The origins of quantum theory are intimately connected to thermodynamics. In addressing the problem of blackbody radiation, Max Planck introduced the hypothesis of energy quantization, thereby establishing the foundations of quantum physics. This review revisits the deep conceptual links between quantum theory and the thermodynamics of small systems, where classical thermodynamic assumptions—particularly extensivity and additivity—break down. Particular emphasis is placed on the role of the subdivision potential in accounting for non-extensive contributions and on Tsallis entropy as a generalized framework capable of capturing the intrinsic non-extensivity of finite and strongly correlated systems, thereby extending thermodynamic reasoning beyond the macroscopic limit.
The origins of quantum theory are intimately connected to thermodynamics. In addressing the problem of blackbody radiation, Max Planck introduced the hypothesis of energy quantization, thereby establishing the foundations of quantum physics. This review revisits the deep conceptual links between quantum theory and the thermodynamics of small systems, where classical thermodynamic assumptions—particularly extensivity and additivity—break down. Particular emphasis is placed on the role of the subdivision potential in accounting for non-extensive contributions and on Tsallis entropy as a generalized framework capable of capturing the intrinsic non-extensivity of finite and strongly correlated systems, thereby extending thermodynamic reasoning beyond the macroscopic limit.
Thermodynamics is commonly presented as a theory of macroscopic systems in stable equilibrium, built upon assumptions of extensivity and scaling with system size. In this paper, we present a universal formulation of the elementary foundations of thermodynamics, in which entropy and energy are defined and employed beyond equilibrium and without assuming extensivity. The formulation applies to all systems -- large and small, with many or few particles -- and to all states, whether equilibrium or nonequilibrium, by relying on carefully stated operational definitions and existence principles rather than macroscopic idealizations. Key thermodynamic concepts, including adiabatic availability and available energy, are developed and illustrated using the energy-entropy diagram representation of nonequilibrium states, which provides geometric insight into irreversibility and the limits of work extraction for systems of any size. A substantial part of the paper is devoted to the analysis of entropy transfer in non-work interactions, leading to precise definitions of heat interactions and heat-and-diffusion interactions of central importance in mesoscopic continuum theories of nonequilibrium behavior in simple and complex solids and fluids. As a direct consequence of this analysis, Clausius inequalities and the Clausius statement of the second law are derived in forms explicitly extended to nonequilibrium processes. The resulting framework presents thermodynamics as a universal theory whose concepts apply uniformly to all systems, large and small, and provides a coherent foundation for both teaching and modern applications.
The origin of the relativity theories was closely bound up with the development of electromagnetic concepts, a development that approached a coherent fieldtheoretical formulation, according to which all actions may vary in a continuous manner. In contrast, quantum theory arose out of the development of atomic concepts, a development that was characterized by the acknowledgment of a fundamental limitation to classical physical ideas when applied to atomic phenomena. This restriction was expressed in the so-called quantum postulate, which attributed to any atomic process an essential discontinuity that was symbolized by Planck's quantum of action. Quantum field theory (QFT) is a later phase of the conceptual developments of quantum theory, and has the old quantum theory and non-relativistic quantum mechanics, essentially the preliminary analyses of the interactions between atoms and radiation, as its predecessors. This chapter will review some features of quantum physics that are relevant to the rise of QFT. The quantization of motion In solving the problem of the equilibrium between matter and radiation, Max Planck (1900) showed that the laws of heat radiation demanded an element of discontinuity in the description of atomic processes. In the statistical behavior of atoms represented, in Planck's description, by linear resonators in their interactions with radiation, only states of vibration should be taken into account whose energy was an integral multiple of a quantum, hv , where h is Planck's constant and v is the frequency of the resonator. Planck himself believed that the discontinuity of energy was only a property of atoms, and was not ready to apply the idea of energy quantization to radiation itself.
