Lévy walk is a popular and more `physical' model to describe the phenomena of superdiffusion, because of its finite velocity. The movements of particles are under the influences of external potentials almost at anytime and anywhere. In this paper, we establish a Langevin system coupled with a subordinator to describe the Lévy walk in the time-dependent periodic force field. The effects of external force are detected and carefully analyzed, including nonzero first moment (even though the force is periodic), adding an additional dispersion on the particle position, the consistent influence on the ensemble- and time-averaged mean-squared displacement, etc. Besides, the generalized Klein-Kramers equation is obtained, not only for the time-dependent force but also for space-dependent one.
Continuous time random walks and Langevin equations are two classes of stochastic models for describing the dynamics of particles in the natural world. While some of the processes can be conveniently characterized by both of them, more often one model has significant advantages (or has to be used) compared with the other one. In this paper, we consider the weakly damped Langevin system coupled with a new subordinator|$α$-dependent subordinator with $1<α<2$. We pay attention to the diffusion behaviour of the stochastic process described by this coupled Langevin system, and find the super-ballis…
Memory effects, sometimes, can not be neglected. In the framework of continuous time random walk, memory effect is modeled by the correlated waiting times. In this paper, we derive the two-point probability distribution of the stochastic process with correlated increments as well as the one of its inverse process, and present the Langevin description of Lévy walk with memory, i.e., correlated waiting times. Based on the built Langevin picture, the properties of aging and nonstationary are discussed. The Langevin system exhibits sub-ballistic superdiffusion if the friction force is involved, wh…
Lévy walk is a practical model and has wide applications in various fields. Here we focus on the effect of an external constant force on the Lévy walk with the exponent of the power-law distributed flight time $α\in(0,2)$. We add the term $Fη(s)$ ($η(s)$ is the Lévy noise) on a subordinated Langevin system to characterize such a constant force, being effective on the velocity process for all physical time after the subordination. We clearly show the effect of the constant force $F$ on this Langevin system and find this system is like the continuous limit of the collision model. The first momen…
Systems living in complex non equilibrated environments often exhibit subdiffusion characterized by a sublinear power-law scaling of the mean square displacement. One of the most common models to describe such subdiffusive dynamics is the continuous time random walk (CTRW). Stochastic trajectories of a CTRW can be described mathematically in terms of a subordination of a normal diffusive process by an inverse Levy-stable process. Here, we propose a simpler Langevin formulation of CTRWs without subordination. By introducing a new type of non-Gaussian noise, we are able to express the CTRW dynam…
Dynamical random walk of classical particle in thermodynamically equilibrium fluctuating medium, - Gaussian random potential field, - is considered in the framework of explicit stochastic representation of deterministic interactions. We discuss corresponding formally exact Langevin equations for the particle's trajectory and show that Marcovian kinetic equation approximation to them is inadequate, - even (and especially) in case of spatially-temporally short-correlated field, - since ignores such actual effects of exponential instability of the trajectory (in respect to small perturbations) as…
Expanding medium is very common in many different fields, such as biology and cosmology. It brings a nonnegligible influence on particle's diffusion, which is quite different from the effect of an external force field. The dynamic mechanism of particle's motion in expanding medium has only been investigated in the framework of continuous-time random walk. To focus on more diffusion processes and physical observables, we build the Langevin picture of anomalous diffusion in expanding medium, and conduct detailed analyses in the framework of Langevin equation. With the help of a subordinator, bot…
We discuss general multi-dimensional stochastic processes driven by a system of Langevin equations with multiplicative white noise. In particular, we address the problem of how time reversal diffusion processes are affected by the variety of conventions available to deal with stochastic integrals. We present a functional formalism to built up the generating functional of correlation functions without any type of discretization of the Langevin equations at any intermediate step. The generating functional is characterized by a functional integration over two sets of commuting variables as well a…
We investigate the time average mean square displacement $\overline{δ^2}(x(t))=\int_0^{t-Δ}[x(t^\prime+Δ)-x(t^\prime)]^2 dt^\prime/(t-Δ)$ for fractional Brownian and Langevin motion. Unlike the previously investigated continuous time random walk model $\overline{δ^2}$ converges to the ensemble average $<x^2 > \sim t^{2 H}$ in the long measurement time limit. The convergence to ergodic behavior is however slow, and surprisingly the Hurst exponent $H=3/4$ marks the critical point of the speed of convergence. When $H<3/4$, the ergodicity breaking parameter ${EB} = {Var} (\overline{δ^2}) / < \over…
Motivated by subdiffusive motion of bio-molecules observed in living cells we study the stochastic properties of a non-Brownian particle whose motion is governed by either fractional Brownian motion or the fractional Langevin equation and restricted to a finite domain. We investigate by analytic calculations and simulations how time-averaged observables (e.g., the time averaged mean squared displacement and displacement correlation) are affected by spatial confinement and dimensionality. In particular we study the degree of weak ergodicity breaking and scatter between different single trajecto…
