21 research outputs found
Escape from the potential well: competition between long jumps and long waiting times
Within a concept of the fractional diffusion equation and subordination, the
paper examines the influence of a competition between long waiting times and
long jumps on the escape from the potential well. Applying analytical arguments
and numerical methods, we demonstrate that the presence of long waiting times
distributed according to a power-law distribution with a diverging mean leads
to very general asymptotic properties of the survival probability. The observed
survival probability asymptotically decays like a power-law whose form is not
affected by the value of the exponent characterizing the power-law jump length
distribution. It is demonstrated that this behavior is typical of and generic
for systems exhibiting long waiting times. We also show that the survival
probability has a universal character not only asymptotically but also at small
times. Finally, it is indicated which properties of the first passage time
density are sensitive to the exact value of the exponent characterizing the
jump length distribution.Comment: 8 pages, 10 figure
Non-Gaussian, non-dynamical stochastic resonance
The archetypal system demonstrating stochastic resonance is nothing more than
a threshold triggered device. It consists of a periodic modulated input and
noise. Every time an output crosses the threshold the signal is recorded. Such
a digitally filtered signal is sensitive to the noise intensity. There exist
the optimal value of the noise intensity resulting in the "most" periodic
output. Here, we explore properties of the non-dynamical stochastic resonance
in non-equilibrium situations, i.e. when the Gaussian noise is replaced by an
-stable noise. We demonstrate that non-equilibrium -stable
noises, depending on noise parameters, can either weaken or enhance the
non-dynamical stochastic resonance.Comment: 5 pages, 6 figurure
Stationary states in 2D systems driven by bi-variate L\'evy noises
Systems driven by -stable noises could be very different from their
Gaussian counterparts. Stationary states in single-well potentials can be
multimodal. Moreover, a potential well needs to be steep enough in order to
produce stationary states. Here, it is demonstrated that 2D systems driven by
bi-variate -stable noises are even more surprising than their 1D
analogs. In 2D systems, intriguing properties of stationary states originate
not only due to heavy tails of noise pulses, which are distributed according to
-stable densities, but also because of properties of spectral measures.
Consequently, 2D systems are described by a whole family of Langevin and
fractional diffusion equations. Solutions of these equations bear some common
properties but also can be very different. It is demonstrated that also for 2D
systems potential wells need to be steep enough in order to produce bounded
states. Moreover, stationary states can have local minima at the origin. The
shape of stationary states reflects symmetries of the underlying noise, i.e.
its spectral measure. Finally, marginal densities in power-law potentials also
have power-law asymptotics.Comment: 9 pages, 8 figure
Activation process driven by strongly non-Gaussian noises
The constructive role of non-Gaussian random fluctuations is studied in the
context of the passage over the dichotomously switching potential barrier. Our
attention focuses on the interplay of the effects of independent sources of
fluctuations: an additive stable noise representing non-equilibrium external
random force acting on the system and a fluctuating barrier. In particular, the
influence of the structure of stable noises on the mean escape time and on the
phenomenon of resonant activation (RA) is investigated. By use of the numerical
Monte Carlo method it is documented that the suitable choice of the barrier
switching rate and random external fields may produce resonant phenomenon
leading to the enhancement of the kinetics and the shortest, most efficient
reaction time.Comment: 11 pages, 8 figure
Resonant activation driven by strongly non-Gaussian noises
The constructive role of non-Gaussian random fluctuations is studied in the
context of the passage over the dichotomously switching potential barrier. Our
attention focuses on the interplay of the effects of independent sources of
fluctuations: an additive stable noise representing non-equilibrium external
random force acting on the system and a fluctuating barrier. In particular, the
influence of the structure of stable noises on the mean escape time and on the
phenomenon of resonant activation (RA) is investigated. By use of the numerical
Monte Carlo method it is documented that the suitable choice of the barrier
switching rate and random external fields may produce resonant phenomenon
leading to the enhancement of the kinetics and the shortest, most efficient
reaction time.Comment: 9 pages, 7 figures, RevTeX
Subordinated diffusion and CTRW asymptotics
Anomalous transport is usually described either by models of continuous time
random walks (CTRW) or, otherwise by fractional Fokker-Planck equations (FFPE).
