517 research outputs found
Fractional Curve Flows and Solitonic Hierarchies in Gravity and Geometric Mechanics
Methods from the geometry of nonholonomic manifolds and Lagrange-Finsler
spaces are applied in fractional calculus with Caputo derivatives and for
elaborating models of fractional gravity and fractional Lagrange mechanics. The
geometric data for such models are encoded into (fractional) bi-Hamiltonian
structures and associated solitonic hierarchies. The constructions yield
horizontal/vertical pairs of fractional vector sine-Gordon equations and
fractional vector mKdV equations when the hierarchies for corresponding curve
fractional flows are described in explicit forms by fractional wave maps and
analogs of Schrodinger maps.Comment: latex2e, 11pt, 21 pages; the variant accepted to J. Math. Phys.; new
and up--dated reference
Non-Markovian diffusion equations and processes: analysis and simulations
In this paper we introduce and analyze a class of diffusion type equations
related to certain non-Markovian stochastic processes. We start from the
forward drift equation which is made non-local in time by the introduction of a
suitable chosen memory kernel K(t). The resulting non-Markovian equation can be
interpreted in a natural way as the evolution equation of the marginal density
function of a random time process l(t). We then consider the subordinated
process Y(t)=X(l(t)) where X(t) is a Markovian diffusion. The corresponding
time evolution of the marginal density function of Y(t) is governed by a
non-Markovian Fokker-Planck equation which involves the memory kernel K(t). We
develop several applications and derive the exact solutions. We consider
different stochastic models for the given equations providing path simulations.Comment: 43 pages, 19 figures, in press on Physica A (2008
Fractional Equations of Curie-von Schweidler and Gauss Laws
The dielectric susceptibility of most materials follows a fractional
power-law frequency dependence that is called the "universal" response. We
prove that in the time domain this dependence gives differential equations with
derivatives and integrals of noninteger order. We obtain equations that
describe "universal" Curie-von Schweidler and Gauss laws for such dielectric
materials. These laws are presented by fractional differential equations such
that the electromagnetic fields in the materials demonstrate "universal"
fractional damping. The suggested fractional equations are common (universal)
to a wide class of materials, regardless of the type of physical structure,
chemical composition or of the nature of the polarization.Comment: 11 pages, LaTe
Some Insights in Superdiffusive Transport
In this paper we deal with high-order corrections for the Fractional
Derivative approach to anomalous diffusion, in super-diffusive regime, which
become relevand whenever one attempts to describe the behavior of particles
close to normal diffusion.Comment: 14 pages, 7 figure
Polymer translocation through a nanopore - a showcase of anomalous diffusion
The translocation dynamics of a polymer chain through a nanopore in the
absence of an external driving force is analyzed by means of scaling arguments,
fractional calculus, and computer simulations. The problem at hand is mapped on
a one dimensional {\em anomalous} diffusion process in terms of reaction
coordinate (i.e. the translocated number of segments at time ) and shown
to be governed by an universal exponent whose
value is nearly the same in two- and three-dimensions. The process is described
by a {\em fractional} diffusion equation which is solved exactly in the
interval with appropriate boundary and initial conditions. The
solution gives the probability distribution of translocation times as well as
the variation with time of the statistical moments: , and which provide full description of the diffusion process. The
comparison of the analytic results with data derived from extensive Monte Carlo
(MC) simulations reveals very good agreement and proves that the diffusion
dynamics of unbiased translocation through a nanopore is anomalous in its
nature.Comment: 5 pages, 3 figures, accepted for publication in Phys. Rev.
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
Subordination model of anomalous diffusion leading to the two-power-law relaxation responses
We derive a general pattern of the nonexponential, two-power-law relaxation
from the compound subordination theory of random processes applied to anomalous
diffusion. The subordination approach is based on a coupling between the very
large jumps in physical and operational times. It allows one to govern a
scaling for small and large times independently. Here we obtain explicitly the
relaxation function, the kinetic equation and the susceptibility expression
applicable to the range of experimentally observed power-law exponents which
cannot be interpreted by means of the commonly known Havriliak-Negami fitting
function. We present a novel two-power relaxation law for this range in a
convenient frequency-domain form and show its relationship to the
Havriliak-Negami one.Comment: 5 pages; 3 figures; corrected versio
Distributed Order Derivatives and Relaxation Patterns
We consider equations of the form , ,
where , is a distributed order derivative, that is the
Caputo-Dzhrbashyan fractional derivative of order , integrated in
with respect to a positive measure . Such equations are
used for modeling anomalous, non-exponential relaxation processes. In this work
we study asymptotic behavior of solutions of the above equation, depending on
properties of the measure
(2+1)-Dimensional Quantum Gravity as the Continuum Limit of Causal Dynamical Triangulations
We perform a non-perturbative sum over geometries in a (2+1)-dimensional
quantum gravity model given in terms of Causal Dynamical Triangulations.
Inspired by the concept of triangulations of product type introduced
previously, we impose an additional notion of order on the discrete, causal
geometries. This simplifies the combinatorial problem of counting geometries
just enough to enable us to calculate the transfer matrix between boundary
states labelled by the area of the spatial universe, as well as the
corresponding quantum Hamiltonian of the continuum theory. This is the first
time in dimension larger than two that a Hamiltonian has been derived from such
a model by mainly analytical means, and opens the way for a better
understanding of scaling and renormalization issues.Comment: 38 pages, 13 figure
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