The paper presents a multidimensional model for nonlinear Markovian random walks that generalizes one we developed previously (Phys. Rev. E v.79, 011110, 2009) in order to describe the Levy type stochastic processes in terms of continuous trajectories of walker motion. This approach may open a way to treat Levy flights or Levy random walks in inhomogeneous media or systems with boundaries in the future.
The proposed model assumes the velocity of a wandering particle to be affected by a linear friction and a nonlinear Langevin force whose intensity is proportional to the magnitude of the velocity for its large values. Based on the singular perturbation technique the corresponding Fokker-Planck equation is analyzed and the relationship between the system parameters and the Levy exponent is found. Following actually the previous paper we demonstrate also that anomalously long displacements of the wandering particle are caused by extremely large fluctuations in the particle velocity whose duration is determined by the system parameters rather than the duration of the observation interval. In this way we overcome the problem of ascribing to Levy random walks non-Markov properties.
We provide explicit formulas for asymptotic densities of the 2- and 3-dimensional ballistic Lévy walks. It turns out that in the 3D case the densities are given by elementary functions. The densities of the 2D Lévy walks are expressed in terms of hypergeometric functions and the right-side Riemann-Liouville fractional derivative which allows to efficiently evaluate them numerically. The theoretical results agree with Monte-Carlo simulations. The obtained functions solve certain differential equations with the fractional material derivative.
We study the diffusion equation with a position-dependent, power-law diffusion coefficient. The equation possesses the Riesz-Weyl fractional operator and includes a memory kernel. It is solved in the diffusion limit of small wave numbers. Two kernels are considered in detail: the exponential kernel, for which the problem resolves itself to the telegrapher's equation, and the power-law one. The resulting distributions have the form of the Lévy process for any kernel. The renormalized fractional moment is introduced to compare different cases with respect to the diffusion properties of the syste…
We consider a Markovian jumping process with two absorbing barriers, for which the waiting-time distribution involves a position-dependent coefficient. We solve the Fokker-Planck equation with boundary conditions and calculate the mean first passage time (MFPT) which appears always finite, also for the subdiffusive case. Then, for the case of the jumping-size distribution in form of the Lévy distribution, we determine the probability density distributions and MFPT by means of numerical simulations. Dependence of the results on process parameters, as well as on the Lévy distribution width, is d…
We consider a previously devised model describing Levy random walks (Phys. Rev E 79, 011110; 80, 031148, (2009)). It is demonstrated numerically that the given model describes Levy random walks with superdiffusive, ballistic, as well as superballistic dynamics. Previously only the superdiffusive regime has been analyzed. In this model the walker velocity is governed by a nonlinear Langevin equation. Analyzing the crossover from small to large time scales we find the time scales on which the velocity correlations decay and the walker motion essentially exhibits Levy statistics. Our analysis is …
We obtain the first passage time density for a Lévy flight random process from a subordination scheme. By this method, we infer the asymptotic behavior directly from the Brownian solution and the Sparre Andersen theorem, avoiding explicit reference to the fractional diffusion equation. Our results corroborate recent findings for Markovian Lévy flights and generalize to broad waiting times.
We derive a functional equation for the mean first-passage time (MFPT) of a generic self-similar Markovian continuous process to a target in a one-dimensional domain and obtain its exact solution. We show that the obtained expression of the MFPT for continuous processes is actually different from the large system size limit of the MFPT for discrete jump processes allowing leapovers. In the case considered here, the asymptotic MFPT admits non-vanishing corrections, which we call residual MFPT. The case of L/'evy flights with diverging variance of jump lengths is investigated in detail, in parti…
The integro-differential wave equation for the probability density function for a classical one-dimensional Lévy walk with continuous sample paths has been derived. This equation involves a classical wave operator together with memory integrals describing the spatio-temporal coupling of the Lévy walk. It is valid for any running time PDF and it does not involve any long-time large-scale approximations. It generalizes the well-known telegraph equation obtained from the persistent random walk. Several non-Markovian cases are considered when the particle's velocity alternates at the gamma and pow…
A Levy walk is a non-Markovian stochastic process in which the elementary steps of the walker consist of motion with constant speed in randomly chosen directions and for a random period of time. The time of flight is chosen from a long-tailed distribution with a finite mean but an infinite variance. Here we consider an open system with boundary injection and removal of particles, at prescribed rates, and study the steady state properties of the system. In particular, we compute density profiles, current and current fluctuations in this system. We also consider the case of a finite density of L…
The Caldeira-Leggett model of quantum Brownian motion is generalized using a generic velocity-dependent coupling. That leads to the description of a set of models able to capture Markovian and non-Markovian versions of Brownian and Lévy statistics, depending on the functional form of the coupling and on the spectral function of the reservoir. One specific coupling force is found that establishes a connection with Lévy statistics of cold atoms in \textit{Sisyphus} laser cooling. In the low-velocity limit, this also gives rise to additional inertia of the Brownian particle, reproducing the Abrah…
We study Markovian continuous-time random walk models for Lévy flights and we show an example in which the convergence to stable densities is not guaranteed when jumps follow a bi-modal power-law distribution that is equal to zero in zero. The significance of this result is two-fold: i) with regard to the probabilistic derivation of the fractional diffusion equation and also ii) with regard to the concept of site fidelity in the framework of Lévy-like motion for wild animals.
