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Aleksei V. Chechkin

Publications and source records attributed to Aleksei V. Chechkin.

44 records · Page 3Linked to original sources
arXiv2009

Matrix approach to discrete fractional calculus II: partial fractional differential equations

A new method that enables easy and convenient discretization of partial differential equations with derivatives of arbitrary real order (so-called fractional derivatives) and delays is presented and illustrated on numerical solution of various types of fractional diffusion equation. The suggested method is the development of Podlubny's matrix approach (Fractional Calculus and Applied Analysis, vol. 3, no. 4, 2000, 359--386). Four examples of numerical solution of fractional diffusion equation with various combinations of time/space fractional derivatives (integer/integer, fractional/integer, integer/fractional, and fractional/fractional) with respect to time and to the spatial variable are provided in order to illustrate how simple and general is the suggested approach. The fifth example illustrates that the method can be equally simply used for fractional differential equations with delays. A set of MATLAB routines for the implementation of the method as well as sample code used to solve the examples have been developed.

math.NA↗
arXiv2008

Bulk-mediated surface diffusion on a cylinder: propagators and crossovers

We consider the effective surface motion of a particle that freely diffuses in the bulk and intermittently binds to that surface. From an exact approach we derive various regimes of the effective surface motion characterized by physical rates for binding/unbinding and the bulk diffusivity. We obtain a transient regime of superdiffusion and, in particular, a saturation regime characteristic for the cylindrical geometry. This saturation, however, in a finite system is not terminal but eventually turns over to normal surface diffusion. The first passage behavior of particles to the cylinder surface is derived. Consequences for actual systems are discussed.

cond-mat.stat-mech↗
arXiv2007

Leapover lengths and first passage time statistics for Lévy flights

Exact results for the first passage time and leapover statistics of symmetric and one-sided Levy flights (LFs) are derived. LFs with stable index alpha are shown to have leapover lengths, that are asymptotically power-law distributed with index alpha for one-sided LFs and, surprisingly, with index alpha/2 for symmetric LFs. The first passage time distribution scales like a power-law with index 1/2 as required by the Sparre Andersen theorem for symmetric LFs, whereas one-sided LFs have a narrow distribution of first passage times. The exact analytic results are confirmed by extensive simulations.

cond-mat.stat-mech↗
arXiv2007

Fractal Properties of Anomalous Diffusion in Intermittent Maps

An intermittent nonlinear map generating subdiffusion is investigated. Computer simulations show that the generalized diffusion coefficient of this map has a fractal, discontinuous dependence on control parameters. An amended continuous time random walk theory well approximates the coarse behavior of this quantity in terms of a continuous function. This theory also reproduces a full suppression of the strength of diffusion, which occurs at the dynamical phase transition from one type of diffusive behavior to another. Similarly, the probability density function of this map exhibits a nontrivial fine structure while its coarse functional form is governed by a time fractional diffusion equation. A more detailed understanding of the irregular structure of the generalized diffusion coefficient is provided by an anomalous Taylor-Green-Kubo formula establishing a relation to de Rham-type fractal functions.

nlin.CD↗
arXiv2006

Barrier crossing driven by Levy noise: Universality and the Role of Noise Intensity

We study the barrier crossing of a particle driven by white symmetric Levy noise of index $α$ and intensity $DD for three different generic types of potentials: (a) a bistable potential; (b) a metastable potential; and (c) a truncated harmonic potential. For the low noise intensity regime we recover the previously proposed algebraic dependence on $D$ of the characteristic escape time, $T_{\mathrm{esc}}\simeq C(α)/D^{μ(α)}$, where $C(α)$ is a coefficient. It is shown that the exponent $μ(α)$ remains approximately constant, $μ\approx 1$ for $0<α<2$; at $α=2$ the power-law form of $T_{\mathrm{esc}}$ changes into the known exponential dependence on 1/D; it exhibits a divergence-like behavior as $α$ approaches 2. In this regime we observe a monotonous increase of the escape time $T_{\mathrm{esc}}$ with increasing $α$ (keeping the noise intensity $D$ constant). The probability density of the escape time decays exponentially. In addition, for low noise intensities the escape times correspond to barrier crossing by multiple Levy steps. For high noise intensities, the escape time curves collapse for all values of $α$. At intermediate noise intensities, the escape time exhibits non-monotonic dependence on the index $α$, while still retaining the exponential form of the escape time density.

cond-mat.stat-mech↗
arXiv2003

First passage and arrival time densities for Lévy flights and the failure of the method of images

We discuss the first passage time problem in the semi-infinite interval, for homogeneous stochastic Markov processes with L{é}vy stable jump length distributions $λ(x)\sim\ell^α/|x|^{1+α}$ ($|x|\gg\ell$), namely, L{é}vy flights (LFs). In particular, we demonstrate that the method of images leads to a result, which violates a theorem due to Sparre Andersen, according to which an arbitrary continuous and symmetric jump length distribution produces a first passage time density (FPTD) governed by the universal long-time decay $\sim t^{-3/2}$. Conversely, we show that for LFs the direct definition known from Gaussian processes in fact defines the probability density of first arrival, which for LFs differs from the FPTD. Our findings are corroborated by numerical results.

cond-mat.stat-mech↗
arXiv2003

Lévy flights in a steep potential well

Lévy flights in steeper than harmonic potentials have been shown to exhibit finite variance and a critical time at which a bifurcation from an initial mono-modal to a terminal bimodal distribution occurs (Chechkin et al., Phys. Rev. E {\bf 67}, 010102(R) (2003)). In this paper, we present a detailed study of Lévy flights in potentials of the type U(x)∝|x|^c with c>2. Apart from the bifurcation into bimodality, we find the interesting result that for c>4 a trimodal transient exists due to the temporal overlap between the decay of the central peak around the initial $δ$-condition and the building up of the two emerging side-peaks, which are characteristic for the stationary state. Thus, for certain system parameters there exists a transient tri-modal distribution of the Lévy flight. These properties of LFs in external potentials of the power-law type can be represented by certain phase diagrams. We also present details about the proof of multi-modality and the numerical procedures to establish the probability distribution of the process.

cond-mat.stat-mech↗