A general quantum theory encompassing Mechanics, Thermodynamics and irreversible dynamics is presented in two parts. The first part is concerned exclusively with the description of the states of any individual physical system. It is based on a new nonlinear quantum equation of motion, which reduces to the Schroedinger equation of motion of motion of conventional quantum dynamics only under special conditions. It accounts for the implications of the laws of Thermodynamics as well as for irreversible phenomena, such as the natural tendency of an isolated system to transit from any non-equilibrium state to an equilibrium state of higher entropy. Conversely, the laws of Thermodynamics and irreversibility emerge as manifestations of the fundamental quantum dynamical behaviour of the elementary constituents of any material system. We call this part Quantum Thermodynamics. The second part of the theory, which contains the first as a special case, is concerned with the description of stochastic distributions of states in an ensemble of identical physical systems each of which individually obeys the laws of Quantum Thermodynamics. It is based on a new measure-theoretic description of ensembles. It accounts unambiguously for the essential distinction between two types of uncertainties that are generally present in an ensemble, namely, quantal uncertainties due to the inherent quantal nature of the states of each individual member system and nonquantal uncertainties due to the stochastic distribution of states. We call this part Quantum Statistical Thermodynamics.
In the framework of the recent quest for well-behaved nonlinear extensions of the traditional Schrödinger-von Neumann unitary dynamics that could provide fundamental explanations of recent experimental evidence of loss of quantum coherence at the microscopic level, a recent paper [Gheorghiu-Svirschevski 2001 Phys. Rev. A 63 054102] reproposes the nonlinear equation of motion proposed by the present author [see Beretta G P 1987 Found. Phys. 17 365 and references therein] for quantum (thermo)dynamics of a single isolated indivisible constituent system, such as a single particle, qubit, qudit, spin or atomic system, or a Bose-Einstein or Fermi-Dirac field. As already proved, such nonlinear dynamics entails a fundamental unifying microscopic proof and extension of Onsager's reciprocity and Callen's fluctuation-dissipation relations to all nonequilibrium states, close and far from thermodynamic equilibrium. In this paper we propose a brief but self-contained review of the main results already proved, including the explicit geometrical construction of the equation of motion from the steepest-entropy-ascent ansatz and its exact mathematical and conceptual equivalence with the maximal-entropy-generation variational-principle formulation presented in Gheorghiu-Svirschevski S 2001 Phys. Rev. A 63 022105. Moreover, we show how it can be extended to the case of a composite system to obtain the general form of the equation of motion, consistent with the demanding requirements of strong separability and of compatibility with general thermodynamics principles. The irreversible term in the equation of motion describes the spontaneous attraction of the state operator in the direction of steepest entropy ascent, thus implementing the maximum entropy production principle in quantum theory. The time rate at which the path of steepest entropy ascent is followed has so far been left unspecified. As a step towards the identification of such rate, here we propose a possible, well-behaved and intriguing, general closure of the dynamics, compatible with the nontrivial requirements of strong separability. Based on the time–energy Heisenberg uncertainty relation, we derive a lower bound to the internal-relaxation-time functionals that determine the rate of entropy generation. This bound entails an upper bound to the rate of entropy generation. By this extreme maximal-entropy-generation-rate ansatz, each indivisible subsystem follows the direction of steepest locally perceived entropy ascent at the highest rate compatible with the time– energy uncertainty principle.
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.