We propose a variety of models of random walk, discrete in space and time, suitable for simulating stable random variables of arbitrary index $α$ ($0< α\le 2$), in the symmetric case. We show that by properly scaled transition to vanishing space and time steps our random walk models converge to the corresponding continuous Markovian stochastic processes, that we refer to as Levy-Feller diffusion processes.
The paper is devoted to the relationship between the continuous Markovian description of Levy flights developed previously and their equivalent representation in terms of discrete steps of a wandering particle, a certain generalization of continuous time random walks. Our consideration is confined to the one-dimensional model for continuous random motion of a particle with inertia. Its dynamics governed by stochastic self-acceleration is described as motion on the phase plane {x,v} comprising the position x and velocity v=dx/dt of the given particle. A notion of random walks inside a certain n…
We introduce a model for directed percolation with a long-range temporal diffusion, while the spatial diffusion is kept short ranged. In an interpretation of directed percolation as an epidemic process, this non-Markovian modification can be understood as incubation times, which are distributed accordingly to a Levy distribution. We argue that the best approach to find the effective action for this problem is through a generalization of the Cardy-Sugar method, adding the non-Markovian features into the geometrical properties of the lattice. We formulate a field theory for this problem and reno…
We study the kinetics of protein folding via statistical energy landscape theory. We concentrate on the local-connectivity case, where the configurational changes can only occur among neighboring states, with the folding progress described in terms of an order parameter given by the fraction of native conformations. The non-Markovian diffusion dynamics is analyzed in detail and an expression for the mean first-passage time (MFPT) from non-native unfolded states to native folded state is obtained. It was found that the MFPT has a V-shaped dependence on the temperature. We also find that the MFP…
A continuous Markovian model for truncated Levy random walks is proposed. It generalizes the approach developed previously by Lubashevsky et al. Phys. Rev. E 79, 011110 (2009); 80, 031148 (2009), Eur. Phys. J. B 78, 207 (2010) allowing for nonlinear friction in wondering particle motion and saturation of the noise intensity depending on the particle velocity. Both the effects have own reason to be considered and individually give rise to truncated Levy random walks as shown in the paper. The nonlinear Langevin equation governing the particle motion was solved numerically using an order 1.5 str…
Some specific features and extensions of the continuous time random walk (CTRW) approach are analyzed in detail within the Markovian representation (MR) and CTRW-based non-Markovian stochastic Liouville equation (SLE). In the MR CTRW processes are represented by multidimensional Markovian ones. In this representation the probability distribution function (PDF) W(t) of fluctuation renewals is associated with that of reoccurrences in a certain jump state of some Markovian controlling process. Within the MR the non-Markovian SLE, which describes the effect of CTRW-like noise on relaxation of dyna…
We introduce a new method to accurately and efficiently estimate the effective dynamics of collective variables in molecular simulations. Such reduced dynamics play an essential role in the study of a broad class of processes, ranging from chemical reactions in solution to conformational changes in biomolecules or phase transitions in condensed matter systems. The standard Markovian approximation often breaks down due to the lack of a proper separation of time scales and memory effects must be taken into account. Using a parametrization based on hidden auxiliary variables, we obtain a generali…
