arXiv subjects

Aleksei Chechkin

Publications and source records attributed to Aleksei Chechkin.

28 records · Page 2Linked to original sources

arXiv2018

Conservative random walks in confining potentials

Lévy walks are continuous time random walks with spatio-temporal coupling of jump lengths and waiting times, often used to model superdiffusive spreading processes such as animals searching for food, tracer motion in weakly chaotic systems, or even the dynamics in quantum systems such as cold atoms. In the simplest version Lévy walks move with a finite speed. Here, we present an extension of the Lévy walk scenario for the case when external force fields influence the motion. The resulting motion is a combination of the response to the deterministic force acting on the particle, changing its velocity according to the principle of total energy conservation, and random velocity reversals governed by the distribution of waiting times. For the fact that the motion stays conservative, that is, on a constant energy surface, our scenario is fundamentally different from thermal motion in the same external potentials. In particular, we present results for the velocity and position distributions for single well potentials of different steepness. The observed dynamics with its continuous velocity changes enriches the theory of Lévy walk processes and will be of use in a variety of systems, for which the particles are externally confined.

cond-mat.stat-mech↗

arXiv2018

Finite-energy Lévy-type motion through heterogeneous ensemble of Brownian particles

Complex systems display anomalous diffusion, whose signature is a space/time scaling $x\sim t^δ$ with $δ\ne 1/2$ in the Probability Density Function (PDF). Anomalous diffusion can emerge jointly with both Gaussian, e.g., fractional Brownian motion, and power-law decaying distributions, e.g., Lévy Flights (LFs) or Lévy Walks (LWs). LFs get anomalous scaling, but also infinite position variance and also infinite energy and discontinuous velocity. LWs are based on random trapping events, resemble a Lévy-type power-law distribution that is truncated in the large displacement range and have finite moments, finite energy and discontinuous velocity. However, both LFs and LWs cannot describe friction-diffusion processes. We propose and discuss a model describing a Heterogeneous Ensemble of Brownian Particles (HEBP) based on a linear Langevin equation. We show that, for proper distributions of relaxation time and velocity diffusivity, the HEBP displays features similar to LWs, in particular power-law decaying PDF, long-range correlations and anomalous diffusion, at the same time keeping finite position moments and finite energy. The main differences between the HEBP model and two LWs are investigated, finding that, even if the PDFs are similar, they differ in three main aspects: (i) LWs are biscaling, while HEBP is monoscaling; (ii) a transition from anomalous ($δ\ne 1/2$) to normal ($δ= 1/2$) diffusion in the long-time regime; (iii) the power-law index of the position PDF and the space/time diffusion scaling are independent in the HEBP, while they both depend on the scaling of the inter-event time PDF in LWs. The HEBP model is derived from a friction-diffusion process, it has finite energy and it satisfies the fluctuation-dissipation theorem.

cond-mat.stat-mech↗

arXiv2016

Comb model with slow and ultraslow diffusion

We consider a generalised diffusion equation in two dimensions for modeling diffusion on a comb-like structures. We analyse the probability distribution functions and we derive the mean squared displacement in $x$ and $y$ directions. Different forms of the memory kernels (Dirac delta, power-law, and distributed order) are considered. It is shown that anomalous diffusion may occur along both $x$ and $y$ directions. Ultraslow diffusion and some more general diffusive processes are observed as well. We give the corresponding continuous time random walk model for the considered two dimensional diffusion-like equation on a comb, and we derive the probability distribution functions which subordinate the process governed by this equation to the Wiener process.

cond-mat.stat-mech↗

arXiv2014

Codifference as a practical tool to measure interdependence

Correlation and spectral analysis represent the standard tools to study interdependence in statistical data. However, for the stochastic processes with heavy-tailed distributions such that the variance diverges, these tools are inadequate. The heavy-tailed processes are ubiquitous in nature and finance. We here discuss codifference as a convenient measure to study statistical interdependence, and we aim to give a short introductory review of its properties. By taking different known stochastic processes as generic examples, we present explicit formulas for their codifferences. We show that for the Gaussian processes codifference is equivalent to covariance. For processes with finite variance these two measures behave similarly with time. For the processes with infinite variance the covariance does not exist, however, the codifference is relevant. We demonstrate the practical importance of the codifference by extracting this function from simulated as well as from real experimental data. We conclude that the codifference serves as a convenient practical tool to study interdependence for stochastic processes with both infinite and finite variances as well.

cond-mat.stat-mech↗

arXiv2013

Generalized Elastic Model: Fractional Langevin Description, Fluctuation Relation, and Linear Response