In recent years we have witnessed a concentrated effort to make sense of thermodynamics for small-scale systems. One of the main difficulties is to capture a suitable notion of work that models realistically the purpose of quantum machines, in an analogous way to the role played, for macroscopic machines, by the energy stored in the idealisation of a lifted weight. Despite several attempts to resolve this issue by putting forward specific models, these are far from realistically capturing the transitions that a quantum machine is expected to perform. In this work, we adopt a novel strategy by considering arbitrary kinds of systems that one can attach to a quantum thermal machine and defining work quantifiers . These are functions that measure the value of a transition and generalise the concept of work beyond those models familiar from phenomenological thermodynamics. We do so by imposing simple operational axioms that any reasonable work quantifier must fulfil and by deriving from them stringent mathematical condition with a clear physical interpretation. Our approach allows us to derive much of the structure of the theory of thermodynamics without taking the definition of work as a primitive. We can derive, for any work quantifier , a quantitative second law in the sense of bounding the work that can be performed using some non-equilibrium resource by the work that is needed to create it. We also discuss in detail the role of reversibility and correlations in connection with the second law. Furthermore, we recover the usual identification of work with energy in degrees of freedom with vanishing entropy as a particular case of our formalism. Our mathematical results can be formulated abstractly and are general enough to carry over to other resource theories than quantum thermodynamics.
This paper presents a comprehensive exposition of quantum thermodynamics, aiming to bridge the foundational concepts of classical thermodynamics and quantum mechanics.Beginning with a rigorous overview of classical thermodynamic laws and statistical ensembles, the work transitions into the quantum domain by introducing quantum states, density operators, and von Neumann entropy.Core thermodynamic quantities such as work, heat, energy, and entropy are redefined in the quantum regime, with emphasis on operational frameworks like the twopoint measurement scheme.The study explores the role of quantum resource theories in formulating generalizations of the second law and delves into the intricacies of nonequilibrium thermodynamics through fluctuation theorems, such as the Jarzynski equality and Crooks theorem.Special attention is given to the influence of coherence, entanglement, and measurement backaction, which contribute uniquely to quantum thermodynamic behavior.The treatise concludes with a discussion on quantum heat engines, thermodynamic consistency of quantum master equations, and information-theoretic approaches, establishing quantum thermodynamics as a vital framework for understanding energy and information flow at the quantum scale.
Planck found, when attempting to describe the way in which hot bodies glow, that energy at microscopic scales often comes in discrete chunks. Thus began the long and intimate relationship between the field of thermodynamics, which explores our ability to manipulate heat and other energy transfers between macroscopic systems, and quantum mechanics, which explains the dynamics of individual microscopic systems. Even as both our technology and our theoretical investigations have extended to ever-smaller devices, our understanding of quantum effects on thermodynamics has remained almost exclusively limited to the quantized nature of energy. There is much more, however, to quantum theory than energy quantization. In this thesis we focus on the property of quantum coherence, the ability of quantum systems to emulate Schrodinger’s cat and somehow be neither dead nor alive, but something completely different altogether. We provide a simple and elegant formulation for the processing of coherence in thermodynamics, relying on the fact that thermodynamical processes possess an underlying time-translation symmetry. This fact allows us to quantify the ways in which coherence can play an active role, facilitating otherwise impossible thermodynamic transformations. We argue that coherence should be thought of as a distinctly quantum-mechanical thermodynamic resource. By considering an isolated quantum system connected, for some period of time, to a thermal bath, we give fundamental limitations on how coherence can be irreversibly manipulated. These limitations are related to those dictated on energy transfer by the second law of thermodynamics. Others topics that are investigated here is what is the role of correlations at the smallest scales and the conversion of quantum coherence into work. It has long been appreciated that thermodynamics is subtly interlinked with the notion of information. This work provides evidence that, to apply the laws of thermodynamics to the smallest systems around us, we must develop deeper insights into the nature of quantum information.
Entropy Beyond the Second Law presents a coherent formulation of all aspects of thermodynamics and statistical mechanics with entropy as the unifying theme. This includes formulating equilibrium theory and explaining the role of the Second Law in establishing the equilibrium state. It also establishes the foundations of non-equilibrium theory by generalizing the concept of entropy in an appropriate way. Finally, it addresses the relationship between classical and quantum statistical mechanics by invoking the collapse of the wave function into entropy states. A secondary purpose is to increase the understanding of entropy and the Second Law, and to correct misapprehensions and errors that have arisen over the years.