Quasi-Set-Theoretical Foundations of Statistical Mechanics: A Research Program

Quasi-set theory provides us a mathematical background for dealing with collections of indistinguishable elementary particles. In this paper, we show how to obtain the usual statistics (Maxwell–Boltzmann, Bose–Einstein, and Fermi–Dirac) into the scope of quasi-set theory. We also show that, in order to derive Maxwell–Boltzmann statistics, it is not necessary to assume that the particles are distinguishable or individuals. In other words, Maxwell–Boltzmann statistics is possible even in an ensamble of indistinguishable particles, at least from the theoretical point of view. The main goal of this paper is to provide the mathematical grounds of a quasi-set theoretical framework for statistical mechanics.     

Einstein's theory of quantum radiation

The stationary probability distribution of the one-step process corresponding to Einstein's theory of absorption and emission of radiation is derived. Gauss' principle is used to identify the entropy, and the second law gives the dynamical equilibrium condition or Planck's radiation law. This condition is in disagreement with Einstein's criterion of dynamical equilibrium. The physical consequences of the new condition are investigated.     

Beyond Einstein's radiation theory

Einstein's radiation mechanism is generalized to account for the possibility of population inversion by placing a nonlinear bound on the growth of an unstable perturbation. The nonstationary linear mechanism of relaxation to blackbody radiation below threshold is studied. The nonstationary photon distribution is the negative binomial distribution, and casting it as a law of error, for which the most probable value is the mean value, gives the expression for the statistical entropy. The second law yields a nonequilibrium generalization of Planck's radiation law. The nonlinear mechanism leading to the transformation from the negative binomial probability distribution, for chaotic light, to a Poisson probability distribution, for coherent light, is then analyzed. A criterion for lasing is given in terms of the chemical potential of radiation which is compared to the inequality for the transition from quantum to classical statistics.     

Large numbers hypothesis. III. Kinetic theory,statistical physics,and thermodynamics

Dirac's large numbers hypothesis (LNH) is incorporated into kinetic theory, statistical physics, and thermodynamics using the self-consistent formalism of units covariance. The ingeodesic equation and matter creation introduce modifications of the most fundamental laws of the subject. Liouville's theorem no longer holds, the Boltzmann equation is modified, as is theH-theorem. This affects the second law of thermodynamics in that for canonical LNH neither reversible nor adiabatic processes are possible (as expected). A significant result is that the collision terms have the same form as in standard physics. This means that equilibrium distribution functions are identical to those of standard physics, as required for self-consistency with the precepts of LNH. The net effect of LNH is as though all matter in our Universe were weakly coupled to a large heat bath.NAS-NRC Senior Research Associate 1981–1983.     

Are Words the Quanta of Human Language? Extending the Domain of Quantum Cognition

In previous research, we showed that ‘texts that tell a story’ exhibit a statistical structure that is not Maxwell–Boltzmann but Bose–Einstein. Our explanation is that this is due to the presence of ‘indistinguishability’ in human language as a result of the same words in different parts of the story being indistinguishable from one another, in much the same way that ’indistinguishability’ occurs in quantum mechanics, also there leading to the presence of Bose–Einstein rather than Maxwell–Boltzmann as a statistical structure. In the current article, we set out to provide an explanation for this Bose–Einstein statistics in human language. We show that it is the presence of ‘meaning’ in ‘texts that tell a story’ that gives rise to the lack of independence characteristic of Bose–Einstein, and provides conclusive evidence that ‘words can be considered the quanta of human language’, structurally similar to how ‘photons are the quanta of electromagnetic radiation’. Using several studies on entanglement from our Brussels research group, we also show, by introducing the von Neumann entropy for human language, that it is also the presence of ‘meaning’ in texts that makes the entropy of a total text smaller relative to the entropy of the words composing it. We explain how the new insights in this article fit in with the research domain called ‘quantum cognition’, where quantum probability models and quantum vector spaces are used in human cognition, and are also relevant to the use of quantum structures in information retrieval and natural language processing, and how they introduce ‘quantization’ and ‘Bose–Einstein statistics’ as relevant quantum effects there. Inspired by the conceptuality interpretation of quantum mechanics, and relying on the new insights, we put forward hypotheses about the nature of physical reality. In doing so, we note how this new type of decrease in entropy, and its explanation, may be important for the development of quantum thermodynamics. We likewise note how it can also give rise to an original explanatory picture of the nature of physical reality on the surface of planet Earth, in which human culture emerges as a reinforcing continuation of life.     