The Generalized Elastic Model is a linear stochastic model which accounts for the behaviour of many physical systems in nature, ranging from polymeric chains to single-file systems. If an external perturbation is exerted \emph{only} on a single point $\vec{x}^\star$ (\emph{tagged probe}), it propagates throughout the entire system. Within the fractional Langevin equation framework, we study the effect of such a perturbation, in cases of a constant force applied. We report most of the results arising from our previous analysis and, in the present work, we show that the Fox $H$-functions formalism provides a compact, elegant and useful tool for the study of the scaling properties of any observable. In particular we show how the generalized Kubo fluctuation relations can be expressed in terms of $H$-functions.

cond-mat.stat-mech↗

arXiv2012

Recognition of stable distribution with Levy index alpha close to 2

We address the problem of recognizing alpha-stable Levy distribution with Levy index close to 2 from experimental data. We are interested in the case when the sample size of available data is not large, thus the power law asymptotics of the distribution is not clearly detectable, and the shape of empirical probability density function is close to a Gaussian. We propose a testing procedure combining a simple visual test based on empirical fourth moment with the Anderson-Darling and Jarque-Bera statistical tests and we check the efficiency of the method on simulated data. Furthermore, we apply our method to the analysis of turbulent plasma density and potential fluctuations measured in the stellarator type fusion device and demonstrate that the phenomenon of L-H transition occurring in this device is accompanied by the transition from Levy to Gaussian fluctuation statistics.

physics.data-an↗

arXiv2012

Generalized Elastic Model: thermal vs non-thermal initial conditions. Universal scaling, roughening, ageing and ergodicity

We study correlation properties of the generalized elastic model which accounts for the dynamics of polymers, membranes, surfaces and fluctuating interfaces, among others. We develop a theoretical framework which leads to the emergence of universal scaling laws for systems starting from thermal (equilibrium) or non-thermal (non-equilibrium) initial conditions. Our analysis incorporates and broadens previous results such as observables' double scaling regimes, (super)roughening and anomalous diffusion, and furnishes a new scaling behavior for correlation functions at small times (long distances). We discuss ageing and ergodic properties of the generalized elastic model in non-equilibrium conditions, providing a comparison with the situation occurring in continuous time random walk. Our analysis also allows to assess which observable is able to distinguish whether the system is in or far from equilibrium conditions in an experimental set-up.

cond-mat.stat-mech↗

arXiv2012

Correlations in a Generalized Elastic Model: Fractional Langevin Equation Approach

The Generalized Elastic Model (GEM) provides the evolution equation which governs the stochastic motion of several many-body systems in nature, such as polymers, membranes, growing interfaces. On the other hand a probe (\emph{tracer}) particle in these systems performs a fractional Brownian motion due to the spatial interactions with the other system's components. The tracer's anomalous dynamics can be described by a Fractional Langevin Equation (FLE) with a space-time correlated noise. We demonstrate that the description given in terms of GEM coincides with that furnished by the relative FLE, by showing that the correlation functions of the stochastic field obtained within the FLE framework agree to the corresponding quantities calculated from the GEM. Furthermore we show that the Fox $H$-function formalism appears to be very convenient to describe the correlation properties within the FLE approach.

cond-mat.stat-mech↗

arXiv2012

Unusual Response to a Localized Perturbation in a Generalized Elastic Model

The generalized elastic model encompasses several physical systems such as polymers, membranes, single file systems, fluctuating surfaces and rough interfaces. We consider the case of an applied localized potential, namely an external force acting only on a single (tagged) probe, leaving the rest of the system unaffected. We derive the fractional Langevin equation for the tagged probe, as well as for a generic (untagged) probe, where the force is not directly applied. Within the framework of the fluctuation-dissipation relations, we discuss the unexpected physical scenarios arising when the force is constant and time periodic, whether or not the hydrodynamic interactions are included in the model. For short times, in case of the constant force, we show that the average drift is linear in time for long range hydrodynamic interactions and behaves ballistically or exponentially for local hydrodynamic interactions. Moreover, it can be opposite to the direction of external disturbance for some values of the model's parameters. When the force is time periodic, the effects are macroscopic: the system splits into two distinct spatial regions whose size is proportional to the value of the applied frequency. These two regions are characterized by different amplitudes and phase shifts in the response dynamics.

cond-mat.stat-mech↗

arXiv2012

Generalized Elastic Model yields Fractional Langevin Equation

Starting from a generalized elastic model which accounts for the stochastic motion of several physical systems such as membranes, (semi)flexible polymers and fluctuating interfaces among others, we derive the fractional Langevin equation (FLE) for a probe particle in such systems, in the case of thermal initial conditions. We show that this FLE is the only one fulfilling the fluctuation-dissipation (FD) relation within a new family of fractional Brownian motion (FBM) equations. The FLE for the time-dependent fluctuations of the donor-acceptor distance in a protein, is shown to be recovered. When the system starts from non-thermal conditions, the corresponding FLE, which does not fulfill FD relation, is derived.

cond-mat.stat-mech↗