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# Langevin Picture of L,vy Walks and Their Extensions

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## Abstract

In this paper we derive Langevin picture of Lévy walks. Applying recent advances in the theory of coupled continuous time random walks we find a limiting process of the properly scaled Lévy walk. Next, we introduce extensions of Levy walks, in which jump sizes are some functions of waiting times. We prove that under proper scaling conditions, such generalized Lévy walks converge in distribution to the appropriate limiting processes. We also derive the corresponding fractional diffusion equations and investigate behavior of the mean square displacements of the limiting processes, showing that different coupling functions lead to various types of anomalous diffusion.

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... This approach is used in our manuscript. It should be added that LWs can also be defined using subordinated Langevin equations, see [7,17] for the details. Let T i be a sequence of independent, identically distributed (IID) waiting times such that P(T i > x) ≈ Cx −α when x → ∞, α ∈ (0, 1), C > 0. ...
... This is due to the fact that for α > 1 the diffusion limit of LW is the well-known and studied α-stable Lévy motion (Lévy flight) [16]. This is in sharp contrast with the case α ∈ (0, 1), for which the diffusion limit is a completely different, much more complicated process [16,17]. ...
... Now we can define limit processes of Lévy walks. For jump models of Lévy walks we have [17,18] 1 n R(nt) ...
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In this paper we study properties of the diffusion limits of three different models of Lévy walks (LW). Exact asymptotic behavior of their trajectories is found using LePage series representation. We also prove an existing conjecture about total variation of LW sample paths. Based on this conjecture we verify martingale properties of the limit processes for LW. We also calculate their probability density functions and apply this result to determine the potential density of the associated non-symmetric α -stable processes. The obtained theoretical results for continuous LW can be used to recognize and verify this type of processes from anomalous diffusion experimental data. In particular they can be used to estimate parameters from experimental data using maximum likelihood methods.
... Clearly, the condition |J i | = T i is satisfied. The sequence of jumps J i defined above belongs to the domain of attraction of α-stable distribution [13] n −1/α ...
... R(t) is also known as wait-first Lévy walk in the literature [25], since the walker at the beginning of its motion (t = 0) first waits and then performs the jump. As shown in [13], R(t) obeys the following scaling limit in distribution ...
... Additionally, the instants of jumps as well as the respective jump lengths of the processes L α (t) and S α (t) are exactly the same. Also, the probability density function (PDF) p t (x) of X(t) satisfies the following fractional equation [10,13] ...
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In this paper we derive explicit formulas for the densities of Levy walks. Our results cover both jump-first and wait-first scenarios. The obtained densities solve certain fractional differential equations involving fractional material derivative operators. In the particular case, when the stability index is rational, the densities can be represented as an integral of Meijer G function. This allows to efficiently evaluate them numerically. Our results show perfect agreement with the Monte Carlo simulations.
... Recently, we have shown that the distribution of the diffusion coefficients of the TAMSDs in the stored-energy-driven Lévy flight (SEDLF) is different from that in CTRW [6]. The SEDLF is a CTRW with jump lengths correlated with trapping times [24,26,28]. One of the most typical examples for such a correlated motion can be observed in Lévy walk [38]. ...
... In terms of an ensemble average, SEDLF exhibits a whole spectrum of diffusion: sub-, normal-, and super-diffusion, depending on the coupling parameter [6,24,26,28].Because distributional behavior of the time-averaged observables such as the diffusion coefficients in SEDLF is different from that in CTRW, it is important to construct a phase diagram in terms of the power-law exponent of the MSD as well as the form of the distribution function of the TAMSD. Here, we provide the phase diagram for the whole parameters range in SEDLF. ...
... where D t = N t k=0 l 2 k /t. We note that the relation (28) does not hold if the random walker moves with constant speed as in Lévy walk because (x t+ − x t ) 2 is simply zero in SEDLF but not in Lévy walk. In fact, the TAMSD does not increase linearly with time in Lévy walk [16][17][18]. ...
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Phase diagram based on the mean square displacement (MSD) and the distribution of diffusion coefficients of the time-averaged MSD for the stored-energy-driven L\'evy flight (SEDLF) is presented. In the SEDLF, a random walker cannot move while storing energy, and it jumps by the stored energy. The SEDLF shows a whole spectrum of anomalous diffusions including subdiffusion and superdiffusion, depending on the coupling parameter between storing time (trapping time) and stored energy. This stochastic process can be investigated analytically with the aid of renewal theory. Here, we consider two different renewal processes, i.e., ordinary renewal process and equilibrium renewal process, when the mean trapping time does not diverge. We analytically show the phase diagram according to the coupling parameter and the power exponent in the trapping-time distribution. In particular, we find that distributional behavior of time-averaged MSD intrinsically appears in superdiffusive as well as normal diffusive regime even when the mean trapping time does not diverge.
... A class of processes that combines successfully two previously mentioned elements, which are heavy-tailed features in space domain and finite variance of trajectory is called wait-first Lévy walk [16,33,21,40,20]. Mathematically it is simply a coupled CTRWs model with the waiting time equal (up to some multiplicative constant) to the length of the corresponding jump. ...
... It is worth mentioning that both processes belongs to the class of cádlág processes. Another very important difference from the modelling point of view is that wait-first LW has all moments finite, while jump-first LW process has all moments infinite [22,21,40,41,20]. ...
