arXiv subjects

A. V. Chechkin

Publications and source records attributed to A. V. Chechkin.

At least 37 records · Page 2Linked to original sources

arXiv2016

Underdamped scaled Brownian motion: (non-)existence of the overdamped limit in anomalous diffusion

It is quite generally assumed that the overdamped Langevin equation provides a quantitative description of the dynamics of a classical Brownian particle in the long time limit. We establish and investigate a paradigm anomalous diffusion process governed by an underdamped Langevin equation with an explicit time dependence of the system temperature and thus the diffusion and damping coefficients. We show that for this underdamped scaled Brownian motion (UDSBM) the overdamped limit fails to describe the long time behaviour of the system and may practically even not exist at all for a certain range of the parameter values. Thus persistent inertial effects play a non-negligible role even at significantly long times. From this study a general questions on the applicability of the overdamped limit to describe the long time motion of an anomalously diffusing particle arises, with profound consequences for the relevance of overdamped anomalous diffusion models. We elucidate our results in view of analytical and simulations results for the anomalous diffusion of particles in free cooling granular gases.

cond-mat.stat-mech↗

arXiv2016

Ageing underdamped scaled Brownian motion: ensemble and time averaged particle displacements, non-ergodicity, and the failure of the overdamping approximation

We investigate both analytically and by computer simulations the ensemble averaged, time averaged, non-ergodic, and ageing properties of massive particles diffusing in a medium with a time dependent diffusivity. We call this stochastic diffusion process the (ageing) underdamped scaled Brownian motion (UDSBM). We demonstrate how the mean squared displacement (MSD) and the time averaged MSD of UDSBM are affected by the inertial term in the Langevin equation, both at short, intermediate, and even long diffusion times. In particular, we quantify the ballistic regime for the MSD and the time averaged MSD as well as the spread of individual time averaged MSD trajectories. One of the main effects we observe is that---both for the MSD and the time averaged MSD---for superdiffusive UDSBM the ballistic regime is much shorter than for ordinary Brownian motion. In contrast, for subdiffusive UDSBM the ballistic region extends to much longer diffusion times. Therefore, particular care needs to be taken when the overdamped limit indeed provides a correct description, even in the long time limit. We also analyze to what extent ergodicity in the Boltzmann-Khinchin sense in this non-stationary system is broken, both for subdiffusive and superdiffusive UDSBM. Finally, the limiting case of ultraslow UDSBM is considered, with a mixed logarithmic and power law dependence of the ensemble and time averaged MSDs of the particles. In the limit of strong ageing, remarkably, the ordinary UDSBM and the ultraslow UDSBM behave similarly in the short time ballistic limit. The approaches developed here open new ways for considering other stochastic processes under physically important conditions when a finite particle mass and ageing in the system cannot be neglected.

cond-mat.stat-mech↗

arXiv2016

Search reliability and search efficiency of combined Lévy-Brownian motion: long relocations mingled with thorough local exploration

A combined dynamics consisting of Brownian motion and Lévy flights is exhibited by a variety of biological systems performing search processes. Assessing the search reliability of ever locating the target and the search efficiency of doing so economically of such dynamics thus poses an important problem. Here we model this dynamics by a one-dimensional fractional Fokker-Planck equation combining unbiased Brownian motion and Lévy flights. By solving this equation both analytically and numerically we show that the superposition of recurrent Brownian motion and Lévy flights with stable exponent $α<1$, by itself implying zero probability of hitting a point on a line, lead to transient motion with finite probability of hitting any point on the line. We present results for the exact dependence of the values of both the search reliability and the search efficiency on the distance between the starting and target positions as well as the choice of the scaling exponent $α$ of the Lévy flight component.

cond-mat.stat-mech↗

arXiv2015

Fluctuation relations for anomalous dynamics generated by time-fractional Fokker-Planck equations

