arXiv subjects

B. H. Lavenda

Publications and source records attributed to B. H. Lavenda.

At least 19 recordsLinked to original sources

arXiv2013

Geometric Phase, Curvature, and the Monodromy Group

The geometric phase requires the multivaluedness of solutions to Fuchsian second-order equations. The angle, or its complement, is given by half the area of a spherical triangle in the case of three singular points, or half the area of a lune in the case of two singular points. Both are fundamental regions where the automorphic function takes a value only once, and a linear-fractional transformation tessellates the plane in replicas of the fundamental region. The condition that the homologues of the poles, representing vertices, be angles places restrictions on quantum numbers which are no longer integers, for, otherwise, the phase factors would become unity. Restriction must be made to regular singular points for only then will solutions to the differential equation be rational functions so that the covering group will be cyclic and the covering space be a "spiral staircase". Many of the equations of mathematical physics, with essential singularities, become Fuchsian differential equations, with regular singularities, at zero kinetic energy. Examples of geometric phase include the phasor, the Pancharatnam phase of beams of polarized light in different states, the Aharanov-Bohm phase, and angular momentum with centripetal "attraction". In the latter example, the phase is one-half the area of the lune, which disappears when the pole at infinity becomes an essential singularity thereby recovering the Schrödinger equation. The behavior of an automorphic function at a limit point on the boundary is analogous to the confluence of two regular singularities in a linear second-order differential equation to produce an essential singularity at infinity.

physics.gen-ph↗

arXiv2012

Noneuclidean Tessellations and their relation to Reggie Trajectories

The coefficients in the confluent hypergeometric equation specify the Regge trajectories and the degeneracy of the angular momentum states. Bound states are associated with real angular momenta while resonances are characterized by complex angular momenta. With a centrifugal potential, the half-plane is tessellated by crescents. The addition of an electrostatic potential converts it into a hydrogen atom, and the crescents into triangles which may have complex conjugate angles; the angle through which a rotation takes place is accompanied by a stretching. Rather than studying the properties of the wave functions themselves, we study their symmetry groups. A complex angle indicates that the group contains loxodromic elements. Since the domain of such groups is not the disc, hyperbolic plane geometry cannot be used. Rather, the theory of the isometric circle is adapted since it treats all groups symmetrically. The pairing of circles and their inverses is likened to pairing particles with their antiparticles which then go one to produce nested circles, or a proliferation of particles. A corollary to Laguerre's theorem, which states that the euclidean angle is represented by a pure imaginary projective invariant, represents the imaginary angle in the form of a real projective invariant.

hep-th↗

arXiv2012

The Khuri-Jones Threshold Factor as an Automorphic Function

The Khuri-Jones correction to the partial wave scattering amplitude at threshold is an automorphic function for a dihedron. An expression for the partial wave amplitude is obtained at the pole which the upper half-plane maps on to the interior of semi-infinite strip. The Lehmann ellipse exists below threshold for bound states. As the system goes from below to above threshold, the discrete dihedral (elliptic) group of Type 1 transforms into a Type 3 group, whose loxodromic elements leave the fixed points 0 and $\infty$ invariant. The transformation of the indifferent fixed points from -1 and +1 to the source-sink fixed points 0 and $\infty$ is the result of a finite resonance width in the imaginary component of the angular momentum. The change in symmetry of the groups, and consequently their tessellations, can be used to distinguish bound states from resonances.

physics.gen-ph↗

arXiv2011

Errors in the Bag Model of Strings, and Regge Trajectories Represent the Conservation of Angular Momentum in Hyperbolic Space

The MIT bag model is shown to be wrong because the bag pressure cannot be held constant, and the volume can be fixed in terms of it. The bag derivation of Regge's trajectories is invalidated by an integration of the energy and angular momentum over all values of the radius up to $r_0=c/ω$. This gives the absurd result that "total" angular momentum decreases as the frequency increases. The correct expression for the angular momentum is obtained from hyperbolic geometry of constant negative curvature $r_0$. When the square of the relativistic mass is introduced, it gives a negative intercept which is the Euclidean value of the angular momentum. Regge trajectories are simply statements of the conservation of angular momentum in hyperbolic space. The frequencies and values of the angular momentum are in remarkable agreement with experiment.

physics.gen-ph↗

arXiv2011

What's Wrong With Black Hole Thermodynamics?

