Prabhakar-like fractional viscoelasticity

The aim of this paper is to present a linear viscoelastic model based on Prabhakar fractional operators. In particular, we propose a modification of the classical fractional Maxwell model, in which we replace the Caputo derivative with the Prabhakar one. Furthermore, we also discuss how to recover a formal equivalence between the new model and the known classical models of linear viscoelasticity by means of a suitable choice of the parameters in the Prabhakar derivative. Moreover, we also underline an interesting connection between the theory of Prabhakar fractional integrals and the recently introduced Caputo-Fabrizio differential operator.Comment: 9 page

Time-like definition of quaternions in exterior algebra

A formal description of quaternions by means of exterior calculus is provided. Considering a three-dimensional space-time characterized by having three time coordinates, we have been able to consistently recover a suitable formulation of quaternions by means of the properties arising from exterior algebra and calculus. As an application, it is also illustrated how rotations may be written in terms of quaternions according to the exterior-algebraic notation.Comment: 7 page

On the propagation of transient waves in a viscoelastic Bessel medium

In this paper we discuss the uniaxial propagation of transient waves within a semi-infinite viscoelastic Bessel medium. First, we provide the analytic expression for the response function of the material as we approach the wave-front. To do so, we take profit of a revisited version of the so called Buchen-Mainardi algorithm. Secondly, we provide an analytic expression for the long time behavior of the response function of the material. This result is obtained by means of the Tauberian theorems for the Laplace transform. Finally, we relate the obtained results to a peculiar model for fluid-filled elastic tubes.Comment: 14 pages, 4 figure

A class of linear viscoelastic models based on Bessel functions

In this paper we investigate a general class of linear viscoelastic models whose creep and relaxation memory functions are expressed in Laplace domain by suitable ratios of modified Bessel functions of contiguous order. In time domain these functions are shown to be expressed by Dirichlet series (that is infinite Prony series). It follows that the corresponding creep compliance and relaxation modulus turn out to be characterized by infinite discrete spectra of retardation and relaxation time respectively. As a matter of fact, we get a class of viscoelastic models depending on a real parameter $\nu > -1$. Such models exhibit rheological properties akin to those of a fractional Maxwell model (of order $1/2$) for short times and of a standard Maxwell model for long times.Comment: 13 pages, 8 figure

A one parameter class of Fractional Maxwell-like models

In this paper we discuss a one parameter modification of the well known fractional Maxwell model of viscoelasticity. Such models appear to be particularly interesting because they describe the short time asymptotic limit of a more general class of viscoelastic models known in the literature as Bessel models.Comment: 8 pages, 4 figure

An Introduction to Space-Time Exterior Calculus

The basic concepts of exterior calculus for space-time multivectors are presented: interior and exterior products, interior and exterior derivatives, oriented integrals over hypersurfaces, circulation and flux of multivector fields. Two Stokes theorems relating the exterior and interior derivatives with circulation and flux respectively are derived. As an application, it is shown how the exterior-calculus space-time formulation of the electromagnetic Maxwell equations and Lorentz force recovers the standard vector-calculus formulations, in both differential and integral forms.Comment: 19 pages, 5 figure

Storage and Dissipation of Energy in Prabhakar Viscoelasticity

In this paper, after a brief review of the physical notion of quality factor in viscoelasticity, we present a complete discussion of the attenuation processes emerging in the Maxwell-Prabhakar model, recently developed by Giusti and Colombaro. Then, taking profit of some illuminating plots, we discuss some potential connections between the presented model and the modern mathematical modelling of seismic processes

