E. Capelas de Oliveira

Models based on Mittag-Leffler functions for anomalous relaxation in dielectrics

Abstract.We revisit the Mittag-Leffler functions of a real variable t, with one, two and three order-parameters {α,β,γ}, as far as their Laplace transform pairs and complete monotonicity properties are concerned. These functions, subjected to the requirement to be completely monotone for t > 0, are shown to be suitable models for non–Debye relaxation phenomena in dielectrics including as particular cases the classical models referred to as Cole–Cole, Davidson–Cole and Havriliak–Negami. We show 3D plots of the relaxations functions and of the corresponding spectral distributions, keeping fixed one of the three order-parameters.

Covariant, algebraic, and operator spinors

We deal with three different definitions for spinors: (I) thecovariant definition, where a particular kind ofcovariant spinor (c-spinor) is a set of complex variables defined by its transformations under a particular spin group; (II) theideal definition, where a particular kind of algebraic spinor (e-spinor) is defined as an element of a lateral ideal defined by the idempotente in an appropriated real Clifford algebra ℝp,q (whene is primitive we writea-spinor instead ofe-spinor); (III) the operator definition where a particular kind of operator spinor (o-spinor) is a Clifford number in an appropriate Clifford algebra ℝp,q determining a set of tensors by bilinear mappings. By introducing the concept of “spinorial metric” in the space of minimal ideals ofa-spinors, we prove that forp+q≤5 there exists an equivalence from the group-theoretic point of view among covariant and algebraic spinors. We also study in which senseo-spinors are equivalent toc-spinors. Our approach contain the following important physical cases: Pauli, Dirac, Majorana, dotted, and undotted two-component spinors (Weyl spinors). Moreover, the explicit representation of thesec-spinors asa-spinors permits us to obtain a new approach for the spinor structure of space-time and to represent Dirac and Maxwell equations in the Clifford and spin-Clifford bundles over space-time.

Differentiation to fractional orders and the fractional telegraph equation

Using methods of differential and integral calculus, we present and discuss the calculation of a fractional Green function associated with the one-dimensional case of the so-called general fractional telegraph equation with one space variable. This is a fractional partial differential equation with constant coefficients. The equation is solved by means of juxtaposition of transforms, i.e., we introduce the Laplace transform in the time variable and the Fourier transform in the space variable. Several particular cases are discussed in terms of the parameters. Some known results are recovered. As a by-product of our main result, we obtain two new relations involving the two-parameter Mittag–Leffler function.

Dirac and Maxwell equations in the Clifford and spin-Clifford bundles

We show how to write the Dirac and the generalized Maxwell equations (including monopoles) in the Clifford and spin-Clifford bundles (of differential forms) over space-time (either of signaturep=1,q=3 orp=3,q=1). In our approach Dirac and Maxwell fields are represented by objects of the same mathematical nature and the Dirac and Maxwell equations can then be directly compared. We show also that all presentations of the Maxwell equations in (matrix) Dirac-like “spinor” form appearing in the literature can be obtained by choosing particular global idempotents in the bundles referred to above. We investigate also the transformation laws under the action of the Lorentz group of Dirac and Maxwell fields (defined as algebraic spinor sections of the Clifford or spin-Clifford bundles), clearing up several misunderstandings and misconceptions found in the literature. Among the many new results, we exhibit a factorization of the Maxwell field into two-component spinor fields (Weyl spinors), which is important.

Tunneling in fractional quantum mechanics

We study tunneling through delta and double delta potentials in fractional quantum mechanics. After solving the fractional Schrodinger equation for these potentials, we calculate the corresponding reflection and transmission coefficients. These coefficients have a very interesting behavior. In particular, we can have zero energy tunneling when the order of the Riesz fractional derivative is different from 2. For both potentials, the zero energy limit of the transmission coefficient is given by , where α is the order of the derivative (1 < α ≤ 2).

On anomalous diffusion and the fractional generalized Langevin equation for a harmonic oscillator

The fractional generalized Langevin equation (FGLE) is proposed to discuss the anomalous diffusive behavior of a harmonic oscillator driven by a two-parameter Mittag–Leffler noise. The solution of this FGLE is discussed by means of the Laplace transform methodology and the kernels are presented in terms of the three-parameter Mittag–Leffler functions. Recent results associated with a generalized Langevin equation are recovered.

On some fractional Green's functions

In this paper we discuss some fractional Green’s functions associated with the fractional differential equations which appear in several fields of science, more precisely, the so-called wave reaction-diffusion equation and some of its particular cases. The methodology presented is the juxtaposition of integral transforms, in particular, the Laplace and the Fourier integral transforms. Some recent results involving the reaction-diffusion equation are pointed out.

Solution of the fractional Langevin equation and the Mittag–Leffler functions

We introduce the fractional generalized Langevin equation in the absence of a deterministic field, with two deterministic conditions for a particle with unitary mass, i.e., an initial condition and an initial velocity are considered. For a particular correlation function, that characterizes the physical process, and using the methodology of the Laplace transform, we obtain the solution in terms of the three-parameter Mittag–Leffler function. As particular cases, some recent results are also presented.

Superluminal electromagnetic waves in free space

Recently it has been shown that all relativistic wave equations possess families of undistorted progressive waves (UPWs) which can travel with arbitrary speeds 0 ≤ v < ∞. In this paper we present the theory of how to generate UPW solutions of the Maxwell equation and discuss the particular case of the superluminal electromagnetic X-wave (SEXW), clarifying its extraordinary properties. The theory of how it is possible to launch finite aperture approximations for the SEXW in free space is also discussed in detail. The theory is illustrated by computer simulations showing the birth of a finite aperture approximation for a SEXW and its superluminal propagation without appreciable distortion up to a 100 km. We discuss also the experimental evidence available and discuss if SEXWs can be used to transmit information.

Thoughtful comments on ‘Bessel beams and signal propagation’

Abstract In this Letter we present thoughtful comments on the paper ‘Bessel beams and signal propagation’ showing that the main claims of that paper are wrong. Moreover, we take the opportunity to show the nontrivial and indeed surprising result that a scalar pulse (i.e., a wave train of compact support in the time domain) that is solution of the homogeneous wave equation (vector ( E → , B → ) pulse that is solution of Maxwell equations) is such that its wave front in some cases does travel with speed greater than c, the speed of light. In order for a pulse to possess a front that travels with speed c, an additional condition must be satisfied, namely the pulse must have finite energy. When this condition is fulfilled the pulse still can show peaks propagating with superluminal (or subluminal) velocities, but now its wave front travels at speed c. These results are important because they explain several experimental results obtained in recent experiments, where superluminal velocities have been observed, without implying in any breakdown of the principle of relativity.