Number Of Physical Applications Research Articles

Spectral properties of the generalized diluted Wishart ensemble

The celebrated Marčenko–Pastur law, that considers the asymptotic spectral density of random covariance matrices, has found a great number of applications in physics, biology, economics, engineering, among others. Here, using techniques from statistical mechanics of spin glasses, we derive simple formulas concerning the spectral density of generalized diluted Wishart matrices. These are defined as , where X and Y are diluted N × P rectangular matrices, whose entries correspond to the links of doubly-weighted random bipartite Poissonian graphs following the distribution , with the probability density ϱ(x, y) controlling the correlation between the matrices entries of X and Y . Our results cover several interesting cases by varying the parameters of the matrix ensemble, namely, the dilution of the graph d, the rectangularity of the matrices α = N/P, and the degree of correlation of the matrix entries via the density ϱ(x, y). Finally, we compare our findings to numerical diagonalisation showing excellent agreement.

Ultrahigh detection sensitivity exceeding 105 V/W in spin-torque diode

Microwave detection has a huge number of applications in physics and engineering. It has already been shown that biased spin torque diodes have performance overcoming the CMOS counterpart in terms of sensitivity. In this regard, the spin torque diodes are promising candidates for the next generation of microwave detectors. Here, we show that the optimization of the rectification process based on the injection locking mechanism gives an ultrahigh sensitivity exceeding 200 kV/W with an output resistance below 1 kΩ while maintaining the advantages over other mechanisms such as vortex expulsion or non-linear resonance, to work without a bias magnetic field.

Like-charge attraction in a one-dimensional setting: the importance of being odd

From cement cohesion to DNA condensation, a proper statistical physics treatment of systems with long-range forces is important for a number of applications in physics, chemistry, and biology. We compute here the effective force between fixed charged macromolecules, screened by oppositely charged mobile ions (counterions). We treat the problem in a one-dimensional configuration that allows for interesting discussion and derivation of exact results, remaining at a level of mathematical difficulty compatible with an undergraduate course. Emphasis is put on the counterintuitive but fundamental phenomenon of like-charge attraction, which our treatment brings for the first time to the level of undergraduate teaching. The parity of the number of counterions is shown to play a prominent role, which sheds light on the binding mechanism at work when like-charge macromolecules do attract.

Conical : An extended module for computing a numerically satisfactory pair of solutions of the differential equation for conical functions

Conical functions appear in a large number of applications in physics and engineering. In this paper we describe an extension of our module Conical (Gil et al., 2012) for the computation of conical functions. Specifically, the module includes now a routine for computing the function R−12+iτm(x), a real-valued numerically satisfactory companion of the function P−12+iτm(x) for x>1. In this way, a natural basis for solving Dirichlet problems bounded by conical domains is provided.The module also improves the performance of our previous algorithm for the conical function P−12+iτm(x) and it includes now the computation of the first order derivative of the function. This is also considered for the function R−12+iτm(x) in the extended algorithm. Program summaryProgram Title: Module ConicalProgram Files doi:http://dx.doi.org/10.17632/rpw5d8gdkg.1Licensing provisions: CC by 4.0Programming language: Fortran 90External routines/libraries: The module Conical uses a Fortran 90 version of the routine dkia (developed by the authors) for computing the modified Bessel functions Kia(x) and its derivative. This routine is available at http://toms.calgo.org.Nature of problem: These functions are the natural function basis for solving, for example, the Laplace’s problem in spherical coordinates for two intersecting cones or for regions bounded by two intersecting spheres, or by one or two confocal hyperboloids of revolution when using toroidal coordinates. The conical function P−12+iτm(x) is also used in the Mehler–Fock integral transform for problems in potential and heat theory,Solution method: The algorithm uses different methods of computation depending on the function under consideration (P−12+iτm(x) or R−12+iτm(x)) and the values of x, τ and m: numerical quadrature, asymptotic expansions in terms of elementary functions, asymptotic expansions in terms of Bessel functions, asymptotic expansions for tau large and backward/forward recursion of three-term recurrence relations.Restrictions: In order to avoid underflow/overflow problems in standard IEEE double precision arithmetic, the admissible parameter ranges for computing the conical function P−12+iτm(x) in the routine conicp are: −1<x<1,0<τ<=100,0≤m≤401<x≤100,0<τ≤100,0≤m≤100.When using the routines conicr and conicpr, the admissible parameter ranges for computing the functions P−12+iτm(x) and R−12+iτm(x) are 1<x≤100,0<τ≤100,0≤m≤100.

