arXiv subjects

Bartlomiej Dybiec

Publications and source records attributed to Bartlomiej Dybiec.

At least 19 recordsLinked to original sources

arXiv2020

Lévy noise-driven escape from arctan potential wells

The escape from a potential well is an archetypal problem in the study of stochastic dynamical systems, representing real-world situations from chemical reactions to leaving an established home range in movement ecology. Concurrently, L{é}vy noise is a well-established approach to model systems characterized by statistical outliers and diverging higher-order moments, ranging from gene expression control to the movement patterns of animals and humans. Here, we study the problem of Lévy noise-driven escape from an almost rectangular, arctan potential well restricted by two absorbing boundaries. We unveil analogies of the observed transient dynamics to the general properties of stationary states of L{é}vy processes in single-well potentials. The first escape dynamics is shown to exhibit exponential tails. We examine the dependence of the escape on the shape parameters, steepness and height, of the arctan potential. Finally, we explore in detail the behavior of the probability densities of the first-escape time and the last-hitting point.

cond-mat.stat-mech↗

arXiv2020

Energy partition for anharmonic, undamped, classical oscillators

Using stochastic methods, general formulas for average kinetic and potential energies for anharmonic, undamped (frictionless), classical oscillators are derived. It is demonstrated that for potentials of $|x|^ν$ ($ν>0$) type energies are equipartitioned for the harmonic potential only. For potential wells weaker than parabolic potential energy dominates, while for potentials stronger than parabolic kinetic energy prevails. Due to energy conservation, the variances of kinetic and potential energies are equal. In the limiting case of the infinite rectangular potential well ($ν\to\infty$) the whole energy is stored in the form of the kinetic energy and variances of energy distributions vanish.

cond-mat.stat-mech↗

arXiv2019

Deterministic and randomized motions in single-well potentials

Newtonian, undamped motion in single-well potentials belong to a class of well-studied conservative systems. Here, we investigate and compare long-time properties of fully deterministic motions in single-well potentials with analogous randomized systems. We consider a special type of energy-conserving randomization process: the deterministic motion is interrupted by hard velocity reversals $\vec{v}(t_i)\to-\vec{v}(t_i)$ at random time instants $t_i$. In the 1D case, for fixed initial conditions, the differences in probability distributions disappear in the long-time limit making asymptotic densities insensitive to the selection of random time instants when velocity is reversed. Substantially different probability distributions can be obtained, for instance, through the additional randomization of initial conditions. Analogously, in 2D setups, the probability distributions asymptotically are insensitive to velocity reversals.

cond-mat.stat-mech↗

arXiv2017

Underdamped stochastic harmonic oscillator

We investigate stationary states of the linear damped stochastic oscillator driven by Lévy 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.

cond-mat.stat-mech↗

arXiv2017

Lévy flights versus Lévy walks in bounded domains

Lévy flights and Lévy 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 Lévy flights is associated with their pathological physical properties, which in turn are resolved by the concept of Lévy walks. Here, we explore Lévy flights and Lévy 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.

cond-mat.stat-mech↗

arXiv2015

Escape from bounded domains driven by multi-variate $α$-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 $α$-stable Lévy 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 $α$ describing jumps' length asymptotics both for spherical and Cartesian Lévy flights. Finally, we study escape from $d$-dimensional hyper-sphere showing that $d$-dimensional escape process can be used to discriminate between various types of multi-variate $α$-stable noises, especially spherical and Cartesian Lévy flights.

cond-mat.stat-mech↗

arXiv2014

Resonant activation in 2D and 3D systems driven by multi-variate Lévy noises

Resonant activation is one of classical effects demonstrating constructive role of noise. In resonant activation cooperative action of barrier modulation process and noise lead to the optimal escape kinetics as measured by the mean first passage time. Resonant activation has been observed in versatilities of systems for various types of barrier modulation processes and noise types. Here, we show that resonant activation is also observed in 2D and 3D systems driven by bi-variate and tri-variate $α$-stable noises. Strength of resonant activation is sensitive to the exact value of the noise parameters. In particular, the decrease in the stability index $α$ results in the disappearance of the resonant activation.

cond-mat.stat-mech↗

arXiv2014

Stationary states in 2D systems driven by bi-variate Lévy noises

Systems driven by $α$-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 $α$-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 $α$-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.

cond-mat.stat-mech↗

arXiv2013

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 $α$-stable noise. We demonstrate that non-equilibrium $α$-stable noises, depending on noise parameters, can either weaken or enhance the non-dynamical stochastic resonance.

cond-mat.stat-mech↗

arXiv2012

Information spreading and development of cultural centers

The historical interplay between societies are governed by many factors, including in particular spreading of languages, religion and other symbolic traits. Cultural development, in turn, is coupled to emergence and maintenance of information spreading. Strong centralized cultures exist thanks to attention from their members, which faithfulness in turn relies on supply of information. Here, we discuss a culture evolution model on a planar geometry that takes into account aspects of the feedback between information spreading and its maintenance. Features of model are highlighted by comparing it to cultural spreading in ancient and medieval Europe, where it in particular suggests that long lived centers should be located in geographically remote regions.

