arXiv subjects

Igor M. Sokolov

Publications and source records attributed to Igor M. Sokolov.

At least 55 records · Page 3Linked to original sources

arXiv2018

Reconstruction of substrate's diffusion landscape by the wavelet analysis of single particle diffusion tracks

We propose an approach to analysing single trajectories of a particle, which moves randomly on a landscape distinct parts of which result in sufficiently various diffusion coefficients. The method based on the mapping the cumulative sum of step-wise elementary displacement squared into a complex oscillating functions with the subsequent continuous wavelet analysis of the later allows the localisation of relatively homogeneous sub-regions and determining values of the diffusion coefficient within each such sub-region. This approach is applied to demonstrable test examples as well as to the reconstruction of the biological membrane's properties from the real data of single molecule walk tracing.

physics.bio-ph↗

arXiv2018

A second order scheme for a Robin boundary condition in random walk algorithms

Random Walk (RW) is a common numerical tool for modeling the Advection-Diffusion equation. In this work, we develop a second order scheme for incorporating a heterogeneous reaction (i.e., a Robin boundary condition) in the RW model. In addition, we apply the approach in two test cases. We compare the second order scheme with the first order one as well as with analytical and other numerical solution. We show that the new scheme can reduce the computational error significantly, relative to the first order scheme. This reduction comes at no additional computational cost.

physics.comp-ph↗

arXiv2017

Adiabatic elimination of inertia of the stochastic microswimmer driven by $α-$stable noise

We consider a microswimmer that moves in two dimensions at a constant speed and changes the direction of its motion due to a torque consisting of a constant and a fluctuating component. The latter will be modeled by a symmetric Lévy-stable ($α$-stable) noise. The purpose is to develop a kinetic approach to eliminate the angular component of the dynamics in order to find a coarse grained description in the coordinate space. By defining the joint probability density function of the position and of the orientation of the particle through the Fokker-Planck equation, we derive transport equations for the position-dependent marginal density, the particle's mean velocity and the velocity's variance. At time scales larger than the relaxation time of the torque $τ_ϕ$ the two higher moments follow the marginal density, and can be adiabatically eliminated. As a result, a closed equation for the marginal density follows. This equation which gives a coarse-grained description of the microswimmer's positions at time scales $t\gg τ_ϕ$, is a diffusion equation with a constant diffusion coefficient depending on the properties of the noise. Hence, the long time dynamics of a microswimmer can be described as a normal, diffusive, Brownian motion with Gaussian increments.

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

Transport in the cat's eye flow on intermediate time scales

We consider the advection-diffusion transport of tracers in a one-parameter family of plane periodic flows where the patterns of streamlines feature regions of confined circulation in the shape of "cat's eyes", separated by meandering jets with ballistic motion inside them. By varying the parameter, we proceed from the regular two-dimensional lattice of eddies without jets to the sinusoidally modulated shear flow without eddies. When a weak thermal noise is added, i.e. at large Péclet numbers, several intermediate time scales arise, with qualitatively and quantitatively different transport properties: depending on the parameter of the flow, the initial position of a tracer, and the aging time, motion of the tracers ranges from subdiffusive to super-ballistic. Extensive numerical simulations of the aged mean squared displacement for different initial conditions are compared to a theory based on a Lévy walk that describes the intermediate-time ballistic regime. Interplay of the walk process with internal circulation dynamics in the trapped state results at intermediate time scale in non-monotonic characteristics of aging.

physics.flu-dyn↗

arXiv2017

Anomalous diffusion in run-and-tumble motion

A random walk scheme, consisting of alternating phases of regular Brownian motion and Lévy walks, is proposed as a model for run-and-tumble bacterial motion. Within the continuous-time random walk approach we obtain the long-time and short-time behavior of the mean squared displacement of the walker as depending on the properties of dwelling time distribution in each phase. Depending on these distributions, normal diffusion, superdiffusion and ballistic spreading may arise.

physics.bio-ph↗

arXiv2017

Time averages in continuous time random walks

We investigate the time averaged squared displacement (TASD) of continuous time random walks with respect to the number of steps $N$, which the random walker performed during the data acquisition time $T$. We prove that the TASD, and as well the apparent diffusion constant, grow linearly with $N$, provided the steps possess a fourth moment and can not accumulate in small intervals. Consequently, the fluctuations of the latter are dominated by the fluctuations of $N$, and fluctuations of the walker's thermal history are irrelevant. Furthermore, we show that the relative scatter decays as $1/\sqrt{N}$, which suppresses all non-linear features in a plot of the TASD against the lag time. Parts of our arguments also hold for continuous time random walks with correlated steps.