Efficient computation of Laguerre polynomials

An efficient algorithm and a Fortran 90 module (LaguerrePol) for computing Laguerre polynomials Ln(α)(z) are presented. The standard three-term recurrence relation satisfied by the polynomials and different types of asymptotic expansions valid for n large and α small, are used depending on the parameter region.Based on tests of contiguous relations in the parameter α and the degree n satisfied by the polynomials, we claim that a relative accuracy close to or better than 10−12 can be obtained using the module LaguerrePol for computing the functions Ln(α)(z) in the parameter range z≥0, −1<α≤5, n≥0. Program summaryProgram Title: Module LaguerrePolProgram Files doi:http://dx.doi.org/10.17632/3jkk659cn8.1Licensing provisions: CC by 4.0Programming language: Fortran 90Nature of problem: Laguerre polynomials Ln(α)(z) appear in a vast number of applications in physics, such as quantum mechanics, plasma physics, etc.Solution method: The algorithm uses asymptotic expansions or recurrence relations for computing the function values depending on the range of parameters.Restrictions: The admissible input parameter ranges for computing the Laguerre Ln(α)(z) are z≥0, −1<α≤5, n≥0.

Handling large variations in mechanics: Some applications

There is a need to use probability distributions with power-law decaying tails to describe the large variations exhibited by some of the physical phenomena. The Weierstrass Random Walk (WRW) shows promise for modeling such phenomena. The theory of anomalous diffusion is now well established. It has found number of applications in Physics, Chemistry and Biology. However, its applications are limited in structural mechanics in general, and structural engineering in particular. The aim of this paper is to present some mathematical preliminaries related to WRW that would help in possible applications. In the limiting case, it represents a diffusion process whose evolution is governed by a fractional partial differential equation. Three applications of superdiffusion processes in mechanics, illustrating their effectiveness in handling large variations, are presented.

Doubly periodic meromorphic solutions of autonomous nonlinear differential equations

The problem of constructing and classifying elliptic solutions of nonlinear differential equations is studied. An effective method enabling one to find any elliptic solution of an autonomous nonlinear ordinary differential equation is described. The method does not require integrating additional differential equations. Much attention is being paid to the case of elliptic solutions with several poles in a parallelogram of periods. With the help of the method we find elliptic solutions up to the fourth order inclusively of an ordinary differential equation with a number of physical applications. The method admits a natural generalization and can be used to find elliptic solutions satisfying systems of ordinary differential equations.

Doubly Periodic Meromorphic Solutions of Autonomous Nonlinear Differential Equations

The problem of constructing and classifying elliptic solutions of nonlinear differential equations is studied. An effective method enabling one to find an elliptic solution of an autonomous nonlinear ordinary differential equation is described. The method does not require integrating additional differential equations. Much attention is paid to the case of elliptic solutions with several poles inside a parallelogram of periods. With the help of the method we find elliptic solutions up to the fourth order inclusively of an ordinary differential equation with a number of physical applications. The method admits a natural generalization and can be used to find elliptic solutions satisfying systems of ordinary differential equations.