Not only is the Bekenstein expression for the entropy of a black hole a convex function of the energy, rather than being a concave function as it must be, it predicts a final equilibrium temperature given by the harmonic mean. This violates the third law, and the principle of maximum work. The property that means are monotonically increasing functions of their argument underscores the error of transferring from temperature means to means in the internal energy when the energy is not a monotonically increasing function of temperature. Whereas the former leads to an increase in entropy, the latter lead to a decrease in entropy thereby violating the second law. The internal energy cannot increase at a slower rate than the temperature itself.

physics.gen-ph↗

arXiv2011

The Lineage of String Theory

The Regge trajectories, upon which string theory is based, behave as rigid rotators rather than vibrating strings. The same relation, between the angular momentum, and the square of the mass, can be found in gravity, the electroweak, and strong interactions. The angle deficit for cosmic strings is shown to be an angle excess that is related to the increase of the circumference of a uniformly rotating disc. Schrödinger's time independent equation with a centrifugal barrier gives an automorphic function that can be constructed as the ratio of its two independent solutions for values of the angular momentum lying outside of their positive, integer values. If the fixed points 0 and $\infty$ in the z-plane correspond to $-i$ and $+i$ in the $ω$-plane, then elliptic substitutions tessellate the $ω$-plane in the form of cresents, while if the fixed points correspond to -1 and +1 in the $ω$-plane are source and sink, just like the lines of force between positive and negative charges. The unit disc undergoes a stretching by a Lorentz transform.

physics.gen-ph↗

arXiv2008

Aberration and radiation pressure in the Klein and Poincare models

Aberration and radiation pressure reflected by a moving mirror are examples of the Klein and Poincaré models of hyperbolic geometry, respectively. Reflection at a moving mirror produces a two-way Dopper shift. Its one-way counterpart, aberration, has nothing to do with the radiation pressure on a moving mirror, but, rather with the pressure on a completely absorbing surface. Both pressures vanish when the angle of parallelism is reached. Two-way, second-order Doppler shifts can be used to establish experimentally the existence of an angle of parallelism.

physics.gen-ph↗

arXiv2008

The relativity of hyperbolic space

The longitudinal Doppler shift is a measure of hyperbolic distance. Transformations of uniform motion are determined by the Doppler shift, while its square root transforms to a uniformly accelerated frame. A time-velocity space metric is derived, by magnifying the Beltrami coordinates with the geometric time, which is similar to the one obtained by Friedmann using Einstein's equations in which the mass tensor describes a universe of dust at zero pressure. No such assumption nor any approximation in which the coordinates increase with time (i.e., constant velocities) need be made. The hyperbolic velocities are related to the sides of a Lambert quadrilateral. In the limit when the acute angle becomes an ideal point, the case of uniform acceleration arises. The relations to Hubble's law, and to the exponential red shift, are discussed.

physics.gen-ph↗

arXiv2008

Hyperbolic nature of uniformly rotating systems and their relation to gravity

Special relativity corresponds to hyperbolic geometry at constant velocity while the so-called general relativity corresponds to hyperbolic geometry of uniformly accelerated systems. Generalized expressions for angular momentum, centrifugal and Coriolis forces are found in hyperbolic space, which reduce to the usual expressions of Euclidean space when the absolute constant tends to infinity. Gravity enters only in the specification of the absolute constant. A uniformly rotating disc corresponds exactly to hyperbolic geometry with a constant negative Gaussian curvature. The angle defect is related to Lorentz contraction of objects normal to the radial direction. Lobachevsky's angle of parallelism accounts for the apparent relativistic distortion of moving objects and would provide a testing ground to measure a positive defect by replacing large distances by high speeds that are comparable with that of light.

gr-qc↗

arXiv2006

Inconsistencies in the MIT bag model of hadrons

It is shown that what is commonly referred to as the MIT `bag' model of hadrons is thermodynamically wrong: The adiabatic conditions between pressure and temperature, and between pressure and volume imply the third, an adiabatic relation between temperature and volume. Consequently, the bag model is destitute of any predictive power since it reduces to a single adiabatic state. The virial theorems proposed by the MIT group are shown to be the result of the normal power density of states of a non-degenerate gas and not the exponential density of states of the Hagedorn mass spectrum. A number of other elementary misconceptions and inaccuracies are also pointed out.

hep-th↗

arXiv2006

The thermodynamics of the Hagedorn mass spectrum

No bootstrap assumption is needed to derive the exponential growth of the Hagedorn hadron mass spectrum: It is a consequence of the second law applied to a relativistic gas, and the relativistic equivalence between inertial mass and its heat content. The Hagedorn temperature occurs in the limit as the number of particles and their internal energy diverge such that their ratio remains constant. The divergences in the $N$ particle entropy, energy, and free energy result when this condition is imposed upon a mixture of ideal gases, one conserving particle number and the other not. The analogy with a droplet in the presence of vapor explains why the pressure of the droplet continues to increase as the temperature rises finally leading to its break up when the Hagedorn temperature is reached. The adiabatic condition relating the particle volume to the Hagedorn temperature is asymptotic. Since it is a limiting temperature, and not a critical one, there can be no phase transition of whatever kind, and the original density of states used to derive such a phase transition is not thermodynamically admissible because its partition function does not exist.

