In recent years, the study of heat to work conversion has been re-invigorated by nanotechnology. Steady-state devices do this conversion without any macroscopic moving parts, through steady-state flows of microscopic particles such as electrons, photons, phonons, etc. This review aims to introduce some of the theories used to describe these steady-state flows in a variety of mesoscopic or nanoscale systems. These theories are introduced in the context of idealized machines which convert heat into electrical power (heat-engines) or convert electrical power into a heat flow (refrigerators). In this sense, the machines could be categorized as thermoelectrics, although this should be understood to include photovoltaics when the heat source is the sun. As quantum mechanics is important for most such machines, they fall into the field of quantum thermodynamics. In many cases, the machines we consider have few degrees of freedom, however the reservoirs of heat and work that they interact with are assumed to be macroscopic. This review discusses different theories which can take into account different aspects of mesoscopic and nanoscale physics, such as coherent quantum transport, magnetic-field induced effects (including topological ones such as the quantum Hall effect), and single electron charging effects. It discusses the efficiency of thermoelectric conversion, and the thermoelectric figure of merit. More specifically, the theories presented are (i) linear response theory with or without magnetic fields, (ii) Landauer scattering theory in the linear response regime and far from equilibrium, (iii) Green-Kubo formula for strongly interacting systems within the linear response regime, (iv) rate equation analysis for small quantum machines with or without interaction effects, (v) stochastic thermodynamic for fluctuating small systems. In all cases, we place particular emphasis on the fundamental questions about the bounds on ideal machines. Can magnetic-fields change the bounds on power or efficiency? What is the relationship between quantum theories of transport and the laws of thermodynamics? Does quantum mechanics place fundamental bounds on heat to work conversion which are absent in the thermodynamics of classical systems?

We show that for systems with broken time-reversal symmetry the maximum efficiency and the efficiency at maximum power are both determined by two parameters: a "figure of merit" and an asymmetry parameter. In contrast to the time-symmetric case, the figure of merit is bounded from above; nevertheless the Carnot efficiency can be reached at lower and lower values of the figure of merit and far from the so-called strong coupling condition as the asymmetry parameter increases. Moreover, the Curzon-Ahlborn limit for efficiency at maximum power can be overcome within linear response. Finally, always within linear response, it is allowed to have simultaneously Carnot efficiency and non-zero power. The understanding of the fundamental limits that thermodynamics imposes on the efficiency of thermal machines is a central issue in physics and is becoming more and more practically relevant in the future society. In particular due to the need of providing a sustainable supply of energy and to strong concerns about the environmental impact of the combustion of fossil fuels, there is an increasing pressure to find best thermoelectric materials [1][2][3][4].A cornerstone result goes back to Carnot [5]. In a cycle between two reservoirs at temperatures T 1 and T 2 (T 1 > T 2 ), the efficiency η, defined as the ratio of the performed work W over the heat Q 1 extracted from the high temperature reservoir, is bounded by the so-called Carnot efficiency η C :The Carnot efficiency is obtained for a quasi static transformation which requires infinite time and therefore the extracted power, in this limit, reduces to zero. For this reason the notion of efficiency at maximum power has been introduced. An upper bound for the efficiency at maximum power has been proposed long ago by several authors [6][7][8][9] and is commonly referred to as Curzon-Ahlborn upper bound:The range of validity of this bound has been widely discussed in several interesting papers [10][11][12][13][14][15]. For the thermoelectric power generation and refrigeration, within linear response and for systems with time-reversal symmetry, both the maximum efficiency and the efficiency at maximum power, are governed by a single parameter, the dimensionless figure of meritwhere σ is the electric conductivity, S is the thermoelectric power (Seebeck coefficient), κ is the thermal conductivity, and T is the temperature. The maximum efficiency is given bywhere η C is the Carnot efficiency; the efficiency η(ω max ) at maximum power ω max reads [10]The only restriction imposed by thermodynamics is ZT ≥ 0, so that η max ≤ η C and η(ω max ) ≤ η (l) CA , where η (l) CA = η C /2 is the Curzon-Alhborn efficiency in the linear response regime. The upper bounds η C and η (l) CA are reached when the figure of merit ZT → ∞. This limit corresponds to the so-called strong coupling condition, for which the Onsager matrix L becomes singular (that is, det L = 0) and therefore the ratio J q /J ρ , with J q heat currrent and J ρ electric (particle) current, is independent of the applied temp...

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Principles of Quantum Computation and Information About the CoverThis acrostic is the famous sator formula. It can be translated as:'Arepo the sower holds the wheels at work'The text may be read in four different ways:(i) horizontally, from left to right (downward) and from right to left (upward); (ii) vertically, downward (left to right) and upward (right to left).The resulting phrase is always the same.It has been suggested that it might be a form of secret message.This acrostic was unearthed during archeological excavation work at Pompeii, which was buried, as well known, by the eruption of Vesuvius in 79 A.D. The formula can be found throughout the Roman Empire, probably also spread by legionnaires. Moreover, it has been found in Mesopotamia, Egypt, Cappadocia, Britain and Hungary.The sator acrostic may have a mystical significance and might have been used as a means for persecuted Christians to recognize each other (it can be rearranged into the form of a cross, with the opening words of the Lord's prayer, A Paternoster O, both vertically and horizontally, intersecting at the letter N, the Latin letters A and O corresponding to the Greek letters alpha and omega, beginning and end of all things). Preface Purpose of the bookThis book is addressed to undergraduate and graduate students in physics, mathematics and computer science. It is written at a level comprehensible to readers with the background of a student near the end of an undergraduate course in one of the above three disciplines. Note that no prior knowledge of either quantum mechanics or classical computation is required to follow this book. Indeed, the first two chapters are a simple introduction to classical computation and quantum mechanics. Our aim is that these chapters should provide the necessary background for an understanding of the subsequent chapters. The book is divided into two volumes. In volume I, after providing the necessary background material in classical computation and quantum mechanics, we develop the basic principles and discuss the main results of quantum computation and information. Volume I would thus be suitable for a one-semester introductory course in quantum information and computation, for both undergraduate and graduate students. It is also our intention that volume I be useful as a general education for other readers who wish to learn the basic principles of quantum computation and information and who have the basic background in physics and mathematics acquired in undergraduate courses in physics, mathematics or computer science.Volume II deals with various important aspects, both theoretical and experimental, of quantum computation and information. The areas include quantum data compression, accessible information, entanglement concentration, limits to quantum computation due to decoherence, quantum error correction, and the first experimental implementations of quantum information protocols. This volume also includes a selection of special topics: Principles of Quantum Computation and Informationchaos and the ...