Generalized exterior-algebraic electromagnetism in (k, n)-dimensional spacetime

This doctoral thesis aims to find a connection between two different disciplines. On the one hand, the electromagnetic theory, one of the most known and most applied theories in physics, properly described by the famous Maxwell equations. On the other hand, the theory of information and communication, provided of a mathematical structure which mainly includes the concepts of probability and statistics. In order to establish a contact point between the two, we first decided to develop a suitable mathematical framework, which could accomodate the two theories in the appropriate context. We therefore chose to use the mathematical theory of exterior algebra, because it allows to combine a simple and intuitive method coming from a classical vector conception, with more advanced but equally effective mathematical tools. Having to build a theory from the beginning, we have opted to consider a procedure as general as possible and, therefore, we proceed in a space-time with arbitrary dimensions, both as regards time and as regards space. We formulate our theory in this space-time, utilizing multivector fields, also of arbitrary degree, in order to broaden the classical concept of vector field. The electromagnetic theory is thus generalized through these multivectors and the fact of having several free parameters, such as the dimensions of space-time and the grade of the multivectoral field, allows to identify various models, obtaining the known ones and opening the doors to new horizons. In practice, to build our theory we can follow two distinct but complementary approaches. In the first place, making an analogy with the classical theory, we can directly use the generalized definitions of exterior algebra to postulate a natural extension of the electromagnetic theory in arbitrary dimensions. Secondly, we have developed in parallel a dynamical theory, so-called Lagrangian, purposely built for multivector fields of arbitrary grade. Regardless of the path chosen, we have obtained a consistent theory that presents its equations of motion, corresponding to the generalized Maxwell equations, and all the equivalent physical quantities resulting from new conservation laws, which identify the quantities of the system that remain unchanged, such as energy and momentum. The connection point with the theory of communication emerges in dealing with the electromagnetic waves coming from the solutions of Maxwell equations. Studied from the multivectoral point of view of exterior algebra, these waves might open the doors to a new interpretation of the transmission of signals from a different perspective.Esta tesis doctoral tiene el objetivo de encontrar un vínculo entre dos disciplinas diferentes. Por un lado, la teoría electromagnética, una de las teorías más conocidas y aplicadas de la física y debidamente descrita por las famosas ecuaciones de Maxwell. Por otro lado, la teoría de la información y de la comunicación, dotada de una estructura matemática que comprende mayormente los conceptos de probabilidad y estadística. Para establecer un punto de encuentro entre las dos, primero decidimos desarrollar una estructura matemática apropiada, que pudiera conciliar las dos teorías en el contexto adecuado. Por lo tanto, decidimos utilizar la teoría matemática del álgebra exterior, porque es capaz de combinar un método simple e intuitivo proveniente de una concepción vectorial clásica, con herramientas matemáticas más avanzadas pero igualmente efectivas. Al tener que construir una teoría desde el principio, hemos optado por considerar un tratamiento lo más general posible y, por tanto, procedemos en un espacio-tiempo con dimensiones arbitrarias, tanto en el tiempo como en el espacio. Construimos nuestra teoría en este espacio-tiempo, basándonos en campos multivectoriales, también de grado arbitrario, para ampliar el concepto clásico de campo vectorial. La teoría electromagnética se generaliza a través de estos multivectores y el hecho de tener varios parámetros libres, como las dimensiones del espacio-tiempo y el grado del campo multivectorial, permite identificar varios modelos, obteniendo los ya conocidos y abriendo las puertas a nuevos horizontes. Operacionalmente, para construir nuestra teoría podemos seguir dos enfoques distintos pero complementarios. En primer lugar, haciendo una analogía con la teoría clásica, podemos utilizar directamente las definiciones generalizadas del álgebra exterior para postular una extensión natural de la teoría electromagnética en dimensiones arbitrarias. En segundo lugar, hemos desarrollado, en paralelo, una teoría dinámica, llamada Lagrangiana, construida a propósito para campos multivectoriales de grado arbitrario. Independientemente del enfoque elegido, hemos obtenido una teoría consistente que presenta sus ecuaciones de movimiento, es decir, las ecuaciones de Maxwell generalizadas, y todas las cantidades físicas equivalentes resultantes de las nuevas leyes de conservación, que identifican las cantidades del sistema que permanecen sin alterar, tales como energía y momento. El punto de conexión con la teoría de la comunicación surge al estudiar las ondas electromagnéticas provenientes de las soluciones de las ecuaciones de Maxwell. Consideradas desde el punto de vista multivectorial del álgebra exterior, estas ondas pueden abrir las puertas a una nueva interpretación de la transmisión de señales desde una perspectiva diferente

Dynamical Casimir effect and the structure of vacuum in quantum field theory

Some of the most interesting phenomena that arise from the developments of the modern physics are surely vacuum fluctuations. They appear in different branches of physics, such as Quantum Field Theory, Cosmology, Condensed Matter Physics, Atomic and Molecular Physics, and also in Mathematical Physics. One of the most important of these vacuum fluctuations, sometimes called "zero-point energy", as well as one of the easiest quantum effect to detect, is the so-called Casimir effect. The purposes of this thesis are: - To propose a simple retarded approach for dynamical Casimir effect, thus a description of this vacuum effect when we have moving boundaries. - To describe the behaviour of the force acting on a boundary, due to its self-interaction with the vacuum

Le rappresentazioni in meccanica quantistica

Lo scopo di questo elaborato è compiere un viaggio virtuale attraverso le tappe principali dello sviluppo della teoria dei quanti e approfondirla nelle sue diverse rappresentazioni, quella di Erwin Schrodinger, quella di Werner Karl Heisenberg e quella di Paul Adrien Maurice Dirac, fino ad arrivare, nella fase conclusiva, a diverse applicazione delle rappresentazioni, sfiorando marginalmente la Teoria dei Campi e, di conseguenza, introducendo un parziale superamento della stessa Teoria Quantistica