In a recent article, Alli and Sewell [J. Math. Phys. 36, 5598 (1995)] formulated a new version of the Dicke–Hepp–Lieb laser model in terms of quantum dynamical semigroups, and thereby extended the macroscopic picture of the model. In the present article, we complement that picture with a corresponding microscopic one, which carries the following new results. (a) The local microscopic dynamics of the model is piloted by the classical, macroscopic field, generated by the collective action of its components; (b) the global state of the system carries no correlations between its constituent atoms after transient effects have died out; and (c) in the latter situation, the state of the system at any time t maximizes its entropy density, subject to the constraints imposed by the instantaneous values of its macroscopic variables.
In a recent article, Alli and Sewell [J. Math. Phys. 36, 5598 (1995)] formulated a new version of the Dicke–Hepp–Lieb laser model in terms of quantum dynamical semigroups, and thereby extended the macroscopic picture of the model. In the present article, we complement that picture with a corresponding microscopic one, which carries the following new results. (a) The local microscopic dynamics of the model is piloted by the classical, macroscopic field, generated by the collective action of its components; (b) the global state of the system carries no correlations between its constituent atoms after transient effects have died out; and (c) in the latter situation, the state of the system at any time t maximizes its entropy density, subject to the constraints imposed by the instantaneous values of its macroscopic variables.
In a recent article, Alli and Sewell formulated a new version of the Dicke-Hepp-Lieb laser model in terms of quantum dynamical semigroups, and thereby extended the macroscopic picture of the model. In the present article, we complement that picture with a corresponding microscopic one, which carries the following new results. (a) The local microscopic dynamics of the model is piloted by the classical, macroscopic field, generated by the collective action of its components; (b) the global state of the system carries no correlations between its constituent atoms after transient effects have died out; and (c) in the latter situation, the state of the system at any time t maximises its entropy density, subject to the constraints imposed by the instantaneous values of its macroscopic variables.
The second law of thermodynamics, for quantum systems, is formulated, on the microscopic level. As for classical systems, such a formulation is only possible when specific conditions are satisfied (continuous spectrum, nonvanishing of the collision operator, etc.). The unitary dynamical group can then be mapped into two contractive semigroups, reaching equilibrium either for t --> +infinity or for t --> -infinity. The second law appears as a symmetry-breaking selection principle, limiting the observables and density functions to the class that tends to thermodynamic equilibrium in the future (for t --> +infinity). The physical content of the dynamical structure is now displayed in terms of the appropriate semigroup, which is realized through a nonunitary transformation. The superposition principle of quantum mechanics has to be reconsidered as irreversible processes transform pure states into mixtures and unitary transformations are limited by the requirement that entropy remains invariant. In the semigroup representation, interacting fields lead to units that behave incoherently at equilibrium. Inversely, nonequilibrium constraints introduce correlations between these units.
In thermodynamics and statistical mechanics entropy is the fundamental physical quantity to describe the evolution of a statistical ensemble. Its microscopic definition was provided by Boltzmann through the celebrated expression S = kB ln Γ, where kB is the Boltzmann constant and Γ is the number of microstates exploiting the given macroscopic properties. In this context, it is well known that in classical mechanics the entropy: i) allows the violation of the uncertainty principle [1], [2]; ii) can be considered as a special case of the so-called Boltzmann-Gibbs-Shannon entropy that enables one to apply results of information theory to physics [1], [3]. In particular, by introducing the principle of maximum entropy (MEP) it was found possible to derive rigorous hydrodynamic (HD) models based on the moments of the distribution function to all orders of a power expansion and including appropriate closure conditions [4], [5]. Accordingly, making use of the Lagrange multipliers technique, it was found possible to construct the set of evolution equations for the macro-variables of interest. Apart from some partial attempts [3], [6], this is no longer the case in quantum mechanics. Here, the main difficulties concern with: i) the definition of a proper quantum entropy that includes particle indistinguishability; ii) the formulation of a global quantum MEP (QMEP) that allows one to obtain a quantum distribution function both for thermodynamic equilibrium and nonequilibrium configurations. From one hand, in the framework of a nonlocal quantum theory, the generalization of the corresponding Lagrange multipliers is also an open problem. Fron another hand, a rigorous formulation of quantum HD (QHD) closed models is a demanding issue for many kinds of problems in quantum systems like, interacting fermionic and bosonic gases, quantum turbulence, quantum fluids, quantized vortices, nuclear physics, confined carrier transport in semiconductor heterostrucures, phonon and electron transport in nanostructures, nanowires and thin layers. The aim of this talk is to address the above drawbacks by defining a quantum entropy in terms of the reduced density matrix, thus explicitly incorporating quantum statistics into problems involving a system of identical particles. As a further step, with respect to the uncertainty principle, starting from the Wigner representation we formulate a quantum maximum entropy principle which, in the framework of information theory, allows us to obtain a non-local theory for the system under study. As a final step, we determine a closed quantum hydrodynamic model for the macroscopic variables used as constraints in the QMEP approach.
