arXiv subjects

Bartłomiej Dybiec

Publications and source records attributed to Bartłomiej Dybiec.

At least 19 recordsLinked to original sources

arXiv2026

Hitting the blinking target under stochastic resetting

The first hitting times of a stochastic process, i.e., the first time a process reaches a particular level, are of significant interest across various scientific disciplines, including biology, chemistry, and economics. We modify the standard setup by allowing the target to spontaneously switch between two states, either active or inactive, and investigate the distribution of first hitting times accrued while the target is active. For this setup, we provide closed formulas for the distribution of the first hitting time. Additionally, we can introduce stochastic resetting to the underlying process and, utilizing our results, derive the formulas for the first time the active target is hit by the process under stochastic resetting. Interestingly, we show that resetting in this setup still leaves some memory; the system is no longer Markovian, which prevents a straightforward application of standard techniques. The analytical results are accompanied by computer simulations of Langevin dynamics.

cond-mat.stat-mech↗

arXiv2024

Role of long jumps in Lévy noise induced multimodality

Lévy noise is a paradigmatic noise used to describe out-of-equilibrium systems. Typically, properties of Lévy noise driven systems are very different from their Gaussian white noise driven counterparts. In particular, under action of Lévy noise, stationary states in single-well, super-harmonic, potentials are no longer unimodal. Typically, they are bimodal however for fine-tuned potentials the number of modes can be further increased. The multimodality arises as a consequence of the competition between long displacements induced by the non-equilibrium stochastic driving and action of the deterministic force. Here, we explore robustness of bimodality in the quartic potential under action of the Lévy noise. We explore various scenarios of bounding long jumps and assess their ability to weaken and destroy multimodality. In general, we demonstrate that despite its robustness it is possible to destroy the bimodality, however it requires drastic reduction in the length of noise-induced jumps.

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arXiv2024

Lévy flights and Lévy walks under stochastic resetting

Stochastic resetting is a protocol of starting anew, which can be used to facilitate the escape kinetics. We demonstrate that restarting can accelerate the escape kinetics from a finite interval restricted by two absorbing boundaries also in the presence of heavy-tailed, Lévy type, $α$-stable noise. However, the width of the domain where resetting is beneficial depends on the value of the stability index $α$ determining power-law decay of jump length distribution. For heavier (smaller $α$) distributions the domain becomes narrower in comparison to lighter tails. Additionally, we explore connections between Lévy flights and Lévy walks in presence of stochastic resetting. First of all, we show that for Lévy walks, the stochastic resetting can be beneficial also in the domain where coefficient of variation is smaller than 1. Moreover, we demonstrate that in the domain where LW are characterized by a finite mean jump duration/length, with the increasing width of the interval LW start to share similarities with LF under stochastic resetting.

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arXiv2023

Optimization of escape kinetics by reflecting and resetting

Stochastic restarting is a strategy of starting anew. Incorporation of the resetting to the random walks can result in the decrease of the mean first passage time, due to the ability to limit unfavorably meandering, sub-optimal trajectories. In the following manuscript we examine how stochastic resetting influences escape dynamics from the $(-\infty,1)$ interval in the presence of the single-well power-law $|x|^κ$ potentials with $κ>0$. Examination of the mean first passage time is complemented by the analysis of the coefficient of variation, which provides a robust and reliable indicator assessing efficiency of stochastic resetting. The restrictive nature of resetting is compared to placing a reflective boundary in the system at hand. In particular, for each potential, the position of the reflecting barrier giving the same mean first passage time as the optimal resetting rate is determined. Finally, in addition to reflecting, we compare effectiveness of other resetting strategies with respect to optimization of the mean first passage time.

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arXiv2023

Weighted Axelrod model: different but similar

The Axelrod model is a cellular automaton which can be used to describe the emergence and development of cultural domains, where culture is represented by a fixed number of cultural features taking a discrete set of possible values (traits). The Axelrod model is based on two sociological phenomena: homophily (a tendency for individuals to form bonds with people similar to themselves) and social influence (the way how individuals change their behavior due to social pressure). However, the Axelrod model does not take into account the fact that cultural attributes may have different significance for a given individual. This is a limitation in the context of how the model reflects mechanisms driving the evolution of real societies. The study aims to modify the Axelrod model by giving individual features different weights that have a decisive impact on the possibility of aligning cultural traits between (interacting) individuals. The comparison of the results obtained for the classic Axelrod model and its modified version shows that introduced weights have a significant impact on the course of the system development, in particular, increasing the final polarization of the system and increasing the time needed to reach the final state.