Lattice quantum chromodynamics at the physical point and beyond

We review the work of the PACS-CS Collaboration, which aimed to realize lattice quantum chromodynamics (QCD) calculations at the physical point, i.e., those with quark masses set at physical values. This has been a long-term goal of lattice QCD simulation since its inception in 1979. After reviewing the algorithmic progress, which played a key role in this development, we summarize the simulations that explored the quark mass dependence of hadron masses down to values close to the physical point. In addition to allowing a reliable determination of the light hadron mass spectrum, this work provided clues on the validity range of chiral perturbation theory, which is widely used in phenomenology. We then describe the application of the technique of quark determinant reweighting, which enables lattice QCD calculations exactly on the physical point. The physical quark masses and the strong coupling constants are fundamental constants of the strong interaction. We describe a non-perturbative Schrödinger functional approach to figure out the non-perturbative renormalization needed to calculate them. There are a number of physical applications that can benefit from lattice QCD calculations carried out either near or at the physical point. We take up three illustrative examples: calculation of the physical properties of the ρ meson as a resonance, the electromagnetic form factor and charge radius of the pion, and charmed meson spectroscopy. Bringing single hadron properties under control opens up a number of new areas for serious lattice QCD research. One such area is electromagnetic effects in hadronic properties. We discuss the combined QCD plus QED simulation strategy and present results on electromagnetic mass difference. Another area is multi-hadron states, or nuclei. We discuss the motivations and difficulties in this area, and describe our work for deuteron and helium as our initial playground. We conclude with a brief discussion on the future perspective of lattice QCD.

The residence time equation

The residence time of a diffusing particle in a prescribed spatial region recently finds an increasing number of applications in physics and chemistry. A partial differential equation for the residence time moments is derived, as a generalization of the ordinary differential equation for the first passage time moments. When one seeks to calculate only the mean residence time and not its full distribution, this “residence time equation” constitutes a significant simplification over the conventional Feynman–Kac approach. We demonstrate this for a ball in d-dimensional space with an infinite observation time. The present approach may be useful also for other Brownian functionals.

Kundt spacetimes

Kundt spacetimes are of great importance in general relativity in four dimensions and have a number of physical applications in higher dimensions in the context of string theory. The degenerate Kundt spacetimes have many special and unique mathematical properties, including their invariant curvature structure and their holonomy structure. We provide a rigorous geometrical kinematical definition of the general Kundt spacetime in four dimensions; essentially a Kundt spacetime is defined as one admitting a null vector that is geodesic, expansion-free, shear-free and twist-free. A Kundt spacetime is said to be degenerate if the preferred kinematic and curvature null frames are all aligned. The degenerate Kundt spacetimes are the only spacetimes in four dimensions that are not -non-degenerate, so that they are not determined by their scalar polynomial curvature invariants. We first discuss the non-aligned Kundt spacetimes, and then turn our attention to the degenerate Kundt spacetimes. The degenerate Kundt spacetimes are classified algebraically by the Riemann tensor and its covariant derivatives in the aligned kinematic frame; as an example, we classify Riemann type D degenerate Kundt spacetimes in which ∇(Riem), ∇(2)(Riem) are also of type D. We discuss other local characteristics of the degenerate Kundt spacetimes. Finally, we discuss degenerate Kundt spacetimes in higher dimensions.

Spectral asymptotics in eigenvalue problems with nonlinear dependence on the spectral parameter

We study the asymptotic distribution of eigenvalues ω of a quadratic operator polynomial of the form (ω2 − L(ω))ϕω = 0, where L(ω) is a second-order differential positive elliptic operator with quadratic dependence on the spectral parameter ω. We derive asymptotics of the spectral density in this problem and show how to compute coefficients of its asymptotic expansion from the coefficients of the asymptotic expansion of the trace of the heat kernel of L(ω). The leading term in the spectral asymptotics is the same as that for a Laplacian in a cavity. The results have a number of physical applications. We illustrate them by examples of field equations in external stationary gravitational and gauge backgrounds.

New Exact Solutions to the Fujita's Equation

New exact solutions to the Fujita's equation, which has quite a large number of applications in physics and mathematics, are obtained by the generalized conditional symmetry approach. The behavior of these solutions is illustrated graphically with particular reference to the phenomena of blow-up and asymptotical behavior.