Mechanism of blackbody radiation

The law of error for Bose statistics is not unique; the family of probability distributions differ insofar as zero-point energy is concerned. This is traced back to the spontaneous emission term in the Einstein mechanism of emission and absorption of radiation. It is argued that the spontaneous emission term is unimportant for blackbody radiation and an alternative mechanism is proposed in which thermal equilibrium is secured through a constraint on the number of quanta in any given mode of the radiation field. Both mechanisms predict a modification of the Maxwell velocity distribution at high frequencies and are compared in relation to Doppler broadening and their low-temperature behavior.     

Classical and Quantum H-Theorem Revisited: Variational Entropy and Relaxation Processes

We propose a novel framework to describe the time-evolution of dilute classical and quantum gases, initially out of equilibrium and with spatial inhomogeneities, towards equilibrium. Briefly, we divide the system into small cells and consider the local equilibrium hypothesis. We subsequently define a global functional that is the sum of cell H-functionals. Each cell functional recovers the corresponding Maxwell–Boltzmann, Fermi–Dirac, or Bose–Einstein distribution function, depending on the classical or quantum nature of the gas. The time-evolution of the system is described by the relationship dH/dt≤0, and the equality condition occurs if the system is in the equilibrium state. Via the variational method, proof of the previous relationship, which might be an extension of the H-theorem for inhomogeneous systems, is presented for both classical and quantum gases. Furthermore, the H-functionals are in agreement with the correspondence principle. We discuss how the H-functionals can be identified with the system’s entropy and analyze the relaxation processes of out-of-equilibrium systems.     

A Kinetic Theory Approach to Quantum Gravity

We describe a kinetic theory approach to quantum gravity by which we mean a theory of the microscopic structure of space-time, not a theory obtained by quantizing general relativity. A figurative conception of this program is like building a ladder with two knotty poles: quantum matter field on the right and space-time on the left. Each rung connecting the corresponding knots represents a distinct level of structure. The lowest rung is hydrodynamics and general relativity; the next rung is semiclassical gravity, with the expectation value of quantum fields acting as source in the semiclassical Einstein equation. We recall how ideas from the statistical mechanics of interacting quantum fields helped us identify the existence of noise in the matter field and its effect on metric fluctuations, leading to the establishment of the third rung: stochastic gravity, described by the Einstein–Langevin equation. Our pathway from stochastic to quantum gravity is via the correlation hierarchy of noise and induced metric fluctuations. Three essential tasks beckon: (1) deduce the correlations of metric fluctuations from correlation noise in the matter field; (2) reconstituting quantum coherence—this is the reverse of decoherence—from these correlation functions; and (3) use the Boltzmann–Langevin equations to identify distinct collective variables depicting recognizable metastable structures in the kinetic and hydrodynamic regimes of quantum matter fields and how they demand of their corresponding space-time counterparts. This will give us a hierarchy of generalized stochastic equations—call them the Boltzmann–Einstein hierarchy of quantum gravity—for each level of space-time structure, from the the macroscopic (general relativity) through the mesoscopic (stochastic gravity) to the microscopic (quantum gravity).     

Connecting Blackbody Radiation,Relativity, and Discrete Charge in Classical Electrodynamics

It is suggested that an understanding of blackbody radiation within classical physics requires the presence of classical electromagnetic zero-point radiation, the restriction to relativistic (Coulomb) scattering systems, and the use of discrete charge. The contrasting scaling properties of nonrelativistic classical mechanics and classical electrodynamics are noted, and it is emphasized that the solutions of classical electrodynamics found in nature involve constants which connect together the scales of length, time, and energy. Indeed, there are analogies between the electrostatic forces for groups of particles of discrete charge and the van der Waals forces in equilibrium thermal radiation. The differing Lorentz- or Galilean-transformation properties of the zero-point radiation spectrum and the Rayleigh-Jeans spectrum are noted in connection with their scaling properties. Also, the thermal effects of acceleration within classical electromagnetism are related to the existence of thermal equilibrium within a gravitational field. The unique scaling and phase-space properties of a discrete charge in the Coulomb potential suggest the possibility of an equilibrium between the zero-point radiation spectrum and matter which is universal (independent of the particle mass), and an equilibrium between a universal thermal radiation spectrum and matter where the matter phase space depends only upon the ratio mc ²/k _B T. The observations and qualitative suggestions made here run counter to the ideas of currently accepted quantum physics.     