Anomalous dynamics characterized by non-Gaussian probability distributions (PDFs) and/or temporal long-range correlations can cause subtle modifications of conventional fluctuation relations. As prototypes we study three variants of a generic time-fractional Fokker-Planck equation with constant force. Type A generates superdiffusion, type B subdiffusion and type C both super- and subdiffusion depending on parameter variation. Furthermore type C obeys a fluctuation-dissipation relation whereas A and B do not. We calculate analytically the position PDFs for all three cases and explore numerically their strongly non-Gaussian shapes. While for type C we obtain the conventional transient work fluctuation relation, type A and type B both yield deviations by featuring a coefficient that depends on time and by a nonlinear dependence on the work. We discuss possible applications of these types of dynamics and fluctuation relations to experiments.

cond-mat.stat-mech↗

arXiv2015

Aging Scaled Brownian Motion

Scaled Brownian motion (SBM) is widely used to model anomalous diffusion of passive tracers in complex and biological systems. It is a highly non-stationary process governed by the Langevin equation for Brownian motion, however, with a power-law time dependence of the noise strength. Here we study the aging properties of SBM for both unconfined and confined motion. Specifically, we derive the ensemble and time averaged mean squared displacements and analyze their behavior in the regimes of weak, intermediate, and strong aging. A very rich behavior is revealed for confined aging SBM depending on different aging times and whether the process is sub- or superdiffusive. We demonstrate that the information on the aging factorizes with respect to the lag time and exhibits a functional form, that is identical to the aging behavior of scale free continuous time random walk processes. While SBM exhibits a disparity between ensemble and time averaged observables and is thus weakly non-ergodic, strong aging is shown to effect a convergence of the ensemble and time averaged mean squared displacement. Finally, we derive the density of first passage times in the semi-infinite domain that features a crossover defined by the aging time.

cond-mat.stat-mech↗

arXiv2014

Scaled Brownian motion: a paradoxical process with a time dependent diffusivity for the description of anomalous diffusion

Anomalous diffusion is frequently described by scaled Brownian motion (SBM), a Gaussian process with a power-law time dependent diffusion coefficient. Its mean squared displacement is $\langle x^2(t)\rangle\simeq\mathscr{K}(t)t$ with $\mathscr{K}(t)\simeq t^{α-1}$ for $0<α<2$. SBM may provide a seemingly adequate description in the case of unbounded diffusion, for which its probability density function coincides with that of fractional Brownian motion. Here we show that free SBM is weakly non-ergodic but does not exhibit a significant amplitude scatter of the time averaged mean squared displacement. More severely, we demonstrate that under confinement, the dynamics encoded by SBM is fundamentally different from both fractional Brownian motion and continuous time random walks. SBM is highly non-stationary and cannot provide a physical description for particles in a thermalised stationary system. Our findings have direct impact on the modelling of single particle tracking experiments, in particular, under confinement inside cellular compartments or when optical tweezers tracking methods are used.

cond-mat.stat-mech↗

arXiv2013

Anomalous fluctuation relations

We study Fluctuation Relations (FRs) for dynamics that are anomalous, in the sense that the diffusive properties strongly deviate from the ones of standard Brownian motion. We first briefly review the concept of transient work FRs for stochastic dynamics modeled by the ordinary Langevin equation. We then introduce three generic types of dynamics generating anomalous diffusion: Lévy flights, long-time correlated Gaussian stochastic processes and time-fractional kinetics. By combining Langevin and kinetic approaches we calculate the work probability distributions in the simple nonequilibrium situation of a particle subject to a constant force. This allows us to check the transient FR for anomalous dynamics. We find a new form of FRs, which is intimately related to the validity of fluctuation-dissipation relations. Analogous results are obtained for a particle in a harmonic potential dragged by a constant force. We argue that these findings are important for understanding fluctuations in experimentally accessible systems. As an example, we discuss the anomalous dynamics of biological cell migration both in equilibrium and in nonequilibrium under chemical gradients.