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arXiv2023

Resetting induced multimodality

Properties of stochastic systems are defined by the noise type and deterministic forces acting on the system. In out-of-equilibrium setups, e.g., for motions under action of Lévy noises, the existence of the stationary state is not only determined by the potential but also by the noise. Potential wells need to be steeper than parabolic in order to assure existence of stationary states. The existence of stationary states, in sub-harmonic potential wells, can be restored by stochastic resetting, which is the protocol of starting over at random times. Herein we demonstrate that the combined action of Lévy noise and Poissonian stochastic resetting can result in the phase transition between non-equilibrium stationary states of various multimodality in the overdamped system in super-harmonic potentials. Fine-tuned resetting rates can increase the modality of stationary states, while for high resetting rates the multimodality is destroyed as the stochastic resetting limits the spread of particles.

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arXiv2023

Stochastic kinetics under combined action of two noise sources

We are exploring two archetypal noise induced escape scenarios: escape from a finite interval and from the positive half-line under the action of the mixture of Lévy and Gaussian white noises in the overdamped regime, for the random acceleration process and higher order processes. In the case of escape from finite intervals, mixture of noises can result in the change of value of the mean first passage time in comparison to the action of each noise separately. At the same time, for the random acceleration process on the (positive) half-line, over the wide range of parameters, the exponent characterizing the power-law decay of the survival probability is equal to the one characterizing the decay of the survival probability under action of the (pure) Lévy noise. There is a transient region, width of which increases with stability index $α$, when the exponent decreases from the one for Lévy noise to the one corresponding to the Gaussian white noise driving.

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arXiv2022

Drifted escape from the finite interval

Properties of the noise-driven escape kinetics are mainly determined by the stochastic component of the system dynamics. Nevertheless, the escape dynamics is also sensitive to deterministic forces. Here, we are exploring properties of the overdamped drifted escape from finite intervals under the action of symmetric $α$-stable noises. We show that the properly rescaled mean first passage time follows the universal pattern as a function of the generalized Pécklet number, which can be used to efficiently discriminate between domains where drift or random force dominate. Stochastic driving of the $α$-stable type is capable of diminishing the significance of the drift in the regime when the drift prevails.

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arXiv2022

Interplay of noise induced stability and stochastic resetting

Stochastic resetting and noise-enhanced stability are two phenomena which can affect the lifetime and relaxation of nonequilibrium states. They can be considered as measures of controlling the efficiency of the completion process when a stochastic system has to reach a desired state. Here, we study interaction of random (Poissonian) resetting and stochastic dynamics in unstable potentials. Unlike noise-induced stability which increases the relaxation time, the stochastic resetting may eliminate winding trajectories contributing to the lifetime and accelerate the escape kinetics from unstable states. In the paper we present a framework to analyze compromises between the two contrasting phenomena in a noise-driven kinetics subject to random restarts.

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arXiv2021

Inertial Lévy flights in bounded domains

The escape from a given domain is one of the fundamental problems in statistical physics and the theory of stochastic processes. Here, we explore properties of the escape of an inertial particle driven by Lévy noise from a bounded domain, restricted by two absorbing boundaries.Presence of two absorbing boundaries assures that the escape process can be characterized by the finite mean first passage time. The detailed analysis of escape kinetics shows that properties of the mean first passage time for the integrated Ornstein--Uhlenbeck process driven by Lévy noise are closely related to properties of the integrated Lévy motions which, in turn, are close to properties of the integrated Wiener process. The extensive studies of the mean first passage time were complemented by examination of the escape velocity and energy along with their sensitivity to initial conditions.

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arXiv2021

Random acceleration process on finite intervals under stochastic restarting

The escape of the randomly accelerated undamped particle from the finite interval under action of stochastic resetting is studied. The motion of such a particle is described by the full Langevin equation and the particle is characterized by the position and velocity. We compare three resetting protocols, which restarts velocity or position (partial resetting) and the whole motion (position and velocity -- full resetting). Using the mean first passage time we assess efficiency of restarting protocols in facilitating or suppressing the escape kinetics. There are fundamental differences between partial resetting scenarios which restart velocity or position, as in the limit of very frequent resets only the position resetting (regardless of initial velocity) can trap the particle in the finite domain of motion. The velocity resetting or the simultaneous position and velocity restarting provide a possibility of facilitating the undamped escape kinetics.

