877 research outputs found
Fractional Diffusion Equation for a Power-Law-Truncated Levy Process
Truncated Levy flights are stochastic processes which display a crossover
from a heavy-tailed Levy behavior to a faster decaying probability distribution
function (pdf). Putting less weight on long flights overcomes the divergence of
the Levy distribution second moment. We introduce a fractional generalization
of the diffusion equation, whose solution defines a process in which a Levy
flight of exponent alpha is truncated by a power-law of exponent 5 - alpha. A
closed form for the characteristic function of the process is derived. The pdf
of the displacement slowly converges to a Gaussian in its central part showing
however a power law far tail. Possible applications are discussed
Computer Simulation Study of the Levy Flight Process
Random walk simulation of the Levy flight shows a linear relation between the
mean square displacement and time. We have analyzed different aspects of
this linearity. It is shown that the restriction of jump length to a maximum
value (lm) affects the diffusion coefficient, even though it remains constant
for lm greater than 1464. So, this factor has no effect on the linearity. In
addition, it is shown that the number of samples does not affect the results.
We have demonstrated that the relation between the mean square displacement and
time remains linear in a continuous space, while continuous variables just
reduce the diffusion coefficient. The results are also implied that the
movement of a levy flight particle is similar to the case the particle moves in
each time step with an average length of jumping . Finally, it is shown that
the non-linear relation of the Levy flight will be satisfied if we use time
average instead of ensemble average. The difference between time average and
ensemble average results points that the Levy distribution may be a non-ergodic
distribution.Comment: 14 pages, 7 figure
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
Scaling detection in time series: diffusion entropy analysis
The methods currently used to determine the scaling exponent of a complex
dynamic process described by a time series are based on the numerical
evaluation of variance. This means that all of them can be safely applied only
to the case where ordinary statistical properties hold true even if strange
kinetics are involved. We illustrate a method of statistical analysis based on
the Shannon entropy of the diffusion process generated by the time series,
called Diffusion Entropy Analysis (DEA). We adopt artificial Gauss and L\'{e}vy
time series, as prototypes of ordinary and anomalus statistics, respectively,
and we analyse them with the DEA and four ordinary methods of analysis, some of
which are very popular. We show that the DEA determines the correct scaling
exponent even when the statistical properties, as well as the dynamic
properties, are anomalous. The other four methods produce correct results in
the Gauss case but fail to detect the correct scaling in the case of L\'{e}vy
statistics.Comment: 21 pages,10 figures, 1 tabl
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
Superdiffusion in Decoupled Continuous Time Random Walks
Continuous time random walk models with decoupled waiting time density are
studied. When the spatial one jump probability density belongs to the Levy
distribution type and the total time transition is exponential a generalized
superdiffusive regime is established. This is verified by showing that the
square width of the probability distribution (appropriately defined)grows as
with when . An important connection
of our results and those of Tsallis' nonextensive statistics is shown. The
normalized q-expectation value of calculated with the corresponding
probability distribution behaves exactly as in the asymptotic
limit.Comment: 9 pages (.tex file), 1 Postscript figures, uses revtex.st
Scaling and Intermittency in Animal Behavior
Scale-invariant spatial or temporal patterns and L\'evy flight motion have
been observed in a large variety of biological systems. It has been argued that
animals in general might perform L\'evy flight motion with power law
distribution of times between two changes of the direction of motion. Here we
study the temporal behaviour of nesting gilts. The time spent by a gilt in a
given form of activity has power law probability distribution without finite
average. Further analysis reveals intermittent eruption of certain periodic
behavioural sequences which are responsible for the scaling behaviour and
indicates the existence of a critical state. We show that this behaviour is in
close analogy with temporal sequences of velocity found in turbulent flows,
where random and regular sequences alternate and form an intermittent sequence.Comment: 10 page
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.
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
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