Transport equations with singularity

Test particle dynamics in the vicinity of singularities of a scattering medium are important for a number of physical applications, one of which is the coupling of kinetic and continuum equations in numerical simulations. Out of several relevant questions arising in this context, the paper concerns boundary conditions for the related hyperbolic system of equations in a slab between two singularities. For a linear model problem, we investigate first exit times from the slab and give a complete characterization of the stochastic particle dynamics, which provides a classification of the hyperbolic systems into Cauchy problems and into those which have to be supplemented with boundary conditions. Properties of the corresponding process like ergodicity, recurrence and asymptotic behaviour are investigated.

Symmetry groups and separation of variables of a class of nonlinear diffusion-convection equations

In this paper, a class of nonlinear diffusion-convection equations, ut = (D(u)uxn)x+P(u)ux , which has quite a large number of physical applications, is analysed by using symmetry group methods which include the classical method, the potential symmetry method and the generalized conditional symmetry method. A complete classification of the functional forms of the diffusion and convection coefficients is presented when the equation admits Lie's point symmetry groups and potential symmetry groups. The separation of variables for the equation is investigated using the generalized conditional symmetry approach. For some interesting cases, exact solutions using the method of separation of variables are discussed in detail.

Energy formulation for electronic, optical and acoustical applications including interfacial properties and irreversible processes

A non-integrable energy formulation lying in the framework of continuum physics with electromagnetic interactions and including interfacial properties is presented. Non-integrability is required in a number of physical applications and particularly when irreversible processes occur. The central concept is that of duality between geometrical and dynamical fields; it allows the distinction between virtual and real states, as well as the use of invariance principles. Here, three invariance principles are basic to the theory: scale-change invariance related to the extension of continuous functions to discontinuous ones, then gauge and rotational invariances associated with electromagnetism and mechanical systems, respectively. The need for such an approach is shown on physical examples where the classical energy and vectorial formulations cease to apply. In brief, after providing a systematic construction of the method in a Galilean and then a Lorentzian framework, the formulation is extended to electromagnetomechanical interactions including interfacial properties and irreversible processes.

The diffraction of SH waves by an arbitrary shaped crack in two dimensions

In this paper we consider the two-dimensional scalar scattering problem for Helmholtz’s equation exterior to a smooth open arc of general shape. The problem has a number of physical applications including the diffraction of sound by a rigid barrier immersed in a compressible fluid and by a crack in an elastic solid which supports a state of anti-plane strain (SH-motion). The mathematical method used here is the crack Green function method introduced by G. R. Wickham. This enables the scattering problem to be reduced to the solution of a Fredholm integral equation of the second kind with a continuous kernel. The numerical solution of this equation is discussed and a number of examples are computed.

Viscous fingering in Hele-Shaw cells

The phenomenon of interfacial motion between two immiscible viscous fluids in the narrow gap between two parallel plates (Hele-Shaw cell) is considered. This flow is currently of interest because of its relation to pattern selection mechanisms and the formation of fractal, structures in a number of physical applications. Attention is concentrated on the fingers that result from the instability when a less-viscous fluid drives a more-viscous one. The status of the problem is reviewed and progress with the thirty-year-old problem of explaining the shape and stability of the fingers is described. The paradoxes and controversies are both mathematical and physical. Theoretical results on the structure and stability of steady shapes are presented for a particular formulation of the boundary conditions at the interface and compared with the experimental phenomenon. Alternative boundary conditions and future approaches are discussed.

Solution of nonlinear boundary value problems—VIII. Evaluation of branching points based on shooting method and GPM technique

General straightforward method for finding branching points of nonlinear boundary value problems is presented. The technique proposed utilizes the equations used in the GPM method. The method can be applied to problems arising in a number of physical applications, the technique is tailored to diffusion and heat conduction problems.

Dual, crossing-symmetric representations with finite-width resonances and Regge asymptotic behaviour

A class of dual, crossing-symmetric representations having finite-width resonances and Regge asymptotic behaviour for all physi cally attainable regions of the crossed-channel variables is obtained. One of these representations is examined in detail. A number of physical applications are also presented including an explanation of the Dalitzplot distribution for\(\bar pn \to 3\pi \), the behaviour of the most important ππ phase shifts and a determination of the ππS-wave scattering lengths.