation of correlation functions 53 A. Summary of diffuse intensities 54 B. Calculation of the C 1 correlation 55 C. Calculation of the C 2 correlation 55 1. Influence of incoming beam profile 56 2. Reflection correlations 57 D. Conductance Fluctuations: C 3 57 E. Calculation of the C 3 correlation function 58 XVII. Third cumulant of the total transmis...

Thermodynamics teaches that if a system initially off-equilibrium is coupled to work sources, the maximum work that it may yield is governed by its energy and entropy. For finite systems this bound is usually not reachable. The maximum extractable work compatible with quantum mechanics ("ergotropy") is derived and expressed in terms of the density matrix and the Hamiltonian. It is related to the property of majorization: more major states can provide more work. Scenarios of work extraction that contrast the thermodynamic intuition are discussed, e.g. a state with larger entropy than another may produce more work, while correlations may increase or reduce the ergotropy.

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The quantum measurement problem, to wit, understanding why a unique outcome is obtained in each individual experiment, is currently tackled by solving models. After an introduction we review the many dynamical models proposed over the years for elucidating quantum measurements. The approaches range from standard quantum theory, relying for instance on quantum statistical mechanics or on decoherence, to quantum-classical methods, to consistent histories and to modifications of the theory. Next, a flexible and rather realistic quantum model is introduced, describing the measurement of the z-component of a spin through interaction with a magnetic memory simulated by a Curie-Weiss magnet, including N 1 spins weakly coupled to a phonon bath. Initially prepared in a metastable paramagnetic state, it may transit to its up or down ferromagnetic state, triggered by its coupling with the tested spin, so that its magnetization acts as a pointer. A detailed solution of the dynamical equations is worked out, exhibiting several time scales. Conditions on the parameters of the model are found, which ensure that the process satisfies all the features of ideal measurements. Various imperfections of the measurement are discussed, as well as attempts of incompatible measurements. The first steps consist in the solution of the Hamiltonian dynamics for the spin-apparatus density matrixD(t). Its off-diagonal blocks in a basis selected by the spin-pointer coupling, rapidly decay owing to the many degrees of freedom of the pointer. Recurrences are ruled out either by some randomness of that coupling, or by the interaction with the bath. On a longer time scale, the trend towards equilibrium of the magnet produces a final stateD(t f ) that involves correlations between the system and the indications of the pointer, thus ensuring registration. AlthoughD(t f ) has the form expected for ideal measurements, it only describes a large set of runs. Individual runs are approached by analyzing the final states associated with all possible subensembles of runs, within a specified version of the statistical interpretation. There the difficulty lies in a quantum ambiguity: There exist many incompatible decompositions of the density matrixD(t f ) into a sum of sub-matrices, so that one cannot infer from its sole determination the states that would describe small subsets of runs. This difficulty is overcome by dynamics due to suitable interactions within the apparatus, which produce a special combination of relaxation and decoherence associated with the broken invariance of the pointer. Any subset of runs thus reaches over a brief delay a stable state which satisfies the same hierarchic property as in classical probability theory; the reduction of the state for each individual run follows. Standard quantum statistical mechanics alone appears sufficient to explain the occurrence of a unique answer in each run and the emergence of classicality in a measurement process. Finally, pedagogical exercises are proposed and lessons for future works on m...

In our recent letter [1] we discussed that thermodynamics is violated in quantum Brownian motion beyond the weak coupling limit. In his comment, Tasaki [2] derives an inequality for the relative entropy and claims, without making any dynamical assumption, that the Clausius inequality is valid, thus contradicting our statements [1]. Here we point out that the claim is unfunded, since the author did not properly identify the concept of heat. Tasaki also applies the inequality to Thomson's formulation of the second law. This application is invalid as well, since the author did not correctly identify the concept of work. Therefore, Tasaki's inequality is perfectly compatible with our findings.

The minimal work principle states that work done on a thermally isolated equilibrium system is minimal for adiabatically slow (reversible) realization of a given process. This principle, one of the formulations of the second law, is studied here for finite (possibly large) quantum systems interacting with macroscopic sources of work. It is shown to be valid as long as the adiabatic energy levels do not cross. If level crossing does occur, counter examples are discussed, showing that the minimal work principle can be violated and that optimal processes are neither adiabatically slow nor reversible. The results are corroborated by an exactly solvable model.

The Schwarzschild-de Sitter and Reissner-Nordström-de Sitter black hole metrics appear as exact solutions in the recently formulated massive gravity of de Rham, Gabadadze and Tolley (dRGT), where the mass term sets the curvature scale. They occur within a two-parameter family of dGRT mass terms. They show no trace of a cloud of scalar graviton modes, and in the limit of vanishing graviton mass they go smoothly to the Schwarzschild and Reissner-Nordström metrics.

A known aspect of the Clausius inequality is that an equilibrium system subjected to a squeezing dS of its entropy must release at least an amount |dQ| = T |dS| of heat. This serves as a basis for the Landauer principle, which puts a lower bound T ln 2 for the heat generated by erasure of one bit of information. Here we show that in the world of quantum entanglement this law is broken.A quantum Brownian particle interacting with its thermal bath can either generate less heat or even adsorb heat during an analogous squeezing process, due to entanglement with the bath. The effect exists even for weak but fixed coupling with the bath, provided that temperature is low enough. This invalidates the Landauer bound in the quantum regime, and suggests that quantum carriers of information can be much more efficient than assumed so far.