357 research outputs found
Generalized Wiener Process and Kolmogorov's Equation for Diffusion induced by Non-Gaussian Noise Source
We show that the increments of generalized Wiener process, useful to describe
non-Gaussian white noise sources, have the properties of infinitely divisible
random processes. Using functional approach and the new correlation formula for
non-Gaussian white noise we derive directly from Langevin equation, with such a
random source, the Kolmogorov's equation for Markovian non-Gaussian process.
From this equation we obtain the Fokker-Planck equation for nonlinear system
driven by white Gaussian noise, the Kolmogorov-Feller equation for
discontinuous Markovian processes, and the fractional Fokker-Planck equation
for anomalous diffusion. The stationary probability distributions for some
simple cases of anomalous diffusion are derived.Comment: 8 pages. in press, Fluctuation and Noise Letters, 200
Critical Phenomena and Diffusion in Complex Systems
Editorial of the International Conference on Critical Phenomena and Diffusion
in Complex Systems held on 5--7 December, 2006 in Nizhniy Novgorod State
University, Russia and was dedicated to the memory and 80th anniversary of
Professor Askold N. Malakhov.Comment: 4 pages, to appear in International Journal of Bifurcation and Chao
Noise Enhanced Stability
The noise can stabilize a fluctuating or a periodically driven metastable
state in such a way that the system remains in this state for a longer time
than in the absence of white noise. This is the noise enhanced stability
phenomenon, observed experimentally and numerically in different physical
systems. After shortly reviewing all the physical systems where the phenomenon
was observed, the theoretical approaches used to explain the effect are
presented. Specifically the conditions to observe the effect: (a) in systems
with periodical driving force, and (b) in random dichotomous driving force, are
discussed. In case (b) we review the analytical results concerning the mean
first passage time and the nonlinear relaxation time as a function of the white
noise intensity, the parameters of the potential barrier, and of the
dichotomous noise.Comment: 18 pages, 6 figures, in press Acta Physica Polonica (2004
Exact Results for Spectra of Overdamped Brownian Motion in Fixed and Randomly Switching Potentials
The exact formulae for spectra of equilibrium diffusion in a fixed bistable
piecewise linear potential and in a randomly flipping monostable potential are
derived. Our results are valid for arbitrary intensity of driving white
Gaussian noise and arbitrary parameters of potential profiles. We find: (i) an
exponentially rapid narrowing of the spectrum with increasing height of the
potential barrier, for fixed bistable potential; (ii) a nonlinear phenomenon,
which manifests in the narrowing of the spectrum with increasing mean rate of
flippings, and (iii) a nonmonotonic behaviour of the spectrum at zero
frequency, as a function of the mean rate of switchings, for randomly switching
potential. The last feature is a new characterization of resonant activation
phenomenon.Comment: in press in Acta Physica Polonica, vol. 35 (4), 200
L\'evy flights versus L\'evy walks in bounded domains
L\'evy flights and L\'evy walks serve as two paradigms of random walks
resembling common features but also bearing fundamental differences. One of the
main dissimilarities are discontinuity versus continuity of their trajectories
and infinite versus finite propagation velocity. In consequence, well developed
theory of L\'evy flights is associated with their pathological physical
properties, which in turn are resolved by the concept of L\'evy walks. Here, we
explore L\'evy flights and L\'evy walks models on bounded domains examining
their differences and analogies. We investigate analytically and numerically
whether and under which conditions both approaches yield similar results in
terms of selected statistical observables characterizing the motion: the
survival probability, mean first passage time and stationary PDFs. It is
demonstrated that similarity of models is affected by the type of boundary
conditions and value of the stability index defining asymptotics of the jump
length distribution.Comment: 15 pages, 13 figure
Noise Enhanced Stability in Fluctuating Metastable States
We derive general equations for the nonlinear relaxation time of Brownian
diffusion in randomly switching potential with a sink. For piece-wise linear
dichotomously fluctuating potential with metastable state, we obtain the exact
average lifetime as a function of the potential parameters and the noise
intensity. Our result is valid for arbitrary white noise intensity and for
arbitrary fluctuation rate of the potential. We find noise enhanced stability
phenomenon in the system investigated: the average lifetime of the metastable
state is greater than the time obtained in the absence of additive white noise.
We obtain the parameter region of the fluctuating potential where the effect
can be observed. The system investigated also exhibits a maximum of the
lifetime as a function of the fluctuation rate of the potential.Comment: 7 pages, 5 figures, to appear in Phys. Rev. E vol. 69 (6),200
L\'{e}vy flights in inhomogeneous environments
We study the long time asymptotics of probability density functions (pdfs) of
L\'{e}vy flights in different confining potentials. For that we use two models:
Langevin - driven and (L\'{e}vy - Schr\"odinger) semigroup - driven dynamics.
It turns out that the semigroup modeling provides much stronger confining
properties than the standard Langevin one. Since contractive semigroups set a
link between L\'{e}vy flights and fractional (pseudo-differential) Hamiltonian
systems, we can use the latter to control the long - time asymptotics of the
pertinent pdfs. To do so, we need to impose suitable restrictions upon the
Hamiltonian and its potential. That provides verifiable criteria for an
invariant pdf to be actually an asymptotic pdf of the semigroup-driven
jump-type process. For computational and visualization purposes our
observations are exemplified for the Cauchy driver and its response to external
polynomial potentials (referring to L\'{e}vy oscillators), with respect to both
dynamical mechanisms.Comment: Major revisio
Heavy-tailed targets and (ab)normal asymptotics in diffusive motion
We investigate temporal behavior of probability density functions (pdfs) of
paradigmatic jump-type and continuous processes that, under confining regimes,
share common heavy-tailed asymptotic (target) pdfs. Namely, we have shown that
under suitable confinement conditions, the ordinary Fokker-Planck equation may
generate non-Gaussian heavy-tailed pdfs (like e.g. Cauchy or more general
L\'evy stable distribution) in its long time asymptotics. For diffusion-type
processes, our main focus is on their transient regimes and specifically the
crossover features, when initially infinite number of the pdf moments drops
down to a few or none at all. The time-dependence of the variance (if in
existence), with , in principle may be
interpreted as a signature of sub-, normal or super-diffusive behavior under
confining conditions; the exponent is generically well defined in
substantial periods of time. However, there is no indication of any universal
time rate hierarchy, due to a proper choice of the driver and/or external
potential.Comment: Major revisio
Linear and nonlinear approximations for periodically driven bistable systems
We analyze periodically driven bistable systems by two different approaches. The first approach is a linearization of the stochastic Langevin equation of our system by the response on small external force. The second one is based on the Gaussian approximation of the kinetic equations for the cumulants. We obtain with the first approach the signal power amplification and output signal-to-noise ratio for a model piece-wise linear bistable potential and compare with the results of linear response approximation. By using the second approach to a bistable quartic potential, we obtain the set of nonlinear differential equations for the first and the second cumulants
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