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arXiv2021

Physics of free climbing

Theory of stochastic processes provides theoretical tools which can be efficiently used to explore properties of noise induced escape kinetics. Since noise facilitated escape over the potential barrier resembles free climbing, one can use the first passage time theory in analysis of rock climbing. We perform the analysis of the mean first passage time in order to answer the question regarding the optimal, i.e., resulting in the fastest climbing, rope length. It is demonstrated that there is a discrete set of favorable rope lengths assuring shortest climbing times, as they correspond to local minima of mean first passage time. Within the set of favorable rope lengths there is the optimal rope giving rise to the shortest climbing time. In particular, more experienced climbers can decrease their climbing time by using longer ropes.

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arXiv2021

Dichotomous flow with thermal diffusion and stochastic resetting

We consider properties of one-dimensional diffusive dichotomous flow and discuss effects of resonant activation in the presence of statistically independent random resetting mechanism. Resonant activation and stochastic resetting are two similar effects, as both of them can optimize the noise induced escape. Our studies show completely different origins of optimization in adapted setups. Efficiency of stochastic resetting relies on elimination of suboptimal trajectories while resonant activation is associated with matching of time scales in the dynamic environment. Consequently, both effects can be easily tracked by studying their asymptotic properties. Finally, stochastic resetting cannot be easily used to further optimization of the resonant activation in symmetric setups.

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arXiv2021

Trapped by the drift

The diffusion type is determined not only by microscopic dynamics but also by the environment properties. For example, the environment's fractal structure is responsible for the emergence of subdiffusive scaling of the mean square displacement in Markovian systems because the presence of non-trivially placed obstacles puts constraints on possible displacements. We investigate how the additional action of drift changes properties of the diffusion in the crowded environment. It is shown that the action of a constant drift increases chances of trapping, which suppresses the persistent ballistic motion. Such a diffusion becomes anisotropic because the drift introduces a preferred direction of motion which is further altered by interactions with obstacles. Moreover, individual trajectories display a high level of variability, which is responsible for the macroscopic properties of the diffusing front. Overall, the interplay between drift, diffusion and crowded environment, as measured by the time-averaged mean square displacement, is responsible for the emergence of superdiffusive and subdiffusive patterns in the very same system. Importantly, in contrast to free motion, the constant drift can enhance signatures of subdiffusive motion.

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arXiv2020

Deterministic force-free resonant activation

Combined action of noise and deterministic force in dynamical systems can induce resonant effects. Here, we demonstrate a minimal, deterministic-force-free, setup allowing for occurrence of resonant, noise induced effects. We show that in the archetypal problem of escape from finite intervals driven by $α$-stale noise with the periodically modulated stability index, depending on the initial direction of the modulation, resonant-activation-like or noise-enhanced-stability-like phenomena can be observed.

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arXiv2020

Underdamped, anomalous kinetics in double-well potentials

The noise driven motion in a bistable potential acts as the archetypal model of various physical phenomena. Here, we contrast the overdamped dynamics with the full (underdamped) dynamics. For the overdamped particle driven by a non-equilibrium, $α$-stable noise the ratio of forward and backward transition rates depends only on the width of a potential barrier separating both minima. Using analytical and numerical methods, we show that in the regime of full dynamics, contrary to the overdamped case, the ratio of transition rates depends both on widths and heights of the potential barrier. The analytical formula for the ratio of transition rates is corroborated by extensive numerical simulations.

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arXiv2020

Nonlinear friction in underdamped anharmonic stochastic oscillators

Stationary states of overdamped anharmonic stochastic oscillators driven by Lévy noise are typically multimodal. The very same situation is recorded for an underdamped Lévy noise driven motion in single-well potentials with linear friction. Within current manuscript we relax the assumption that the friction experienced by a particle is linear. Using computer simulations, we study underdamped motion in single-well potentials in the regime of nonlinear friction. We demonstrate that it is relatively easy to observe multimodality in the velocity distribution as it is determined by the friction itself and it is the same as the multimodality in the overdamped case with the analogous deterministic force. Contrary to the velocity marginal density, it is more difficult to produce multimodality in the position. Nevertheless, for fine-tuned nonlinear friction, the spatial multimodality can be recorded.

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arXiv2020

Bounding energy growth in friction-less stochastic oscillators

The paper presents analytical and numerical results on energetics of non-harmonic, undamped, single-well, stochastic oscillators driven by additive Gaussian white noises. Absence of damping and the action of noise are responsible for lack of stationary states in such systems. We explore properties of average kinetic, potential and total energies along with the generalized equipartition relations. It is demonstrated that in the friction-less dynamics nonequilibrium stationary states can be produced by stochastic resetting. For an appropriate resetting protocol average energies become bounded. If the resetting protocol is not characterized by finite variance of renewal time intervals stochastic resetting can only slow down the growth of average energies but does not bound them. Under special conditions regarding frequency of resets, ratios of average energies follow the generalized equipartition relations.

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