Using Langevin dynamics simulations, we study the hysteresis in unzipping of longer double stranded DNA chains whose ends are subjected to a time dependent periodic force with frequency $ω$ and amplitude $G$ keeping the other end fixed. We find that the area of the hysteresis loop, $A_{loop}$, scales as $1/ω$ at higher frequencies, whereas it scales as $(G-G_c)^αω^β$ with exponents $α=1$ and $β=1.25$ in the low frequency regime. These values are same as the exponents obtained in Monte Carlo simulation studies of a directed self avoiding walk model of a homopolymer DNA [R. Kapri, Phys. Rev. E 9…
We present a heuristic derivation of the first passage time exponent for the integral of a random walk [Y. G. Sinai, Theor. Math. Phys. {\bf 90}, 219 (1992)]. Building on this derivation, we construct an estimation scheme to understand the first passage time exponent for the integral of the integral of a random walk, which is numerically observed to be $0.220\pm0.001$. We discuss the implications of this estimation scheme for the $n{\rm th}$ integral of a random walk. For completeness, we also address the $n=\infty$ case. Finally, we explore an application of these processes to an extended, el…
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 …
The Langevin equation for a particle (`random walker') moving in d-dimensional space under an attractive central force, and driven by a Gaussian white noise, is considered for the case of a power-law force, F(r) = - Ar^{-sigma}. The `persistence probability', P_0(t), that the particle has not visited the origin up to time t, is calculated. For sigma > 1, the force is asymptotically irrelevant (with respect to the noise), and the asymptotics of P_0(t) are those of a free random walker. For sigma < 1, the noise is (dangerously) irrelevant and the asymptotics of P_0(t) can be extracted from a wea…
Anomalous diffusion phenomena have been observed in many complex physical and biological systems. One significant advance recently is the physical extension of particle's motion in static medium to uniformly (and even nonuniformly) expanding medium. The dynamic mechanism of particle's motion in the nonuniformly expanding medium has only been investigated in the framework of continuous-time random walk. To study more physical observables and supplement the theory of the expanding medium problems, we characterize the nonuniformly expanding medium with a spatial-temporal dependent scale factor $a…
The problem of biological motion is a very intriguing and topical issue. Many efforts are being focused on the development of novel modeling approaches for the description of anomalous diffusion in biological systems, such as the very complex and heterogeneous cell environment. Nevertheless, many questions are still open, such as the joint manifestation of statistical features in agreement with different models that can be also somewhat alternative to each other, e.g., Continuous Time Random Walk (CTRW) and Fractional Brownian Motion (FBM). To overcome these limitations, we propose a stochasti…
We derive general expressions for the free energy, entropy production and entropy extraction rates for a Brownian particle that walks in a viscous medium where the dynamics of its motion is governed by the Langevin equation. It is shown that when the system is out of equilibrium, it constantly produces entropy and at the same time extracts entropy out of the system. Its entropy production and extraction rates decrease in time and saturate to a constant value. In long time limit, the rate of entropy production balances the rate of entropy extraction and at equilibrium both entropy production an…
In this article, we generalize the recent Discrete Time Random Walk (DTRW) algorithm, which was introduced for the computation of probability densities of fractional diffusion. Although it has the same computational complexity and shares the same desirable features (consistency, conservation of mass, strictly non-negative solutions), it applies to virtually every conceivable Continuous Time Random Walk (CTRW) limit process, which we define broadly as the limit of a sequence of jump processes with renewals at every jump. Our only restrictive assumption is the boundedness and continuity of coeff…
Many self-propelled objects are large enough to exhibit inertial effects but still suffer from environmental fluctuations. The corresponding basic equations of motion are governed by active Langevin dynamics which involve inertia, friction and stochastic noise for both the translational and orientational degrees of freedom coupled via the self-propulsion along the particle orientation. In this paper, we generalize the active Langevin model to time-dependent parameters and explicitly discuss the effect of time-dependent inertia for achiral and chiral particles. Realizations of this situation ar…
Aging phenomena have been observed in numerous physical systems. Many statistical quantities depend on the aging time $t_a$ for aging anomalous diffusion processes. This paper pays more attention to how an external force field affects the aging Lévy walk. Based on the Langevin picture of Lévy walk and generalized Green-Kubo formula, we investigate the quantities which include the ensemble- and time-averaged mean-squared displacements in both weak aging $t_a\ll t$ and strong aging $t_a\gg t$ cases, and compare them to the quantities in the absence of any force field. Two typical force fields, c…
Random walks on graphs are widely used in all sciences to describe a great variety of phenomena where dynamical random processes are affected by topology. In recent years, relevant mathematical results have been obtained in this field, and new ideas have been introduced, which can be fruitfully extended to different areas and disciplines. Here we aim at giving a brief but comprehensive perspective of these progresses, with a particular emphasis on physical aspects.