A basic formulation of the laser system is given in the form most convenient to study the time-dependent cooperative ( symmetry breaking ) behavior of the system in the far-from-equilibrium states from the microscopic and fully quantum-statistical-mechanical point of view. The quantum mechanical master equation, derived from a microscopic model Hamiltonian, is mapped onto a c-number function space with the use of the Boson and spin coherent states representations. The distribution function (mapped from the density operator) is expanded in terms of the Hermite polynomials and the spherical harmonics to obtain differential equations for the expansion coefficients with which time evolution of the laser system can be determined completely.
The celebrated Evans--Searles, respectively Gallavotti--Cohen, fluctuation theorem concerns certain universal statistical features of the entropy production rate of a classical system in a transient, respectively steady, state. In this paper, we review and compare several possible extensions of these fluctuation theorems to quantum systems. We present, in an introductory way, the approach proposed in a recent series of works [Lett. Math. Phys. 114, 32 (2024): doi:10.1007/s11005-024-01777-0], [Rev. Math. Phys 38, 2461006 (2026): doi:10.1142/S0129055X24610063], [Commun. Math. Phys. 406, 201 (2025): 10.1007/s00220-025-05360-z] and [J. Math. Phys. 67, 022101 (2026): doi:10.1063/5.0311182] including a previously missing extension of the Gallavotti--Cohen (steady state) version of the FT to two-time measurement entropy production in the algebraic setting of quantum dynamical systems. We focus on the open quantum systems, where a small system interacts with several (infinitely extended) reservoirs, and discuss the following approaches: naive, two-time measurement, phase space contraction and ancilla state tomography. We review the connection with modular theory. The latter provides a way to extend the classical notions to the quantum domain. We discuss the quantum version of the principle of regular entropic fluctuations, introduced in the classical context in [Nonlinearity 24, 699 (2011): doi:10.1088/0951-7715/24/3/003]. Finally, we relate the statistical properties of these various notions of entropy production to spectral resonances of quantum transfer operators and apply this approach to a paradigmatic model of non-equilibrium statistical mechanics: the spin-fermion model. The results presented here shed a new light on the nature of entropic fluctuations in quantum statistical mechanics.
I. Introduction.Statistical mechanics is the science of explaining, predicting and understanding the gross, macroscopic attributes of matter (which may be taken to mean mechanical systems with essentially an infinite number of degrees of freedom) in terms of the elementary dynamical laws governing its atomic constituents.The problems that arise are sufficiently complex and intriguing, but at the same time sufficiently well posed, that the subject is nowadays as much a part of mathematics as of physics.The fields of information theory and ergodic theory had their genesis in statistical mechanical modes of thought and are now well established in the mathematics literature; there will be more to come.Ludwig Boltzmann, who died in 1906, was one of the principal founders of statistical mechanics, and his monument in Vienna contains the following eloquent testimonial to his scientific creativity:(1) S = k log W.Surely, this hypothesis of Boltzmann [1] is one of the most important and daring in statistical mechanics, for it relates S, the macroscopic entropy of a system, to W, the number of microscopic states of the system which have the same, given macroscopic properties.The number k is a universal constant, called Boltzmann's constant, and, for our purposes, we can consider it to be 1.In these lectures we shall explore some of the abstract properties of entropy, after first giving a precise formulation of it, and will include some recent results (with M. B. Ruskai) which extend formerly known facts about the strong subadditivity of entropy from the domain of classical mechanics to the quantum-mechanical domain.The presentation here will be sketchy and the reader is referred to the original papers [3], [4], [5] for more details.An expanded version of an invited address delivered before the M.I.T.