cond-mat.stat-mech↗

arXiv2017

Effective Distances for Epidemics Spreading on Complex Networks

We show that the recently introduced logarithmic metrics used to predict disease arrival times on complex networks are approximations of more general network-based measures derived from random walks theory. Using the daily air-traffic transportation data we perform numerical experiments to compare the infection arrival time with this alternative metric that is obtained by accounting for multiple walks instead of only the most probable path. The comparison with direct simulations of arrival times reveals a higher correlation compared to the shortest path approach used previously. In addition our method allows to connect fundamental observables in epidemic spreading with the cumulant generating function of the hitting time for a Markov chain. Our results provides a general and computationally efficient approach to the problem using only algebraic methods.

physics.soc-ph↗

arXiv2016

Diffusion of Active Particles With Stochastic Torques Modeled as $α$-Stable Noise

We investigate the stochastic dynamics of an active particle moving at a constant speed under the influence of a fluctuating torque. In our model the angular velocity is generated by a constant torque and random fluctuations described as a Lévy-stable noise. Two situations are investigated. First, we study white Lévy noise where the constant speed and the angular noise generate a persistent motion characterized by the persistence time $τ_D$. At this time scale the crossover from ballistic to normal diffusive behavior is observed. The corresponding diffusion coefficient can be obtained analytically for the whole class of symmetric $α$-stable noises. As typical for models with noise-driven angular dynamics, the diffusion coefficient depends non-monotonously on the angular noise intensity. As second example, we study angular noise as described by an Ornstein-Uhlenbeck process with correlation time $τ_c$ driven by the Cauchy white noise. We discuss the asymptotic diffusive properties of this model and obtain the same analytical expression for the diffusion coefficient as in the first case which is thus independent on $τ_c$. Remarkably, for $τ_c>τ_D$ the crossover from a non-Gaussian to a Gaussian distribution of displacements takes place at a time $τ_G$ which can be considerably larger than the persistence time $τ_D$.

cond-mat.stat-mech↗

arXiv2016

Anomalous transport in cellular flows: The role of initial conditions and aging

We consider the diffusion-advection problem in two simple cellular flow models (often invoked as examples for subdiffusive tracer's motion) and concentrate on the intermediate time range, in which the tracer's motion indeed may show subdiffusion. We have performed extensive numerical simulations of the systems under different initial conditions, and show that the pure intermediate-time subdiffusion regime is only evident when the particles start at the border between different cells, i.e. at the separatrix, and is less pronounced or absent for other initial conditions. The motion moreover shows quite peculiar aging properties which are also mirrored in the behavior of the time-averaged mean squared displacement for single trajectories. This kind of behavior is due to the complex motion of tracers trapped inside the cell, and is absent in classical models based on continuous time random walks (CTRW) with no dynamics in the trapped state.

physics.flu-dyn↗

arXiv2016

Effective medium approximation for lattice random walks with long-range jumps

We consider the random walk on a lattice with random transition rates and arbitrarily long-range jumps. We employ Bruggeman's effective medium approximation (EMA) to find the disorder averaged (coarse-grained) dynamics. The EMA procedure replaces the disordered system with a cleverly guessed reference system in a self-consistent manner. We give necessary conditions on the reference system and discuss possible physical mechanisms of anomalous diffusion. In case of a power-law scaling between transition rates and distance, lattice variants of Levy-ights emerge as the effective medium, and the problem is solved analytically, bearing the effective anomalous diffusivity. Finally, we discuss several example distributions, and demonstrate very good agreement with numerical simulations.

cond-mat.dis-nn↗

arXiv2016

Reconstructing Interaction Potentials in Thin Films from Real-Space Images

We demonstrate that an inverse Monte Carlo approach allows to reconstruct effective interaction potentials from real-space images. The method is exemplified on monomolecular ethanol-water films imaged with scanning force microscopy (SFM), which provides the spatial distribution of the molecules. Direct Monte Carlo simulations with the reconstructed potential allow for obtaining characteristics of the system which are unavailable in the experiment, such as the heat capacity of the monomolecularly thin film, and for a prediction of the critical temperature of the demixing transition.

