In Chapter 3 the introduction of entropy as a state function was facilitated by the realization that there exist possible and impossible processes, and by an examination of the heat and work effects occurring during these processes. From the formal statement of the Second Law of Thermodynamics, as developed from Classical Thermodynamics arguments, it is difficult to assign a physical significance or a physical quality to entropy. In this respect entropy differs from internal energy in spite of the fact that, within the scope of Classical Thermodynamics, both properties are simply mathematical functions of the state of a system. The ready acceptance of the First Law of Thermodynamics, after its enunciation, was due to the easily understood physical significance of internal energy, whereas the lack of corresponding understanding of entropy caused the acceptance of the Second Law of Thermodynamics to be slow. From the classical viewpoint the Second Law is only valid as a direct result of the fact that, to date, human ingenuity has failed to invent a perpetual motion machine. Thus, within the scope of Classical Thermodynamics, the Second Law is a “law” only because it has not yet been disproved. The physical interpretation of entropy had to await the development of quantum theory and statistical mechanics.
In Chapter 3 the introduction of entropy as a state function was facilitated by the realization that there exist possible and impossible processes, and by an examination of the heat and work effects occurring during these processes. From the formal statement of the Second Law of Thermodynamics, as developed from Classical Thermodynamics arguments, it is difficult to assign a physical significance or a physical quality to entropy. In this respect entropy differs from internal energy in spite of the fact that, within the scope of Classical Thermodynamics, both properties are simply mathematical functions of the state of a system. The ready acceptance of the First Law of Thermodynamics, after its enunciation, was due to the easily understood physical significance of internal energy, whereas the lack of corresponding understanding of entropy caused the acceptance of the Second Law of Thermodynamics to be slow. From the classical viewpoint the Second Law is only valid as a direct result of the fact that, to date, human ingenuity has failed to invent a perpetual motion machine. Thus, within the scope of Classical Thermodynamics, the Second Law is a “law” only because it has not yet been disproved. The physical interpretation of entropy had to await the development of quantum theory and statistical mechanics.
What is the physical significance of entropy? What is the physical origin of irreversibility? Do entropy and irreversibility exist only for complex and macroscopic systems? The bulk of the physics community accepts and teaches that all these fundamental questions are rationalized within statistical mechanics. Indeed, for everyday laboratory physics, the mathematical formalism of statistical mechanics (canonical and grandcanonica1, Boltzmann, Bose-Einstein and Fermi-Dirac distributions) allows a successful description of the thermodynamic equilibrium properties of matter, including entropy values. But an ever growing handful of physicists (Schrodinger among the first) have realized that, even in its explanation of the meaning of entropy, statistical mechanics is impaired by ambiguities and logical inconsistencies. They have started to search for a better theory to eliminate these stumbling blocks while maintaining the mathematical formalism that has been so successful in so many applications. This handful of upstreamers must not be confused with the many schools of physicists that have thrived on the more renowned incompleteness of statistical mechanics, namely, the lack of a quantitative (and the weakness of the qualitative) explanation of the origin of irreversibility. In these studies the thrust is provided by the discovery that the macroscopic dynamics of certain complex systems may be modeled using a few-degrees-of-freedom nonlinear Hamiltonian with singularities that give rise to bifurcations and chaotic behavior. These results have generated successful ways to describe irreversible behavior, but their link to the origin of irreversibility is still only heuristic (what is the connection between the nonlinear model Hamiltonian and the true full Hamiitonian?) and does not provide yet a rigorous resolution of the century-old paradox of the conflict between the irreversibiiity of macroscopic behavior, and the reversibility of the laws of mechanics. To resolve both the problem of the meaning of entropy and that of the origin of irreversibility we have built entropy and irreversibility into the laws of mechanics. The result is a theory that we call quantum thermodynamics that has all the necessary properties to combine mechanics and thermodynamics uniting all the successful results of both theories, eliminating the logical inconsistencies of statistical mechanics and the paradox on irreversibiiity, and providing an entirely new perspective on the microscopic origin of irreversibility, nonlinearity and therefore chaotic behavior. The mathematical formalism of quantum thermodynamics differs from that of statistical mechanics mainly in the equation of motion which is nonlinear but has solutions identical to those of the Schrodinger equation for all the states for which statistical mechanics reduces to quantum mechanics. The physical meaning of the formalism of quantum thermodynamics differs more drastically from that of statistical mechanics. The significance of the state operator of quantum thermodynamics is entirely different from that of the density operator of statistical mechanics, even though the two are mathematically equivalent. Indeed they obey different equations of motion. In particular, quantum thermodynamics is concerned only with those systems for which quantum mechanics would describe the states with vectors in Hilbert space or, equivalently, projection operators. Using a well known jargon, we can say that quantum thermodynamics like quantum mechanics is concerned only with pure quantum states. However, it postulates that the set of pure quantum states of a system is much broader than contemplated by quantum mechanics. Pure quantum states must be described by operators defined by all the features of projection operators except the condition of idempotence. As a result, an operator that within statistical mechanics would describe a mixed quantum state (that is, the average state of a statistical mixture of identical systems in different pure quantum states) in quantum thermodynamics describes a pure quantum state, a state that neither quantum mechanics non statistical mechanics would contemplate. Conceptually, the increased richness of pure quantum states is a new revolutionary postulate of quantum physics. But from the point of view of the statistical mechanics practitioners the new theory is not as traumatic as it seems. Whenever one uses a nonidempotent density operator to describe a thermodynamic equilibrium state one simply has to reinterpret it as one of the new pure quantum states. One even saves the usual ad hoc arguments on thermal baths and reservoirs that are usually required in statistical mechanics to justify the use of a nonidempotent density operator to describe the state of a system. In this paper we discuss the background and formalism of quantum thermodynamics including its nonlinear equation of motion and the main general results. Our objective is to show in a not-too-technical manner that this theory provides indeed a complete and coherent resolution of the century-old dilemma on the meaning of entropy and the origin of irreversibility. As a byproduct, we discuss a long set of criteria that a theory should meet in order to afford the same claim.
