293 research outputs found
Fractal time random walk and subrecoil laser cooling considered as renewal processes with infinite mean waiting times
There exist important stochastic physical processes involving infinite mean
waiting times. The mean divergence has dramatic consequences on the process
dynamics. Fractal time random walks, a diffusion process, and subrecoil laser
cooling, a concentration process, are two such processes that look
qualitatively dissimilar. Yet, a unifying treatment of these two processes,
which is the topic of this pedagogic paper, can be developed by combining
renewal theory with the generalized central limit theorem. This approach
enables to derive without technical difficulties the key physical properties
and it emphasizes the role of the behaviour of sums with infinite means.Comment: 9 pages, 7 figures, to appear in the Proceedings of Cargese Summer
School on "Chaotic dynamics and transport in classical and quantum systems
Accelerating random walks by disorder
We investigate the dynamic impact of heterogeneous environments on
superdiffusive random walks known as L\'evy flights. We devote particular
attention to the relative weight of source and target locations on the rates
for spatial displacements of the random walk. Unlike ordinary random walks
which are slowed down for all values of the relative weight of source and
target, non-local superdiffusive processes show distinct regimes of attenuation
and acceleration for increased source and target weight, respectively.
Consequently, spatial inhomogeneities can facilitate the spread of
superdiffusive processes, in contrast to common belief that external disorder
generally slows down stochastic processes. Our results are based on a novel
type of fractional Fokker-Planck equation which we investigate numerically and
by perturbation theory for weak disorder.Comment: 8 pages, 5 figure
Relativistic Weierstrass random walks
The Weierstrass random walk is a paradigmatic Markov chain giving rise to a
L\'evy-type superdiffusive behavior. It is well known that Special Relativity
prevents the arbitrarily high velocities necessary to establish a
superdiffusive behavior in any process occurring in Minkowski spacetime,
implying, in particular, that any relativistic Markov chain describing
spacetime phenomena must be essentially Gaussian. Here, we introduce a simple
relativistic extension of the Weierstrass random walk and show that there must
exist a transition time delimiting two qualitative distinct dynamical
regimes: the (non-relativistic) superdiffusive L\'evy flights, for ,
and the usual (relativistic) Gaussian diffusion, for . Implications of
this crossover between different diffusion regimes are discussed for some
explicit examples. The study of such an explicit and simple Markov chain can
shed some light on several results obtained in much more involved contexts.Comment: 5 pages, final version to appear in PR
Diffusive behavior of a greedy traveling salesman
Using Monte Carlo simulations we examine the diffusive properties of the
greedy algorithm in the d-dimensional traveling salesman problem. Our results
show that for d=3 and 4 the average squared distance from the origin is
proportional to the number of steps t. In the d=2 case such a scaling is
modified with some logarithmic corrections, which might suggest that d=2 is the
critical dimension of the problem. The distribution of lengths also shows
marked differences between d=2 and d>2 versions. A simple strategy adopted by
the salesman might resemble strategies chosen by some foraging and hunting
animals, for which anomalous diffusive behavior has recently been reported and
interpreted in terms of Levy flights. Our results suggest that broad and
Levy-like distributions in such systems might appear due to dimension-dependent
properties of a search space.Comment: accepted in Phys. Rev.
Levy Flights in Inhomogeneous Media
We investigate the impact of external periodic potentials on superdiffusive
random walks known as Levy flights and show that even strongly superdiffusive
transport is substantially affected by the external field. Unlike ordinary
random walks, Levy flights are surprisingly sensitive to the shape of the
potential while their asymptotic behavior ceases to depend on the Levy index
. Our analysis is based on a novel generalization of the Fokker-Planck
equation suitable for systems in thermal equilibrium. Thus, the results
presented are applicable to the large class of situations in which
superdiffusion is caused by topological complexity, such as diffusion on folded
polymers and scale-free networks.Comment: 4 pages, 4 figure
Anomalous jumping in a double-well potential
Noise induced jumping between meta-stable states in a potential depends on
the structure of the noise. For an -stable noise, jumping triggered by
single extreme events contributes to the transition probability. This is also
called Levy flights and might be of importance in triggering sudden changes in
geophysical flow and perhaps even climatic changes. The steady state statistics
is also influenced by the noise structure leading to a non-Gibbs distribution
for an -stable noise.Comment: 11 pages, 7 figure
Family of generalized random matrix ensembles
Using the Generalized Maximium Entropy Principle based on the nonextensive q
entropy a new family of random matrix ensembles is generated. This family
unifies previous extensions of Random Matrix Theory and gives rise to an
orthogonal invariant stable Levy ensemble with new statistical properties. Some
of them are analytically derived.Comment: 13 pages and 2 figure
Finite Larmor radius effects on non-diffusive tracer transport in a zonal flow
Finite Larmor radius (FLR) effects on non-diffusive transport in a
prototypical zonal flow with drift waves are studied in the context of a
simplified chaotic transport model. The model consists of a superposition of
drift waves of the linearized Hasegawa-Mima equation and a zonal shear flow
perpendicular to the density gradient. High frequency FLR effects are
incorporated by gyroaveraging the ExB velocity. Transport in the direction of
the density gradient is negligible and we therefore focus on transport parallel
to the zonal flows. A prescribed asymmetry produces strongly asymmetric non-
Gaussian PDFs of particle displacements, with L\'evy flights in one direction
but not the other. For zero Larmor radius, a transition is observed in the
scaling of the second moment of particle displacements. However, FLR effects
seem to eliminate this transition. The PDFs of trapping and flight events show
clear evidence of algebraic scaling with decay exponents depending on the value
of the Larmor radii. The shape and spatio-temporal self-similar anomalous
scaling of the PDFs of particle displacements are reproduced accurately with a
neutral, asymmetric effective fractional diffusion model.Comment: 14 pages, 13 figures, submitted to Physics of Plasma
Vanishing Loss Effect on the Effective ac Conductivity behavior for 2D Composite Metal-Dielectric Films At The Percolation Threshold
We study the imaginary part of the effective conductivity as well as its
distribution probability for vanishing losses in 2D composites. This
investigation showed that the effective medium theory provides only
informations about the average conductivity, while its fluctuations which
correspond to the field energy in this limit are neglected by this theory.Comment: 6 pages, 2 figures, submitted to Phys.Rev.
Levy flights in quenched random force fields
Levy flights, characterized by the microscopic step index f, are for f<2 (the
case of rare events) considered in short range and long range quenched random
force fields with arbitrary vector character to first loop order in an
expansion about the critical dimension 2f-2 in the short range case and the
critical fall-off exponent 2f-2 in the long range case. By means of a dynamic
renormalization group analysis based on the momentum shell integration method,
we determine flows, fixed point, and the associated scaling properties for the
probability distribution and the frequency and wave number dependent diffusion
coefficient. Unlike the case of ordinary Brownian motion in a quenched force
field characterized by a single critical dimension or fall-off exponent d=2,
two critical dimensions appear in the Levy case. A critical dimension (or
fall-off exponent) d=f below which the diffusion coefficient exhibits anomalous
scaling behavior, i.e, algebraic spatial behavior and long time tails, and a
critical dimension (or fall-off exponent) d=2f-2 below which the force
correlations characterized by a non trivial fixed point become relevant. As a
general result we find in all cases that the dynamic exponent z, characterizing
the mean square displacement, locks onto the Levy index f, independent of
dimension and independent of the presence of weak quenched disorder.Comment: 27 pages, Revtex file, 17 figures in ps format attached, submitted to
Phys. Rev.
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