cond-mat.stat-mech↗

arXiv2015

Quantifying the non-ergodicity of scaled Brownian motion

We examine the non-ergodic properties of scaled Brownian motion, a non-stationary stochastic process with a time dependent diffusivity of the form $D(t)\simeq t^{α-1}$. We compute the ergodicity breaking parameter EB in the entire range of scaling exponents $α$, both analytically and via extensive computer simulations of the stochastic Langevin equation. We demonstrate that in the limit of long trajectory lengths $T$ and short lag times $Δ$ the EB parameter as function of the scaling exponent $α$ has no divergence at $α=1/2$ and present the asymptotes for EB in different limits. We generalise the analytical and simulations results for the time averaged and ergodic properties of scaled Brownian motion in the presence of ageing, that is, when the observation of the system starts only a finite time span after its initiation. The approach developed here for the calculation of the higher time averaged moments of the particle displacement can be applied to derive the ergodic properties of other stochastic processes such as fractional Brownian motion.

cond-mat.stat-mech↗

arXiv2015

Robust linear regression with broad distributions of errors

We consider the problem of linear fitting of noisy data in the case of broad (say $α$-stable) distributions of random impacts ("noise"), which can lack even the first moment. This situation, common in statistical physics of small systems, in Earth sciences, in network science or in econophysics, does not allow for application of conventional Gaussian maximum-likelihood estimators resulting in usual least-squares fits. Such fits lead to large deviations of fitted parameters from their true values due to the presence of outliers. The approaches discussed here aim onto the minimization of the width of the distribution of residua. The corresponding width of the distribution can either be defined via the interquantile distance of the corresponding distributions or via the scale parameter in its characteristic function. The methods provide the robust regression even in the case of short samples with large outliers, and are equivalent to the normal least squares fit for the Gaussian noises. Our discussion is illustrated by numerical examples.

physics.data-an↗

arXiv2014

Spectral properties of fractional Fokker-Plank operator for the Lévy flight in a harmonic potential

We present a detailed analysis of the eigenfunctions of the Fokker-Planck operator for the Lévy-Ornstein-Uhlenbeck process, their asymptotic behavior and recurrence relations, explicit expressions in coordinate space for the special cases of the Ornstein-Uhlenbeck process with Gaussian and with Cauchy white noise and for the transformation kernel, which maps the fractional Fokker-Planck operator of the Cauchy-Ornstein-Uhlenbeck process to the non-fractional Fokker-Planck operator of the usual Gaussian Ornstein-Uhlenbeck process. We also describe how non-spectral relaxation can be observed in bounded random variables of the Lévy-Ornstein-Uhlenbeck process and their correlation functions.

math-ph↗

arXiv2014

Anomalous Diffusion of Self-Propelled Particles in Directed Random Environments

We theoretically study the transport properties of self-propelled particles on complex structures, such as motor proteins on filament networks. A general master equation formalism is developed to investigate the persistent motion of individual random walkers, which enables us to identify the contributions of key parameters: the motor processivity, and the anisotropy and heterogeneity of the underlying network. We prove the existence of different dynamical regimes of anomalous motion, and that the crossover times between these regimes as well as the asymptotic diffusion coefficient can be increased by several orders of magnitude within biologically relevant control parameter ranges. In terms of motion in continuous space, the interplay between stepping strategy and persistency of the walker is established as a source of anomalous diffusion at short and intermediate time scales.

cond-mat.soft↗

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↗

arXiv2014

Dynamical localization and eigenstate localization in trap models

The one-dimensional random trap model with a power-law distribution of mean sojourn times exhibits a phenomenon of dynamical localization in the case where diffusion is anomalous: The probability to find two independent walkers at the same site, as given by the participation ratio, stays constant and high in a broad domain of intermediate times. This phenomenon is absent in dimensions two and higher. In finite lattices of all dimensions the participation ratio finally equilibrates to a different final value. We numerically investigate two-particle properties in a random trap model in one and in three dimensions, using a method based on spectral decomposition of the transition rate matrix. The method delivers a very effective computational scheme producing numerically exact results for the averages over thermal histories and initial conditions in a given landscape realization. Only a single averaging procedure over disorder realizations is necessary. The behavior of the participation ratio is compared to other measures of localization, as for example to the states' gyration radius, according to which the dynamically localized states are extended. This means that although the particles are found at the same site with a high probability, the typical distance between them grows. Moreover the final equilibrium state is extended both with respect to its gyration radius and to its Lyapunov exponent. In addition, we show that the phenomenon of dynamical localization is only marginally connected with the spectrum of the transition rate matrix, and is dominated by the properties of its eigenfunctions which differ significantly in dimensions one and three.

physics.data-an↗