A unified quantum theory of mechanics and thermodynamics. Part IIa. Available energy

Part II of this three-part paper presents some of the most important theorems that can be deduced from the four postulates of the unified theory discussed in Part I. In Part IIa, it is shown that the maximum energy that can be extracted adiabatically from any system in any state is solely a function of the density operator associated with the state. Moreover, it is shown that for any state of a system, nonequilibrium, equilibrium or stable equilibrium, a unique propertyS exists which is proportional to the total energy of the system minus the maximum energy that can be extracted adiabatically from the system in combination with a reservoir. For statistically independent systems, propertyS is extensive, it is invariant during all reversible processes, and it increases during all irreversible processes.Part I of this paper appeared inFound. Phys. 6(1) (1976). The numbering of the sections, equations, and references in this part of the paper continues from those in Part I.     

Toward a microrealistic version of quantum mechanics. Part II

In this paper, possible objections to the propensity microrealistic version of quantum mechanics proposed in Part I are answered. This version of quantum mechanics is compared with the statistical, particle, microrealistic viewpoint, and a crucial experiment is proposed designed to distinguish between these two microrealistic versions of quantum mechanics.Part I of this paper appeared inFound. Phys. 6, 275 (1976).     

A unified quantum theory of mechanics and thermodynamics. Part III. Irreducible quantal dispersions

This part of the paper concludes the presentation of the unified theory. It is shown that the theory requires the existence of, and applies only to, irreducible quantal dispersions associated with pure or mixed states. Two experimental procedures are given for the operational verification of such dispersions. Because the existence of irreducible dispersions associated with mixed states is required by Postulate 4 of the theory, and because Postulate 4 expresses the basic implications of the second law of classical thermodynamics, it is concluded that the second law is a manifestation of phenomena characteristic of irreducible quantal dispersions associated with the elementary constituents of matter.Parts I, IIa, and IIb of this paper appeared inFound. Phys. 6, 15, 127, 439 (1976), respectively. The numbering of the sections, equations, and references in this part continues from the previous parts.     

A unified quantum theory of mechanics and thermodynamics. Part I. Postulates

A unified axiomatic theory that embraces both mechanics and thermodynamics is presented in three parts. It is based on four postulates; three are taken from quantum mechanics, and the fourth is the new disclosure of the existence of quantum states that are stable (Part I). For nonequilibrium and equilibrium states, the theory provides general original results, such as the relation between irreducible density operators and the maximum work that can be extracted adiabatically (Part IIa). For stable equilibrium states, it shows for the first time that the canonical and grand canonical distributions are the only stable distributions (Part IIb). The theory discloses the incompleteness of the equation of motion of quantum mechanics not only for irreversible processes but, more significantly, for reversible processes (Part IIb). It establishes the operational meaning of an irreducible density operator and irreducible dispersions associated with any state, and reveals the relationship between such dispersions and the second law (Part III).     

Radiation reaction. II

Regularized classical field theory [B. Ritchie,Found. Phys. Lett. 4, 375 (1991)] is used to derive a new model for the Coulomb problem of an electron bound to an infinitely heavy nucleus. The model is solved numerically and interesting results are obtained, including the prediction of a stable ground state.     

Luminosity Distance,Angular Size and Surface Brightness in Cosmological General Relativity

This paper corrects the explanation given by Oliveira and Hartnett, Found. Phys. Lett. 19(6), 519–535, 2006 for the luminosity distance in Cosmological General Relativity. The mathematical expression for the luminosity distance used in that paper is correct but the explanation in Eqs. (22) and (23) is flawed. Expressions for the angular size and surface brightness of sources are also derived. Finally some comment is made about the calculations of χ² values in that paper compared with an earlier paper, Found. Phys. 36(6), 839–861, 2006.     

Klein-Gordon thermal equation for a planck gas

In this paper the quantum hyperbolic equation formulated in our earlier paper [Found. Phys. Lett. 10, 599 (1997)] is applied to the study of the propagation of the initial thermal state of the universe. It is shown that the propagation depends on the barrier height. The Planck wall potential is introduced,V _P = ħ/8t_P = 1.125 10¹⁸ GeV, wheret _P is a Planck time. For the barrier heightV <V _P , the master thermal equation isthe modified telegrapher’sequation, and for barrier heightV >V _P the master equation is theKlein- Gordon equation. The solutions of both type equations for Cauchy boundary conditions are discussed.     