cond-mat.stat-mech↗

arXiv2012

Stationary States in Bistable System Driven by Lévy Noise

We study the properties of the probability density function (PDF) of a bistable system driven by heavy tailed white symmetric Lévy noise. The shape of the stationary PDF is found analytically for the particular case of the Lévy index α= 1 (Cauchy noise). For an arbitrary Lévy index we employ numerical methods based on the solution of the stochastic Langevin equation and space fractional kinetic equation. In contrast with the bistable system driven by Gaussian noise, in the Lévy case the positions of maxima of the stationary PDF do not coincide with the positions of minima of the bistable potential. We provide a detailed study of the distance between the maxima and the minima as a function of the potential's depth and Lévy noise parameters.

cond-mat.stat-mech↗

arXiv2011

First passage behaviour of fractional Brownian motion in two-dimensional wedge domains

We study the survival probability and the corresponding first passage time density of fractional Brownian motion confined to a two-dimensional open wedge domain with absorbing boundaries. By analytical arguments and numerical simulation we show that in the long time limit the first passage time density scales as t**{-1+pi*(2H-2)/(2*Theta)} in terms of the Hurst exponent H and the wedge angle Theta. We discuss this scaling behaviour in connection with the reaction kinetics of FBM particles in a one-dimensional domain.

cond-mat.stat-mech↗

arXiv2009

Continuous time random walk with correlated waiting times

Based on the Langevin description of the Continuous Time Random Walk (CTRW), we consider a generalization of CTRW in which the waiting times between the subsequent jumps are correlated. We discuss the cases of exponential and slowly decaying persistent power-law correlations between the waiting times as two generic examples and obtain the corresponding mean squared displacements as functions of time. In the case of exponential-type correlations the (sub)diffusion at short times is slower than in the absence of correlations. At long times the behavior of the mean squared displacement is the same as in uncorrelated CTRW. For power-law correlations we find subdiffusion characterized by the same exponent at all times, which appears to be smaller than the one in uncorrelated CTRW. Interestingly, in the limiting case of an extremely long power-law correlations, the (sub)diffusion exponent does not tend to zero, but is bounded from below by the subdiffusion exponent corresponding to a short time behavior in the case of exponential correlations.

cond-mat.stat-mech↗

arXiv2009

Fluctuation relations for anomalous dynamics

We consider work fluctuation relations (FRs) for generic types of dynamics generating anomalous diffusion: Levy flights, long-correlated Gaussian processes and time-fractional kinetics. By combining Langevin and kinetic approaches we calculate the probability distributions of mechanical and thermodynamical work in two paradigmatic nonequilibrium situations, respectively: a particle subject to a constant force and a particle in a harmonic potential dragged by a constant force. We check the transient FR for two models exhibiting superdiffusion, where a fluctuation-dissipation relation does not exist, and for two other models displaying subdiffusion, where there is a fluctuation-dissipation relation. In the two former cases the conventional transient FR is not recovered, whereas in the latter two it holds either exactly or in the long-time limit.

cond-mat.stat-mech↗

arXiv2007

Fluctuation-driven directed transport in the presence of Levy flights

Numerical evidence of directed transport driven by symmetric Levy noise in time-independent ratchet potentials in the absence of an external tilting force is presented. The results are based on the numerical solution of the fractional Fokker-Planck equation in a periodic potential and the corresponding Langevin equation with Levy noise. The Levy noise drives the system out of thermodynamic equilibrium and an up-hill net current is generated. For small values of the noise intensity there is an optimal value of the Levy noise index yielding the maximum current. The direction and magnitude of the current can be manipulated by changing the Levy noise asymmetry and the potential asymmetry.