hep-th↗

arXiv2006

The thermodynamics governing 'endoreversible' engines

The thermodynamics of the Curzon-Ahlborn engine, which is a prototype of endoreversible engines, is elucidated. In particular, their criterion for adiabatic equilibration is revised. The so-called irreversibility of endoreversible engines arises from the selection of the coldest reservoir for heat rejection. Rather, if the reservoirs are allowed to come into thermal and mechanical contact, a mean value results which optimizes the work output and heat uptake, and is entirely reversible. The Carnot efficiency cannot be beaten because nothing is as cold as the coldest reservoir.

cond-mat.stat-mech↗

arXiv2005

Geometric Entropies of Mixing (EOM)

Trigonometric and trigonometric-algebraic entropies are introduced. Regularity increases the entropy and the maximal entropy is shown to result when a regular $n$-gon is inscribed in a circle. A regular $n$-gon circumscribing a circle gives the largest entropy reduction, or the smallest change in entropy from the state of maximum entropy which occurs in the asymptotic infinite $n$ limit. EOM are shown to correspond to minimum perimeter and maximum area in the theory of convex bodies, and can be used in the prediction of new inequalities for convex sets. These expressions are shown to be related to the phase functions obtained from the WKB approximation for Bessel and Hermite functions.

cond-mat.stat-mech↗

arXiv2005

Thermodynamics of an ideal generalized gas: I Thermodynamic laws

The equations of state for an ideal generalized gas, like an ideal quantum gas, are expressed in terms of power laws of the temperature. The reduction of an ideal generalized gas to an ideal classical case occurs when the characteristic empirical temperature exponents in the thermal equation of state and in the absolute temperature coincide in contrast to the merger of an ideal quantum gas with an ideal classical gas in the high temperature limit. A corollary to Carnot's theorem is proved asserting that the ratio of the work done over a cycle to the heat absorbed to increase the temperature at constant volume is the same for all bodies at the same volume. As power means, the energy and entropy are incomparable and a new adiabatic potential is introduced by showing that the volume raised to a characteristic exponent is also the integrating factor for the quantity of heat so that the second law can be based on the property that power means are monotonically increasing functions of their order.

cond-mat.stat-mech↗

arXiv2005

Thermodynamics of an ideal generalized gas:II Means of order $α$

The property that power means are monotonically increasing functions of their order is shown to be the basis of the second laws not only for processes involving heat conduction but also for processes involving deformations. In an $L$-potentail equilibration the final state will be one of maximum entropy, while in an entropy equilibrium the final state will be one of minimum $L$. A metric space is connected with the power means, and the distance between means of different order is related to the Carnot efficiency. In the ideal classical gas limit, the average change in the entropy is shown to be proportional to the difference between the Shannon and Rényi entropies for nonextensive systems that are multifractal in nature. The $L$-potential, like the internal energy, is a Schur convex function of the empirical temperature, which satisfies Jensen's inequality, and serves as a measure of the tendency to uniformity in processes involving pure thermal conduction.

cond-mat.stat-mech↗

arXiv2005

Entropies of Mixing and the Lorenz Order

Entropies of mixing can be derived directly from the parent distributions of extreme value theory. They correspond to pseudo-additive entropies in the case of Pareto and power function distributions, while to the Shannon entropy in the case of the exponential distribution.The former tend to the latter when their shape parameters tend to infinity and zero, respectively. Hence processes whose entropies of mixing are pseudo-additive entropies majorize, in the Lorenz order sense, those whose entropy is the Shannon entropy. In the case of the arcsine distribution, maximal properties of regular polygons correspond to maximum entropy of mixing.

cond-mat.stat-mech↗

arXiv2004

Thermodynamics of means

Thermodynamics of power means applies to an ideal quantum gas which may be nonextensive. Transition to an ideal classical gas occurs when the empirical temperature exponents of the internal energy and absolute temperature coalesce. Limiting processes are pure heat conduction and pure deformations. Largest and smallest mean final volumes occur for isothermal and adiabatic processes, respectively. The increment in the heat admits two integrating factors which yield conserved quantities for adiabatic processes. Energy-conserving equilibrations yield the largest final means possible, while the second law follows from the property that the power means are monotonically increasing functions of their order. In the ideal classical gas limit, the change in the average entropy is proportional to the difference between the Shannon and Rényi entropies for nonextensive, isothermal systems that are multifractal in nature.

cond-mat.stat-mech↗

arXiv2004

Fundamental inconsistencies of 'superstatistics'

Fundamental inconsistencies of superstatistics are highlighted. There is no such thing as a superposition of Boltzmann factors; what is actually derived is a generating function and not a normalizable probability density. The beta density of the second kind for the energy is shown to be subordinated to a $χ^2$-probability distribution, and this provides a power-tail distribution for lage values of the energy, comparable to a Pareto distribution. The authors' claim that the Laplace method will only work on the inverse Laplace transform in the opposite limit of small energies is shown to be incorrect. The exact expression is obtained by contour integration at a multipole located at the origin corresponding to the high temperature limit.

cond-mat.stat-mech↗