Thermodynamic Uncertainty Relations Again: A Reply to Lavenda

In a previous paper (Found. Phys. 29, 655 (1999)), we have presented a review of various approaches in the literature towards the derivation of so-called thermodynamic uncertainty relations in statistical thermodynamics. This review has been critical. We have argued that some of these approaches are sound, i.e., they reach a valid conclusion, albeit under restricted conditions, whereas others were found to be incoherent and could not withstand the scrutiny of logical analysis. In the latter category we have included work of Lavenda on this topic. However, in a comment (Found. Phys. Lett. 13, 487 (2000)), Lavenda claims to have uncovered fundamental errors in our paper. In this reply we show that these claims are mistaken.     

Augmented Langevin approach to fluctuations in nonlinear irreversible processes

A Fokker-Planck equation derived from statistical mechanics by M. S. Green [J. Chem. Phys. 20:1281 (1952)] has been used by Grabertet al. [Phys. Rev. A 21:2136 (1980)] to study fluctuations in nonlinear irreversible processes. These authors remarked that a phenomenological Langevin approach would not have given the correct reversible part of the Fokker-Planck drift flux, from which they concluded that the Langevin approach is untrustworthy for systems with partly reversible fluxes. Here it is shown that a simple modification of the Langevin approach leads to precisely the same covariant Fokker-Planck equation as that of Grabertet al., including the reversible drift terms. The modification consists of augmenting the usual nonlinear Langevin equation by adding to the deterministic flow a correction term which vanishes in the limit of zero fluctuations, and which is self-consistently determined from the assumed form of the equilibrium distribution by imposing the usual potential conditions. This development provides a simple phenomenological route to the Fokker-Planck equation of Green, which has previously appeared to require a more microscopic treatment. It also extends the applicability of the Langevin approach to fluctuations in a wider class of nonlinear systems.     

Yes,More Decoherence: A Reply to Critics

Recently I published an article in this journal entitled “Less interpretation and more decoherence in quantum gravity and inflationary cosmology” (Crull in Found Phys 45(9):1019–1045, 2015). This article generated responses from three pairs of authors: Vassallo and Esfeld (Found Phys 45(12):1533–1536, 2015), Okon and Sudarsky (Found Phys 46(7):852–879, 2016) and Fortin and Lombardi (Found Phys, 2017). In what follows, I reply to the criticisms raised by these authors.     

An integral equation and Monte Carlo study of homo- and hetero- nuclear square-well diatomic fluids

Square-well homo-nuclear and hetero-nuclear diatomic fluids are studied using the Ornstein-Zernike equation and the recently proposed RHNC-VM closure. Monte Carlo canonical simulations have been performed to complete recent literature simulation data. The integral equation thermodynamic and structural results are compared with these and literature simulation data at three elongations over a large range of densities and temperatures. The RHNC-VM theory agrees excellently with the simulation thermodynamic and structural results. Its accuracy revealed slight errors in simulation data in work by Lisal and Nezbeda [1999, Molec. Phys., 96, 335]. The data have been re-simulated.     

Explaining the Unobserved—Why Quantum Mechanics Ain’t Only About Information

A remarkable theorem by Clifton et al [Found Phys. 33(11), 1561–1591 (2003)] (CBH) characterizes quantum theory in terms of information-theoretic principles. According to Bub [Stud. Hist. Phil. Mod. Phys. 35 B, 241–266 (2004); Found. Phys. 35(4), 541–560 (2005)] the philosophical significance of the theorem is that quantum theory should be regarded as a “principle” theory about (quantum) information rather than a “constructive” theory about the dynamics of quantum systems. Here we criticize Bub’s principle approach arguing that if the mathematical formalism of quantum mechanics remains intact then there is no escape route from solving the measurement problem by constructive theories. We further propose a (Wigner-type) thought experiment that we argue demonstrates that quantum mechanics on the information-theoretic approach is incomplete.     

