160 research outputs found
Subordination Pathways to Fractional Diffusion
The uncoupled Continuous Time Random Walk (CTRW) in one space-dimension and
under power law regime is splitted into three distinct random walks: (rw_1), a
random walk along the line of natural time, happening in operational time;
(rw_2), a random walk along the line of space, happening in operational
time;(rw_3), the inversion of (rw_1), namely a random walk along the line of
operational time, happening in natural time. Via the general integral equation
of CTRW and appropriate rescaling, the transition to the diffusion limit is
carried out for each of these three random walks. Combining the limits of
(rw_1) and (rw_2) we get the method of parametric subordination for generating
particle paths, whereas combination of (rw_2) and (rw_3) yields the
subordination integral for the sojourn probability density in space-time
fractional diffusion.Comment: 20 pages, 4 figure
Fractional wave equation and damped waves
In this paper, a fractional generalization of the wave equation that
describes propagation of damped waves is considered. In contrast to the
fractional diffusion-wave equation, the fractional wave equation contains
fractional derivatives of the same order both in
space and in time. We show that this feature is a decisive factor for
inheriting some crucial characteristics of the wave equation like a constant
propagation velocity of both the maximum of its fundamental solution and its
gravity and mass centers. Moreover, the first, the second, and the Smith
centrovelocities of the damped waves described by the fractional wave equation
are constant and depend just on the equation order . The fundamental
solution of the fractional wave equation is determined and shown to be a
spatial probability density function evolving in time that possesses finite
moments up to the order . To illustrate analytical findings, results of
numerical calculations and numerous plots are presented.Comment: 21 pages, 10 figure
Spatially fractional-order viscoelasticity, non-locality and a new kind of anisotropy
Spatial non-locality of space-fractional viscoelastic equations of motion is
studied. Relaxation effects are accounted for by replacing second-order time
derivatives by lower-order fractional derivatives and their generalizations. It
is shown that space-fractional equations of motion of an order strictly less
than 2 allow for a new kind anisotropy, associated with angular dependence of
non-local interactions between stress and strain at different material points.
Constitutive equations of such viscoelastic media are determined. Explicit
fundamental solutions of the Cauchy problem are constructed for some cases
isotropic and anisotropic non-locality
Characterizations and simulations of a class of stochastic processes to model anomalous diffusion
In this paper we study a parametric class of stochastic processes to model
both fast and slow anomalous diffusion. This class, called generalized grey
Brownian motion (ggBm), is made up off self-similar with stationary increments
processes (H-sssi) and depends on two real parameters alpha in (0,2) and beta
in (0,1]. It includes fractional Brownian motion when alpha in (0,2) and
beta=1, and time-fractional diffusion stochastic processes when alpha=beta in
(0,1). The latters have marginal probability density function governed by
time-fractional diffusion equations of order beta. The ggBm is defined through
the explicit construction of the underline probability space. However, in this
paper we show that it is possible to define it in an unspecified probability
space. For this purpose, we write down explicitly all the finite dimensional
probability density functions. Moreover, we provide different ggBm
characterizations. The role of the M-Wright function, which is related to the
fundamental solution of the time-fractional diffusion equation, emerges as a
natural generalization of the Gaussian distribution. Furthermore, we show that
ggBm can be represented in terms of the product of a random variable, which is
related to the M-Wright function, and an independent fractional Brownian
motion. This representation highlights the -{\bf sssi} nature of the ggBm
and provides a way to study and simulate the trajectories. For this purpose, we
developed a random walk model based on a finite difference approximation of a
partial integro-differenital equation of fractional type.Comment: 25 pages, 9 figure
Fractional Fokker-Planck Equation for Ultraslow Kinetics
Several classes of physical systems exhibit ultraslow diffusion for which the
mean squared displacement at long times grows as a power of the logarithm of
time ("strong anomaly") and share the interesting property that the probability
distribution of particle's position at long times is a double-sided
exponential. We show that such behaviors can be adequately described by a
distributed-order fractional Fokker-Planck equations with a power-law
weighting-function. We discuss the equations and the properties of their
solutions, and connect this description with a scheme based on continuous-time
random walks
Creep, Relaxation and Viscosity Properties for Basic Fractional Models in Rheology
The purpose of this paper is twofold: from one side we provide a general
survey to the viscoelastic models constructed via fractional calculus and from
the other side we intend to analyze the basic fractional models as far as their
creep, relaxation and viscosity properties are considered. The basic models are
those that generalize via derivatives of fractional order the classical
mechanical models characterized by two, three and four parameters, that we
refer to as Kelvin-Voigt, Maxwell, Zener, anti-Zener and Burgers. For each
fractional model we provide plots of the creep compliance, relaxation modulus
and effective viscosity in non dimensional form in terms of a suitable time
scale for different values of the order of fractional derivative. We also
discuss the role of the order of fractional derivative in modifying the
properties of the classical models.Comment: 41 pages, 8 figure
Uncoupled continuous-time random walks: Solution and limiting behavior of the master equation
A detailed study is presented for a large class of uncoupled continuous-time
random walks (CTRWs). The master equation is solved for the Mittag-Leffler
survival probability. The properly scaled diffusive limit of the master
equation is taken and its relation with the fractional diffusion equation is
discussed. Finally, some common objections found in the literature are
thoroughly reviewed.Comment: Preprint version of an already published paper. 8 page
Retarding Sub- and Accelerating Super-Diffusion Governed by Distributed Order Fractional Diffusion Equations
We propose diffusion-like equations with time and space fractional
derivatives of the distributed order for the kinetic description of anomalous
diffusion and relaxation phenomena, whose diffusion exponent varies with time
and which, correspondingly, can not be viewed as self-affine random processes
possessing a unique Hurst exponent. We prove the positivity of the solutions of
the proposed equations and establish the relation to the Continuous Time Random
Walk theory. We show that the distributed order time fractional diffusion
equation describes the sub-diffusion random process which is subordinated to
the Wiener process and whose diffusion exponent diminishes in time (retarding
sub-diffusion) leading to superslow diffusion, for which the square
displacement grows logarithmically in time. We also demonstrate that the
distributed order space fractional diffusion equation describes super-diffusion
phenomena when the diffusion exponent grows in time (accelerating
super-diffusion).Comment: 11 pages, LaTe
Constant Curvature Coefficients and Exact Solutions in Fractional Gravity and Geometric Mechanics
We study fractional configurations in gravity theories and Lagrange
mechanics. The approach is based on Caputo fractional derivative which gives
zero for actions on constants. We elaborate fractional geometric models of
physical interactions and we formulate a method of nonholonomic deformations to
other types of fractional derivatives. The main result of this paper consists
in a proof that for corresponding classes of nonholonomic distributions a large
class of physical theories are modelled as nonholonomic manifolds with constant
matrix curvature. This allows us to encode the fractional dynamics of
interactions and constraints into the geometry of curve flows and solitonic
hierarchies.Comment: latex2e, 11pt, 27 pages, the variant accepted to CEJP; added and
up-dated reference
Extreme value statistics from the Real Space Renormalization Group: Brownian Motion, Bessel Processes and Continuous Time Random Walks
We use the Real Space Renormalization Group (RSRG) method to study extreme
value statistics for a variety of Brownian motions, free or constrained such as
the Brownian bridge, excursion, meander and reflected bridge, recovering some
standard results, and extending others. We apply the same method to compute the
distribution of extrema of Bessel processes. We briefly show how the continuous
time random walk (CTRW) corresponds to a non standard fixed point of the RSRG
transformation.Comment: 24 pages, 5 figure
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