429 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
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
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.
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
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
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
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
From deterministic dynamics to kinetic phenomena
We investigate a one-dimenisonal Hamiltonian system that describes a system
of particles interacting through short-range repulsive potentials. Depending on
the particle mean energy, , the system demonstrates a spectrum of
kinetic regimes, characterized by their transport properties ranging from
ballistic motion to localized oscillations through anomalous diffusion regimes.
We etsablish relationships between the observed kinetic regimes and the
"thermodynamic" states of the system. The nature of heat conduction in the
proposed model is discussed.Comment: 4 pages, 4 figure
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