Addendum to: “Coupled KdV Equations of Hirota-Satsuma Type” (J. Nonlin. Math. Phys. Vol. 6, No.3 (1999), 255–262)

Abstract

It is shown that one system of coupled KdV equations, found in J. Nonlin. Math. Phys., 1999, V.6, N 3, 255–262 to possess the Painlevé property, is integrable but not new.     

A non self-referential expression of Tsallis' probability distribution function

The canonical probability distribution function (pdf) obtained by optimizing the Tsallis entropy under either the linear mean energy constraint U or the escort mean energy constraint U_q suffer self-referentiality. In a recent paper [Phys. Lett. A 335, 351 (2005)] the authors have shown that the pdfs obtained with either U or U_q are equivalent to the pdf in a non self-referential form. Based on this result we derive an alternative expression for the Tsallis distributions, employing either U or U_q, which is non self-referential.     

Gravitation theory in the spacetimeR×S 3

A geometric formulation of the gravitation theory in the spacetime R × S ³ is given. A linear connection is introduced on the tangent bundle T(R × S ³ ) and then the connection coefficients and the Riemann curvature tensor are calculated. It is shown that their expressions differ from those of Carmeli and Malin [Found. Phys.17, 407 (1987)] by supplementary terms due to the noncommutativity of derivatives used on the spacetime R × S ³ . The Einstein field equations are written as usually and a comparison with other results is given. Finally, some observations about a possible gauge theory of gravitation in the spacetime R × S ³ are made.     

Inhomogeneous Tsallis distributions in the HMF model

We study the maximization of the Tsallis functional at fixed mass and energy in the Hamiltonian Mean Field (HMF) model. We give a thermodynamical and a dynamical interpretation of this variational principle. This leads to q-distributions known as stellar polytropes in astrophysics. We study phase transitions between spatially homogeneous and spatially inhomogeneous equilibrium states. We show that there exists a particular index q _c = 3 playing the role of a canonical tricritical point separating first and second order phase transitions in the canonical ensemble and marking the occurence of a negative specific heat region in the microcanonical ensemble. We apply our results to the situation considered by Antoni and Ruffo [Phys. Rev. E 52, 2361 (1995)] and show that the anomaly displayed on their caloric curve can be explained naturally by assuming that, in this region, the QSSs are polytropes with critical index q _c = 3. We qualitatively justify the occurrence of polytropic (Tsallis) distributions with compact support in terms of incomplete relaxation and inefficient mixing (non-ergodicity). Our paper provides an exhaustive study of polytropic distributions in the HMF model and the first plausible explanation of the surprising result observed numerically by Antoni and Ruffo (1995). In the course of our analysis, we also report an interesting situation where the caloric curve presents both microcanonical first and second order phase transitions.     

Phase Space Optimization of Quantum Representations: Non-Cartesian Coordinate Spaces

In an earlier article [Found. Phys. 30, 1191 (2000)], a quasiclassical phase space approximation for quantum projection operators was presented, whose accuracy increases in the limit of large basis size (projection subspace dimensionality). In a second paper [J. Chem. Phys. 111, 4869 (1999)], this approximation was used to generate a nearly optimal direct-product basis for representing an arbitrary (Cartesian) quantum Hamiltonian, within a given energy range of interest. From a few reduced-dimensional integrals, the method determines the optimal 1D marginal Hamiltonians, whose eigenstates comprise the direct-product basis. In the present paper, this phase space optimized direct-product basis method is generalized to incorporate non-Cartesian coordinate spaces, composed of radii and angles, that arise in molecular applications. Analytical results are presented for certain standard systems, including rigid rotors, and three-body vibrators.     

Popper's Experiment Revisited

Recently, there has been increasing theoretical and experimental interest in Popper's gedanken experiment. We calculate in this paper, using the path integral approach, the diffraction patterns predicted by quantum mechanics for this arrangement. The calculations confirm the narrowing of the width of the pattern in absence of the slit obtained experimentally by Kim and Shih (Y. Kim and Y. Shih, Found. Phys. 29, 1849 (1999)).