Connections Between the Thermodynamics of Classical Electrodynamic Systems and Quantum Mechanical Systems for Quasielectrostatic Operations

The thermodynamic behavior is analyzed of a single classical charged particle in thermal equilibrium with classical electromagnetic thermal radiation, while electrostatically bound by a fixed charge distribution of opposite sign. A quasistatic displacement of this system in an applied electrostatic potential is investigated. Treating the system nonrelativistically, the change in internal energy, the work done, and the change in caloric entropy are all shown to be expressible in terms of averages involving the distribution of the position coordinates alone. A convenient representation for the probability distribution is shown to be the ensemble average of the absolute square value of an expansion over the eigenstates of a Schrödinger-like equation, since the heat flow is shown to vanish for each hypothetical state. Subject to key assumptions highlighted here, the demand that the entropy be a function of state results in statistical averages in agreement with the form in quantum statistical mechanics. Examining the very low and very high temperature situations yields Planck's and Boltzmann's constants. The blackbody radiation spectrum is then deduced. From the viewpoint of the theory explored here, the method in quantum statistical mechanics of statistically counting the states at thermal equilibrium by using the energy eigenvalue structure, is simply a convenient counting scheme, rather than actually representing averages involving physically discrete energy states.     

High-order kinetic flux vector splitting schemes in general coordinates for ideal quantum gas dynamics

A class of high-order kinetic flux vector splitting schemes are presented for solving ideal quantum gas dynamics based on quantum statistical mechanics. The collisionless quantum Boltzmann equation approach is adopted and both Bose–Einstein and Fermi–Dirac gases are considered. The formulas for the split flux vectors are derived based on the general three-dimensional distribution function in velocity space and formulas for lower dimensions can be directly deduced. General curvilinear coordinates are introduced to treat practical problems with general geometry. High-order accurate schemes using weighted essentially non-oscillatory methods are implemented. The resulting high resolution kinetic flux splitting schemes are tested for 1D shock tube flows and shock wave diffraction by a 2D wedge and by a circular cylinder in ideal quantum gases. Excellent results have been obtained for all examples computed.     

Manipulating nonclassical quantum statistical properties of light field by an f-deformed Bose–Einstein condensate

We consider the interaction between an f-deformed Bose–Einstein condensate and a single-mode quantized light field. By using the Gardiner’s phonon operators, we find that there exists a natural deformation in the model which modifies the Bogoliubov approximation under the condition of large but finite number of particles in condensate. This approach introduces an intrinsically deformed Bose–Einstein condensate, where the deformation parameter, well-defined by the particle number N in condensate, controls the strength of the associated nonlinearity. By introducing the deformed Gardiner’s phonon operators we modify the very dilute-gas approximation through including atomic collisions in condensate. The rate of atomic collisions κ, as a new deformation parameter in the deformed Bose–Einstein condensate, controls the nonlinearity related to the atomic collisions. We show that by controlling the nonlinearities in the f-deformed atomic condensate through the two atomic parameters N and κ, it is possible to generate and manipulate the nonclassical quantum statistical properties of radiation field, such as, the sub-Poissonian photon statistics and quadrature squeezing. Also, it is possible to control the collapses and revivals phenomena in the average number of photons by atomic parameters N and κ.     

Four Quantum Conservation Laws for Black Hole Stationary Equilibrium Radiation Processes

The classical first law of thermodynamics for a Kerr–Newman black hole (KNBH)is generalized to a law in quantum form on the event horizon. Then four quantumconservation laws on the KNBH equilibrium radiation process are derived. TheBekenstein–Hawking relation S = A/4 is exactly established. It can be inferredthat the classical entropy of black hole arises from the quantum entropy of fieldquanta or quasiparticles inside the hole.     

The Pauli Principle and the Restricted Primitive Model

The restricted primitive model is an electrically neutral, classical model consisting of hard spheres charged either +q or –q. We show that, by appropriately selecting the diameter of the hard spheres, the pressure when q=0 can be made equal to that for a fluid of Maxwell–Boltzmann point ions and an ideal Fermi gas of electrons. We compare the series expansion of these classical and quantum systems and find that, except for intermediate de Broglie density and moderate to strong electrical interaction strength, the restricted primitive model gives a reasonable representation of the pressure of the corresponding quantum system. Much of the current interest, however, has been focused on the above, excepted region.     