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 veloci…
Considering the dynamics of non-interacting particles randomly moving on a lattice, the occurrence of a discontinuous transition in the values of the lattice parameters (lattice spacing and hopping times) determines the uprisal of two lattice phases. In this Letter we show that the hyperbolic hydrodynamic model obtained by enforcing the boundedness of lattice velocities derived by Giona (2018) correctly describes the dynamics of the system and permits to derive easily the boundary condition at the interface, which, contrarily to the common belief, involves the lattice velocities in the two pha…
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 sam…
We have derived a fractional Fokker-Planck equation for subdiffusion in a general space-and- time-dependent force field from power law waiting time continuous time random walks biased by Boltzmann weights. The governing equation is derived from a generalized master equation and is shown to be equivalent to a subordinated stochastic Langevin equation.
We investigate the dynamics of a particle executing a general Continuous Time Random Walk (CTRW) in three dimensions under the influence of arbitrary time-varying external fields. Contrary to the general approach in recent works, our method invokes neither the Fractional Fokker-Planck equation (FFPE) nor the Stochastic Langevin Equation (SLE). Rather, we use rigorous probability arguments to derive the general expression for moments of all orders of the position probability density of the random walker for arbitrary waiting time density and jump probability density. Closed form expression for …
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…
Waves fail to propagate in random media. First predicted for quantum particles in the presence of a disordered potential, Anderson localization has been observed also in classical acoustics, electromagnetism and optics. Here, for the first time, we report the observation of Anderson localization of pairs of entangled photons in a two-particle discrete quantum walk affected by position dependent disorder. A quantum walk on a disordered lattice is realized by an integrated array of interferometers fabricated in glass by femtosecond laser writing. A novel technique is used to introduce a controll…
Continuous Time Random Walk models (CTRW) of anomalous diffusion are studied, where the anomalous exponent $β(x) \in (0,1)$ varies in space. This type of situation occurs e.g. in biophysics, where the density of the intracellular matrix varies throughout a cell. Scaling limits of CTRWs are known to have probability distributions which solve fractional Fokker-Planck type equations (FFPE). This correspondence between stochastic processes and FFPE solutions has many useful extensions e.g. to nonlinear particle interactions and reactions, but has not yet been sufficiently developed for FFPEs of th…
The comb model is a simplified description for anomalous diffusion under geometric constraints. It represents particles spreading out in a two-dimensional space where the motions in the x-direction are allowed only when the y coordinate of the particle is zero. Here, we propose an extension for the comb model via Langevin-like equations driven by fractional Gaussian noises (long-range correlated). By carrying out computer simulations, we show that the correlations in the y-direction affect the diffusive behavior in the x-direction in a non-trivial fashion, resulting in a quite rich diffusive s…
We will construct a theory which can explain the dynamics toward the steady state self-gravitating systems (SGSs) where many particles interact via the gravitational force. Real examples of SGS in the universe are globular clusters and galaxies. The idea is to represent an interaction by which a particle of the system is affected from the others by a special random force. That is, we will use a special Langevin equation, just as the normal Langevin equation can unveil the dynamics toward the steady state described by the Maxwell-Boltzmann distribution. However, we cannot introduce the randomne…
We study the dynamics of a starving random walk in general spatial dimension $d$. This model represents an idealized description for the fate of an unaware forager whose motion is not affected by the presence or absence of resources. The forager depletes its environment by consuming resources and dies if it wanders too long without finding food. In the exactly-solvable case of one dimension, we explicitly derive the average lifetime of the walk and the distribution for the number of distinct sites visited by the walk at the instant of starvation. We also give a heuristic derivation for the ave…
We investigate the impact of external periodic potentials on superdiffusive random walks known as Levy flights and show that even strongly superdiffusive transport is substantially affected by the external field. Unlike ordinary random walks, Levy flights are surprisingly sensitive to the shape of the potential while their asymptotic behavior ceases to depend on the Levy index $μ$. Our analysis is based on a novel generalization of the Fokker-Planck equation suitable for systems in thermal equilibrium. Thus, the results presented are applicable to the large class of situations in which superdi…
We give an exact solution to the generalized Langevin equation of motion of a charged Brownian particle in a uniform magnetic field that is driven internally by an exponentially-correlated stochastic force. A strong dissipation regime is described in which the ensemble-averaged fluctuations of the velocity exhibit transient oscillations that arise from memory effects. Also, we calculate generalized diffusion coefficients describing the transport of these particles and briefly discuss how they are affected by the magnetic field strength and correlation time. Our asymptotic results are extended …
Double (or parity conserving) branching annihilating random walk, introduced by Sudbury in '90, is a one-dimensional non-attractive particle system in which positive and negative particles perform nearest neighbor hopping, produce two offsprings to neighboring lattice points and annihilate when they meet. Given an odd number of initial particles, positive recurrence as seen from the leftmost particle position was first proved by Belitsky, Ferrari, Menshikov and Popov in '01 and, subsequently in a much more general setup, in the article by Sturm and Swart (Tightness of voter model interfaces) i…
A heterogeneous continuous time random walk is an analytical formalism for studying and modeling diffusion processes in heterogeneous structures on microscopic and macroscopic scales. In this paper we study both analytically and numerically the effects of spatio-temporal heterogeneities onto the diffusive dynamics on different types of networks. We investigate how the distribution of the first passage time is affected by the global topological network properties and heterogeneities in the distributions of the travel times. In particular, we analyze transport properties of random networks and d…