ABSTRACT. What is the physical significance of entropy? What is the physical origin of irreversibility? Do entropy and irreversibility exist only for complex and macroscopic systems? Most physicists still accept and teach that the rationalization of these fundamental ques-tions is given by Statistical Mechanics. Indeed, for everyday laboratory physics, the math-ematical formalism of Statistical Mechanics (canonical and grand-canonical, Boltzmann, Bose-Einstein and Fermi-Dirac distributions) allows a successful description of the ther-modynamic equilibrium properties of matter, including entropy values. However, as al-ready recognized by Schrödinger in 1936, Statistical Mechanics is impaired by conceptual ambiguities and logical inconsistencies, both in its explanation of the meaning of entropy and in its implications on the concept of state of a system. An alternative theory has been developed by Gyftopoulos, Hatsopoulos and the present author to eliminate these stumbling conceptual blocks while maintaining the mathematical formalism so successful in applications. To resolve both the problem of the meaning of entropy and that of the origin of irreversibility we have built entropy and irreversibility
Deriving the laws of thermodynamics from a microscopic picture is a central quest of statistical mechanics.This tutorial focuses on the derivation of the first and second law for isolated and open quantum systems far from equilibrium, where such foundational questions also become practically relevant for emergent nanotechnologies.The derivation is based on a microscopic definition of five essential quantities: internal energy, thermodynamic entropy, work, heat, and temperature.These definitions are shown to satisfy the phenomenological laws of nonequilibrium thermodynamics for a large class of states and processes.The consistency with previous results is demonstrated.The framework applies to multiple baths including particle transport and accounts for processes with, e.g., a changing temperature of the bath, which is determined microscopically.An integral fluctuation theorem for entropy production is satisfied.In summary, this tutorial introduces a consistent and versatile framework to understand and apply the laws of thermodynamics in the quantum regime and beyond.
Entropy is the distinguishing and most important concept of our efforts to understand and regularize our observations of a very large class of natural phenomena, and yet, it is one of the most contentious concepts of physics. In this article, we review two expositions of thermodynamics, one without reference to quantum theory, and the other quantum mechanical without probabilities of statistical mechanics. In the first, we show that entropy is an inherent property of any system in any state, and that its analytical expression must conform to eight criteria. In the second, we recognize that quantum thermodynamics: (i) admits quantum probabilities described either by wave functions or by nonstatistical density operators; and (ii) requires a nonlinear equation of motion that is delimited by but more general than the Schroedinger equation, and that accounts for both reversible and irreversible evolutions of the state of the system in time. Both the more general quantum probabilities, and the equation of motion have been defined, and the three laws of thermodynamics are shown to be theorems of this equation.
Entropy is a fundamental concept which emerged along with other ideas during the development of thermodynamics and statistical mechanics Landau and Lifshitz (1978); Lieb and Yngvason (1999). Entropy has developed foremost out of phenomenological thermodynamical considerations such as the second law of thermodynamics in which it plays a prominent role Wehrl (1978). With the intense interest in the investigation of the physics of matter at the atomic and subatomic quantum levels, it maywell be askedwhether this concept can emerge out of the study of systems at a more fundamental level. In fact, it may be argued that a correct definition is only possible in the framework of quantum mechanics, whereas in classical mechanics, entropy can only be introduced in a rather limited and artificial way. Entropy relates macroscopic and microscopic aspects of nature, and ultimately determines the behavior of macroscopic systems. It is the intention here to present an introduction to this subject in a readable manner from the quantum point of view. There are many reasons for undertaking this. The intense interest in irreversible thermodynamics Grossing (2008), the statistical mechanics of astrophysical objects Padmanabhan (1990); Pathria (1977), quantum gravity and entropy of black holes Peres & al. (2004), testing quantum mechanics Ballentine (1970) and applications to condensed matter and quantum optics Haroche & al. (2006); Raimond & al. (2001) are just a few areas which are directly or indirectly touched on here. Let us begin by introducing the concept of entropy from the quantummechanical perspective, realizing that the purpose is to focus on quantum mechanics in particular. Quantum mechanics makes a clear distinction between observables and states. Observables such as position and momentum are mathematically described by self-adjoint operators in a Hilbert space. States, which are generally mixed, can be described by a density matrix, which is designated by ρ throughout. This operator ρ is Hermitean, has trace one and yields the expectation value of an observable A in the state ρ through the definition
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.