cond-mat.stat-mech↗

arXiv2006

On the First Passage Time and Leapover Properties of Levy Motions

We investigate two coupled properties of Levy stable random motions: The first passage times (FPTs) and the first passage leapovers (FPLs). While, in general, the FPT problem has been studied quite extensively, the FPL problem has hardly attracted any attention. Considering a particle that starts at the origin and performs random jumps with independent increments chosen from a Levy stable probability law $λ_(alpha,beta)(x)$, the FPT measures how long it takes the particle to arrive at or cross a target. The FPL addresses a different question: Given that the first passage jump crosses the target, then how far does it get beyond the target? These two properties are investigated for three subclasses of Levy stable motions: (i) symmetric Levy motions characterized by Levy index $α$ ($0<α<2$) and skewness parameter $β=0$, (ii) one-sided Levy motions with $0<α<1$, $β=1$, and (iii) two-sided skewed Levy motions, the extreme case, $1<α<2$, $β=-1$.

cond-mat.soft↗

arXiv2004

Understanding Anomalous Transport in Intermittent Maps: From Continuous Time Random Walks to Fractals

We show that the generalized diffusion coefficient of a subdiffusive intermittent map is a fractal function of control parameters. A modified continuous time random walk theory yields its coarse functional form and correctly describes a dynamical phase transition from normal to anomalous diffusion marked by strong suppression of diffusion. Similarly, the probability density of moving particles is governed by a time-fractional diffusion equation on coarse scales while exhibiting a specific fine structure. Approximations beyond stochastic theory are derived from a generalized Taylor-Green-Kubo formula.

nlin.CD↗

arXiv2004

Distributed-Order Fractional Kinetics

Fractional diffusion equations are widely used to describe anomalous diffusion processes where the characteristic displacement scales as a power of time. For processes lacking such scaling the corresponding description may be given by distributed-order equations. In the present paper we consider different forms of distributed-order fractional kinetic equations and investigate the effects described by different classes of such equations. In particular, the equations describing accelerating and decelerating subdiffusion, as well as the those describing accelerating and decelerating superdiffusion are presented.

cond-mat.stat-mech↗

arXiv2003

Fractional Diffusion Equation for a Power-Law-Truncated Levy Process

Truncated Levy flights are stochastic processes which display a crossover from a heavy-tailed Levy behavior to a faster decaying probability distribution function (pdf). Putting less weight on long flights overcomes the divergence of the Levy distribution second moment. We introduce a fractional generalization of the diffusion equation, whose solution defines a process in which a Levy flight of exponent alpha is truncated by a power-law of exponent 5 - alpha. A closed form for the characteristic function of the process is derived. The pdf of the displacement slowly converges to a Gaussian in its central part showing however a power law far tail. Possible applications are discussed.

cond-mat↗

arXiv2003

Fractional Fokker-Planck Equation for Ultraslow Kinetics

Several classes of physical systems exhibit ultraslow diffusion for which the mean squared displacement at long times grows as a power of the logarithm of time ("strong anomaly") and share the interesting property that the probability distribution of particle's position at long times is a double-sided exponential. We show that such behaviors can be adequately described by a distributed-order fractional Fokker-Planck equations with a power-law weighting-function. We discuss the equations and the properties of their solutions, and connect this description with a scheme based on continuous-time random walks.

cond-mat.stat-mech↗

arXiv2002

Retarding Sub- and Accelerating Super-Diffusion Governed by Distributed Order Fractional Diffusion Equations

We propose diffusion-like equations with time and space fractional derivatives of the distributed order for the kinetic description of anomalous diffusion and relaxation phenomena, whose diffusion exponent varies with time and which, correspondingly, can not be viewed as self-affine random processes possessing a unique Hurst exponent. We prove the positivity of the solutions of the proposed equations and establish the relation to the Continuous Time Random Walk theory. We show that the distributed order time fractional diffusion equation describes the sub-diffusion random process which is subordinated to the Wiener process and whose diffusion exponent diminishes in time (retarding sub-diffusion) leading to superslow diffusion, for which the square displacement grows logarithmically in time. We also demonstrate that the distributed order space fractional diffusion equation describes super-diffusion phenomena when the diffusion exponent grows in time (accelerating super-diffusion).

cond-mat.stat-mech↗