The asymptotic relation between properly scaled CTRW and fractional diffusion
process has been worked out via various approaches widely discussed in
literature. Here, we focus on a correspondence between CTRWs and time and space
fractional diffusion equation stemming from two different methods aimed to
accurately approximate anomalous diffusion processes. One of them is the Monte
Carlo simulation of uncoupled CTRW with a L\'evy -stable distribution
of jumps in space and a one-parameter Mittag-Leffler distribution of waiting
times. The other is based on a discretized form of a subordinated Langevin
equation in which the physical time defined via the number of subsequent steps
of motion is itself a random variable. Both approaches are tested for their
numerical performance and verified with known analytical solutions for the
Green function of a space-time fractional diffusion equation. The comparison
demonstrates trade off between precision of constructed solutions and
computational costs. The method based on the subordinated Langevin equation
leads to a higher accuracy of results, while the CTRW framework with a
Mittag-Leffler distribution of waiting times provides efficiently an
approximate fundamental solution to the FFPE and converges to the probability
density function of the subordinated process in a long-time limit.Comment: 10 pages, 7 figure
Resonant effects in a voltage-activated channel gating
The non-selective voltage activated cation channel from the human red cells,
which is activated at depolarizing potentials, has been shown to exhibit
counter-clockwise gating hysteresis. We have analyzed the phenomenon with the
simplest possible phenomenological models by assuming discrete
states, i.e. two normal open/closed states with two different states of ``gate
tension.'' Rates of transitions between the two branches of the hysteresis
curve have been modeled with single-barrier kinetics by introducing a
real-valued ``reaction coordinate'' parameterizing the protein's conformational
change. When described in terms of the effective potential with cyclic
variations of the control parameter (an activating voltage), this model
exhibits typical ``resonant effects'': synchronization, resonant activation and
stochastic resonance. Occurrence of the phenomena is investigated by running
the stochastic dynamics of the model and analyzing statistical properties of
gating trajectories.Comment: 12 pages, 9 figure
Escape from bounded domains driven by multi-variate -stable noises
In this paper we provide an analysis of a mean first passage time problem of
a random walker subject to a bi-variate -stable L\'evy type noise from
a 2-dimensional disk. For an appropriate choice of parameters the mean first
passage time reveals non-trivial, non-monotonous dependence on the stability
index describing jumps' length asymptotics both for spherical and
Cartesian L\'evy flights. Finally, we study escape from -dimensional
hyper-sphere showing that -dimensional escape process can be used to
discriminate between various types of multi-variate -stable noises,
especially spherical and Cartesian L\'evy flights.Comment: 8 pages, 5 figure
Implication of Barrier Fluctuations on the Rate of Weakly Adiabatic Electron Transfer
The problem of escape of a Brownian particle in a cusp-shaped metastable
potential is of special importance in nonadiabatic and weakly-adiabatic rate
theory for electron transfer (ET) reactions. Especially, for the
weakly-adiabatic reactions, the reaction follows an adiabaticity criterion in
the presence of a sharp barrier. In contrast to the non-adiabatic case, the ET
kinetics can be, however considerably influenced by the medium dynamics.
In this paper, the problem of the escape time over a dichotomously
fluctuating cusp barrier is discussed with its relevance to the high
temperature ET reactions in condensed media.Comment: RevTeX 4, 14 pages, 3 figures. To be printed in IJMP C. References
corrected and update
Underdamped stochastic harmonic oscillator
We investigate stationary states of the linear damped stochastic oscillator
driven by L\'evy noises. In the long time limit kinetic and potential energies
of the oscillator do not fulfill the equipartition theorem and their
distributions follow the power-law asymptotics. At the same time, partition of
the mechanical energy is controlled by the damping coefficient. We show that in
the limit of vanishing damping a stochastic analogue of the equipartition
theorem can be proposed, namely the statistical properties of potential and
kinetic energies attain distributions characterized by the same width. Finally,
we demonstrate that the ratio of instantaneous kinetic and potential energies
which signifies departure from the mechanical energy equipartition, follows
universal power-law asymptotics.Comment: 8 pages. 3 figure