641 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
Discrete Models of Time-Fractional Diffusion in a Potential Well
Mathematics Subject Classification: 26A33, 45K05, 60J60, 60G50, 65N06, 80-99.By generalization of Ehrenfest’s urn model, we obtain discrete approximations
to spatially one-dimensional time-fractional diffusion processes with
drift towards the origin. These discrete approximations can be interpreted
(a) as difference schemes for the relevant time-fractional partial differential
equation, (b) as random walk models. The relevant convergence questions as
well as the behaviour for time tending to infinity are discussed, and results
of numerical case studies are displayed.
See also, http://www.diss.fu-berlin.de/2004/168/index.htm
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
Anomalous escape governed by thermal 1/f noise
We present an analytic study for subdiffusive escape of overdamped particles
out of a cusp-shaped parabolic potential well which are driven by thermal,
fractional Gaussian noise with a power spectrum. This
long-standing challenge becomes mathematically tractable by use of a
generalized Langevin dynamics via its corresponding non-Markovian,
time-convolutionless master equation: We find that the escape is governed
asymptotically by a power law whose exponent depends exponentially on the ratio
of barrier height and temperature. This result is in distinct contrast to a
description with a corresponding subdiffusive fractional Fokker-Planck
approach; thus providing experimentalists an amenable testbed to differentiate
between the two escape scenarios
V-Langevin Equations, Continuous Time Random Walks and Fractional Diffusion
The following question is addressed: under what conditions can a strange
diffusive process, defined by a semi-dynamical V-Langevin equation or its
associated Hybrid kinetic equation (HKE), be described by an equivalent purely
stochastic process, defined by a Continuous Time Random Walk (CTRW) or by a
Fractional Differential Equation (FDE)? More specifically, does there exist a
class of V-Langevin equations with long-range (algebraic) velocity temporal
correlation, that leads to a time-fractional superdiffusive process? The answer
is always affirmative in one dimension. It is always negative in two
dimensions: any algebraically decaying temporal velocity correlation (with a
Gaussian spatial correlation) produces a normal diffusive process. General
conditions relating the diffusive nature of the process to the temporal
exponent of the Lagrangian velocity correlation (in Corrsin approximation) are
derived.Comment: Latex 69 pages including 23 EPS figure
Nonlinear Abel type integral equation in modelling creep crack propagation
Copyright @ 2011 Birkhäuser BostonA nonlinear Abel-type equation is obtained in this paper to model creep crack time-dependent propagation in the infinite viscoelastic plane. A finite time when the integral equation solution becomes unbounded is obtained analytically as well as the equation parameters when solution blows up for all times. A modification to the Nyström method is introduced to numerically solve the equation and some computational results are presented
(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
L\'evy-Schr\"odinger wave packets
We analyze the time--dependent solutions of the pseudo--differential
L\'evy--Schr\"odinger wave equation in the free case, and we compare them with
the associated L\'evy processes. We list the principal laws used to describe
the time evolutions of both the L\'evy process densities, and the
L\'evy--Schr\"odinger wave packets. To have self--adjoint generators and
unitary evolutions we will consider only absolutely continuous, infinitely
divisible L\'evy noises with laws symmetric under change of sign of the
independent variable. We then show several examples of the characteristic
behavior of the L\'evy--Schr\"odinger wave packets, and in particular of the
bi-modality arising in their evolutions: a feature at variance with the typical
diffusive uni--modality of both the L\'evy process densities, and the usual
Schr\"odinger wave functions.Comment: 41 pages, 13 figures; paper substantially shortened, while keeping
intact examples and results; changed format from "report" to "article";
eliminated Appendices B, C, F (old names); shifted Chapters 4 and 5 (old
numbers) from text to Appendices C, D (new names); introduced connection
between Relativistic q.m. laws and Generalized Hyperbolic law
Memory-induced anomalous dynamics: emergence of diffusion, subdiffusion, and superdiffusion from a single random walk model
We present a random walk model that exhibits asymptotic subdiffusive,
diffusive, and superdiffusive behavior in different parameter regimes. This
appears to be the first instance of a single random walk model leading to all
three forms of behavior by simply changing parameter values. Furthermore, the
model offers the great advantage of analytic tractability. Our model is
non-Markovian in that the next jump of the walker is (probabilistically)
determined by the history of past jumps. It also has elements of intermittency
in that one possibility at each step is that the walker does not move at all.
This rich encompassing scenario arising from a single model provides useful
insights into the source of different types of asymptotic behavior
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
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