physics.soc-ph↗

arXiv2012

Fluctuation-dissipation relations under Levy noises

For systems close to equilibrium, the relaxation properties of measurable physical quantities are described by the linear response theory and the fluctuation-dissipation theorem (FDT). Accordingly, the response or the generalized susceptibility, which is a function of the unperturbed equilibrium system, can be related to the correlation between spontaneous fluctuations of a given conjugate variable. There have been several attempts to extend the FDT far from equilibrium, introducing new terms or using effective temperatures. Recently, Prost, Joanny, and Parrondo [Phys. Rev. Lett. 103, 090601 (2009)] have shown that the FDT can be restored far from equilibrium by choosing the appropriate variables conjugate to the external perturbations. Here, we apply the generalized FDT to a system perturbed by time-dependent deterministic forces and acting under the influence of white alpha-stable noises.

cond-mat.stat-mech↗

arXiv2011

Harmonic oscillator under Levy noise: Unexpected properties in the phase space

A harmonic oscillator under influence of the noise is a basic model of various physical phenomena. Under Gaussian white noise the position and velocity of the oscillator are independent random variables which are distributed according to the bivariate Gaussian distribution with elliptic level lines. The distribution of phase is homogeneous. None of these properties hold in the general Lévy case. Thus, the level lines of the joint probability density are not elliptic. The coordinate and the velocity of the oscillator are strongly dependent, and this dependence is quantified by introducing the corresponding parameter ("width deficit"). The distribution of the phase is inhomogeneous and highly nontrivial.

cond-mat.stat-mech↗

arXiv2010

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 Lévy $α$-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.

cond-mat.stat-mech↗

arXiv2010

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.

cond-mat.stat-mech↗

arXiv2010

Lévy Ratchet in a Weak Noise Limit: Theory and Simulation

We study the motion of a particle embedded in a time independent periodic potential with broken mirror symmetry and subjected to a Lévy noise possessing Lévy stable probability law (Lévy ratchet). We develop analytical approach to the problem based on the asymptotic probabilistic method of decomposition proposed by P. Imkeller and I. Pavlyukevich [J. Phys. A {\bf39}, L237 (2006); Stoch. Proc. Appl. {\bf116}, 611 (2006)]. We derive analytical expressions for the quantities characterizing the particle motion, namely the splitting probabilities of first escape from a single well, the transition probabilities and the particle current. A particular attention is devoted to the interplay between the asymmetry of the ratchet potential and the asymmetry (skewness) of the Lévy noise. Intensive numerical simulations demonstrate a good agreement with the analytical predictions for sufficiently small intensities of the Lévy noise driving the particle.

cond-mat.stat-mech↗

arXiv2010

Stationary states in single-well potentials under symmetric Levy noises

We discuss the existence of stationary states for subharmonic potentials $V(x) \propto |x|^c$, $c<2$, under action of symmetric $α$-stable noises. We show analytically that the necessary condition for the existence of the steady state is $c>2-α$. These states are characterized by heavy-tailed probability density functions which decay as $P(x) \propto x^{-(c+α-1)}$ for $|x| \to \infty$, i.e. stationary states posses a heavier tail than the corresponding $α$-stable law. Monte Carlo simulations confirm the existence of such stationary states and the form of the tails of corresponding probability densities.

cond-mat.stat-mech↗

arXiv2009

Paradoxical diffusion: Discriminating between normal and anomalous random walks

Commonly, normal diffusive behavior is characterized by a linear dependence of the second central moment on time, $< x^2(t) >\propto t$, while anomalous behavior is expected to show a different time dependence, $ < x^2(t) > \propto t^δ$ with $δ<1$ for subdiffusive and $δ>1$ for superdiffusive motions. Here we demonstrate that this kind of qualification, if applied straightforwardly, may be misleading: There are anomalous transport motions revealing perfectly "normal" diffusive character ($< x^2(t) >\propto t$), yet being non-Markov and non-Gaussian in nature. We use recently developed framework \cite[Phys. Rev. E \textbf{75}, 056702 (2007)]{magdziarz2007b} of Monte Carlo simulations which incorporates anomalous diffusion statistics in time and space and creates trajectories of such an extended random walk. For special choice of stability indices describing statistics of waiting times and jump lengths, the ensemble analysis of paradoxical diffusion is shown to hide temporal memory effects which can be properly detected only by examination of formal criteria of Markovianity (fulfillment of the Chapman-Kolmogorov equation).

cond-mat.stat-mech↗

arXiv2009

Anomalous diffusion and generalized Sparre-Andersen scaling

We are discussing long-time, scaling limit for the anomalous diffusion composed of the subordinated Lévy-Wiener process. The limiting anomalous diffusion is in general non-Markov, even in the regime, where ensemble averages of a mean-square displacement or quantiles representing the group spread of the distribution follow the scaling characteristic for an ordinary stochastic diffusion. To discriminate between truly memory-less process and the non-Markov one, we are analyzing deviation of the survival probability from the (standard) Sparre-Andersen scaling.

cond-mat.stat-mech↗