Particle Creation in the Oscillatory Phase of Inflaton

A thermal squeezed state representation of inflaton is constructed for a flat Friedmann–Robertson–Walker (FRW) background metric and the phenomenon of particle creation is examined during the oscillatory phase of inflaton, in the semiclassical theory of gravity. An approximate solution to the semiclassical Einstein equation is obtained in thermal squeezed state formalism perturbatively and is found obey the same power-law expansion as that of classical Einstein equation. In addition to that the solution shows oscillatory in nature except on a particular condition. It is also noted that, the coherently oscillating nonclassical inflaton, in thermal squeezed vacuum state, thermal squeezed state, and thermal coherent state, suffers particle production and the created particles exhibit oscillatory behavior. The present study can account for the postinflation particle creation due to thermal and quantum effects of inflation in a flat FRW universe.     

Phenomenon of gravitational repulsion in the general theory of relativity

Einstein's generalization of Newton's theory of gravitation in the general theory of relativity led not only to small quantitative differences between gravitational effects in the relativistic theory and the Newtonian theory, but also to essentially new phenomena and effects peculiar to the relativistic theory and absent in the Newtonian theory. This difference is so large that the gravitational interaction in Einstein's theory even altered the attraction-only property, characteristic of Newton's theory, the law of universal gravitation, and became both attractive and repulsive. It is notable that the nature of the repulsion in gravitational interaction already appears in the simplest case of a spherically symmetric isolated body. Einstein's equations admit for a spherical body a solution whose physical interpretation uniquely indicates the repulsive nature of a gravitational field inside the body, if the number of particles that make up the body is sufficiently large. The structure of such a body, density distribution of the number of particles, mass, and pressure, is determined in the equilibrium state by the pressure of the substance, the gravitational attraction of peripheral layers toward the center, and the gravitational repulsion of inner layers of matter away from the center. As a result of the gravitational repulsion of matter away from the center inside the body there appears a cavity, free of the matter making up the body and its electromagnetic radiation. If the body is cold, then the volume of the world tube of the cavity can differ from zero. In the opposite case, the world tube of the cavity reduces to the world line of the center, which is inaccessible to particles of matter and to electromagnetic radiation. Gravitational repulsion, on the other hand, is a result of the existence of a field singularity at the center of the body, whose world line is time-like.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 61–67, April, 1981.     

Control of temperature and entropy by frequent quantum measurements

We study deviations from thermal equilibrium between two-level systems (TLS) and a bath by frequent and brief quantum measurements of the TLS energy-states. The resulting entropy and temperature of both the system and the bath are found to be completely determined by the measurement rate, and unrelated to what is expected by standard thermodynamical rules that hold for Markovian baths. These anomalies allow for very fast control heating, cooling and state-purification (entropy reduction) of quantum systems much sooner than their thermal equilibration time.     

Convection-Diffusion Lattice Boltzmann Scheme for Irregular Lattices

In this paper, a lattice Boltzmann (LB) scheme for convection diffusion on irregular lattices is presented, which is free of any interpolation or coarse graining step. The scheme is derived using the axioma that the velocity moments of the equilibrium distribution equal those of the Maxwell–Boltzmann distribution. The axioma holds for both Bravais and irregular lattices, implying a single framework for LB schemes for all lattice types. By solving benchmark problems we have shown that the scheme is indeed consistent with convection diffusion. Furthermore, we have compared the performance of the LB schemes with that of finite difference and finite element schemes. The comparison shows that the LB scheme has a similar performance as the one-step second-order Lax–Wendroff scheme: it has little numerical diffusion, but has a slight dispersion error. By changing the relaxation parameter ω the dispersion error can be balanced by a small increase of the numerical diffusion.     

Energy in a highly ordered universe

A new theory of particles proposed in an earlier paper is now applied to explain energy. Having earlier derived the Rydberg formula for atomic spectra without using the Pauli principle, the authors now derive the photoelectric effect, deflection of light by gravitation, and Planck's law for blackbody radiation without using Planck's assumption on energy quanta or Einstein's theory of general relativity.