S. R. Valluri

The Lambert W function and quantum statistics

We present some applications of the Lambert W function (W function) to the formalism of quantum statistics (QS). We consider the problem of finding extrema in terms of energy for a general QS distribution, which involves the solution of a transcendental equation in terms of the W function. We then present some applications of this formula including Bose–Einstein systems in d dimensions, Maxwell–Boltzmann systems, and black body radiation. We also show that for the appropriate parameter values, this formula reduces to an analytic expression in connection with Wien’s displacement law that was found in a previous study. In addition, we show that for Maxwell–Boltzmann and Bose–Einstein systems, the W function allows us to express the temperature of the system as a function of the thermodynamically relevant chemical potential, the particle density, and other parameters. Finally, we explore an indirect relationship of the W function to the polylogarithm function and to the Lambert transform.

QED Effective Action Revisited

The derivation of a convergent series representation for the quantum electrodynamic effective action obtained by two of us (S.R.V. and D.R.L.) in [Can. J. Phys. 71, 389 (1993)] is reexamined. We pr...

D-dimensional Bose gases and the Lambert W function

The applications of the Lambert W function (also known as the W function) to D-dimensional Bose gases are presented. We introduce two sets of families of logarithmic transcendental equations that occur frequently in thermodynamics and statistical mechanics and present their solution in terms of the W function. The low temperature T behavior of free ideal Bose gases is considered in three and four dimensions. It is shown that near condensation in four dimensions, the chemical potential μ and pressure P can be expressed in terms of T through the W function. The low T behavior of one- and two-dimensional ideal Bose gases in a harmonic trap is studied. In 1D, the W function is used to express the condensate temperature, TC, in terms of the number of particles N; in 2D, it is used to express μ in terms of T. In the low T limit of the 1D hard-core and the 3D Bose gas, T can be expressed in terms of P and μ through the W function. Our analysis allows for the possibility to consider μ, T, and P as complex variabl...

A study of the orbits of the logarithmic potential for galaxies

The logarithmic potential is of great interest and relevance in the study of the dynamics of galaxies. Some small corrections to the work of Contopoulos & Seimenis who used the method of Prendergast to find periodic orbits and bifurcations within such a potential are presented. The solution of the orbital radial equation for the purely radial logarithmic potential is then considered using the precessing ellipse (p-ellipse) method pioneered by Struck. This differential orbital equation is a special case of the generalized Burgers equation. The apsidal angle is also determined, both numerically and analytically by means of the LambertW and the polylogarithmic functions. The use of these functions in computing the gravitational lensing produced by logarithmic potentials is discussed.

Intersection volumes and surface areas of cylinders for geometrical modelling and tolerancing

Abstract Solid geometric modelling is an important medium for communicating part shape-geometry information, and it encourages the study of solid volumes and surface areas. Growing interest in solid-modelling applications which use surface and volume intersection methods is becoming evident. In their previous work, the authors of the paper examined variations of primitive solids to create a means of representing geometric tolerances in a solid modeller. The paper examines the intersection of two cylinders with reference to the parametric tolerance zones that are associated with size, position and orientation tolerances. Intersection surface areas and volumes are examined. This information can be used not only in the building of models with Boolean operations on primitives, but also in the checking of the interference between mating parts which can arise in fittings of various sorts and functional tolerancing. Because of the likely computer implementation of surface-area or volume calculations relating to the intersection of primitives, the inclusion of faceted solids and methods of dealing with solid primitives are considered. The intersection of two cylinders of arbitrary radii at an arbitrary angle, in a complex steam-trap and flash-tank pipe network, is imaged.

The gravitational wave pulsar signal with jovian and lunar perturbations and orbital eccentricity corrections

In this work, we present the analytic treatment of the Fourier transform (FT) of the gravitational wave (GW) signal from a pulsar in a parametrized model discussed by Brady et al (1998 Phys. Rev. D 57 2101). The formalism lends itself to a development of the FT that accounts for the corrections due to the orbital eccentricity, and the Jovian and lunar perturbations in terms of well-known special functions.

Tutorial: The quantum finite square well and the Lambert W function

We present a solution of the quantum mechanics problem of the allowable energy levels of a bound particle in a one-dimensional finite square well. The method is a geometric-analytic technique utilizing the conformal mapping w → z = wew between two complex domains. The solution of the finite square well problem can be seen to be described by the images of simple geometric shapes, lines, and circles, under this map and its inverse image. The technique can also be described using the Lambert W function. One can work in either of the complex domains, thereby obtaining additional insight into the finite square well problem and its bound energy states. This suggests interesting possibilities for the design of materials that are sensitive to minute changes in their environment such as nanostructures and the quantum well infrared photodetector.

A study of the gravitational wave form from pulsars

We present analytical and numerical studies of the Fourier transform (FT) of the gravitational wave (GW) signal from a pulsar, taking into account the rotation and orbital motion of the Earth. We also briefly discuss the Zak– Gelfand integral transform and a special class of the generalized hypergeometric function of potential relevance. The Zak–Gelfand integral transform that arises in our analytic approach has also been useful for Schr¨ odinger operators in periodic potentials in condensed matter physics (Bloch wavefunctions) and holds promise for the study of periodic GW signals for long integration times.

The study of the Heisenberg‐Euler Lagrangian and some of its applications

The Heisenberg‐Euler Lagrangian (HEL) is not only a topic of fundamental interest, but also has a rich variety of diverse applications in astrophysics, nonlinear optics and elementary particle physics etc. We discuss the series representation of this Lagrangian and a few of its applications in this study.

The Carter Constant for Inclined Orbits about a Massive Kerr Black Hole: I. Circular Orbits

In an extreme binary black hole system, an orbit will increase its angle of inclination ( ) as it evolves in Kerr spacetime. We focus our attention on the behaviour of the Carter constant (Q) for near-polar orbits; and develop an analysis that is independent of and complements radiation reaction models. For a Schwarzschild black hole, the polar orbits represent the abutment between the prograde and retrograde orbits at which Q is at its maximum value for given values of latus rectum ( ~) and eccentricity (e). The introduction of spin ( ~ S = jJj=M 2 ) to the massive black hole causes this boundary, or abutment, to be moved towards greater orbital inclination; thus it no longer cleanly separates prograde and retrograde orbits. To characterise the abutment of a Kerr black hole (KBH), we first investigated the last stable orbit (LSO) of a test-particle about a KBH, and then extended this work to general orbits. To develop a better understanding of the evolution of Q we developed analytical formulae for Q in terms of ~, e, and ~ S to describe elliptical orbits at the abutment, polar orbits, and last stable orbits (LSO). By knowing the analytical form of @Q=@~ at the abutment, we were able to test a 2PN flux equation for Q. We also used these formulae to numerically calculate the @=@ ~ of hypothetical circular orbits that evolve along the abutment. From these values we have determined that@=@ ~ = 122:7 ~