Preprint
In this paper we investigate the asymptotic properties of the wait-first and jump-first L\'evy walk with rest, which is a generalization of standard jump-first and jump-first L\'evy walk that assumes each waiting time in the model is a sum of two positive random variables. We investigate the asymptotic properties of the theses new-type waiting times. Next we use the previous results of this paper together with continuous mapping approach to establish the main result, which is a functional convergence in Skorokhod $\mathbb{J}_1$ topology for the L\'evy walks with rests.
... Anomalous superdiffusive processes describe a wide variety of systems arising in different disciplines such as physics and biology. Levy walks is one of the simpler models that lead to this feature [1][2][3][4][5][6]. It is a generalization of the classical Drude model where a particle moves, in successive random directions, with constant velocity during random periods of time. ...
... The corrections to this expression are of order (1/t). On the other hand, we notice that w ± are the asymptotic transition probabilities defined in Eq. (6). For µ ≷ 1/2, for increasing (decreasing) x t the probability P t (k) increase (decrease). ...
Article
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The ensemble properties and time-averaged observables of a memory-induced diffusive-superdiffusive transition are studied. The model consists in a random walker whose transitions in a given direction depend on a weighted linear combination of the number of both right and left previous transitions. Depending on the characteristic memory parameters, the ensemble behavior may be normal, superdiffusive, or ballistic. In contrast, the time-averaged mean squared displacement is equal to that of a normal undriven random walk, which renders the process non-ergodic. In addition, and similarly to Levy walks [Godec and Metzler, Phys. Rev. Lett. 110 , 020603 (2013)], for trajectories of finite duration the time-averaged displacement apparently become random with properties that depend on the measurement time and also on the memory properties. These features are related to the non-stationary power-law decay of the transition probabilities to their stationary values. Time-averaged response to a bias is also calculated. In contrast with Levy walks [Froemberg and Barkai, Phys. Rev. E 87, 030104(R) (2013)], the response always vanishes asymptotically.
... However, Refs. [46,47,50,73,74] report similar results obtained under certain conditions for coupled Levy walks and Levy walks at the finite velocity. In particular, if the resting time in the coupled Levy walk model demonstrates a linear dependence on the length of preceding jump, this model can be described by the same equation [46,74] and exhibits the same root mean square deviation of particles [47] as in the velocity Levy walk model. ...
... In particular, if the resting time in the coupled Levy walk model demonstrates a linear dependence on the length of preceding jump, this model can be described by the same equation [46,74] and exhibits the same root mean square deviation of particles [47] as in the velocity Levy walk model. However, the asymptotic distributions are different [50,73]. A more detailed description of the approaches describing Levy walks and systems where they appear is given in review [75]. ...
... In fact the OLW process consists of 'jump-wait' events (also called 'first jump' events), while LW process consists of 'wait-jump' events ('first wait' events). One of the main differences between LW and OLW processes is that the second moment of the LW process is finite, while it is infinite for OLW process [13,14,15,16] 2.2. Asymptotic properties of Lévy walks and overshooting Lévy walks. ...
... We emphasize that the difference between scaling limits of LW and OLW processes is due to the fact that the LW process is composed of 'wait-jump' events, while OLW -from 'jump-wait' events. For an extended discussion on this matter see [13,14,15,16,21]. ...
Article
In this paper we present stochastic foundations of fractional dynamics driven by fractional material derivative of distributed order-type. Before stating our main result we present the stochastic scenario which underlies the dynamics given by fractional material derivative. Then we introduce a Levy walk process of distributed-order type to establish our main result, which is the scaling limit of the considered process. It appears that the probability density function of the scaling limit process fulfills, in a weak sense, the fractional diffusion equation with material derivative of distributed-order type.
... Such annealed CTRWs give rise to the celebrated space-time fractional Fokker-Planck equations [5,19]. Another important class of CTRW models satisfying the renewal assumption, with additional property of coupling (strong dependence) between jumps and rests, are Lévy walks [20][21][22][23][24][25]. Applications of Lévy walks in the modeling of real-life phenomena include: fluid flow in a rotating annulus [6], blinking nanocrystals [26], Lévy-Lorentz gas [27], human travel [28,29], epidemic spreading [30,31], foraging of animals [32,33], transport of light in optical materials [34]. ...
... as well as (23) for any b > 0 and ε > 0. ...
... In the wait-first model the particle jumps at the end of the waiting time, whereas in the jump-first model the particle jumps and then rests for the corresponding waiting time. Such random walks were previously considered in the literature [18][19][20][21][22][23][24][25][26], and recently also with a method of infinite densities [27] in the sub-ballistic superdiffusive regime (flight or waiting times with finite mean, 1 < γ < 2). However, in general the exact analyt-ical solutions describing the density of random walking particles are rare and can be obtained only for some particular values of γ [16,28]. ...
... By Eqs. (20), (22) and (23) we find the scaling solutions in original spacetime domain: ...
Article
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We propose an analytical method to determine the shape of density profiles in the asymptotic long time limit for a broad class of coupled continuous time random walks which operate in the ballistic regime. In particular, we show that different scenarios of performing a random walk step, via making an instantaneous jump penalized by a proper waiting time or via moving with a constant speed, dramatically effect the corresponding propagators, despite the fact that the end points of the steps are identical. Furthermore, if the speed during each step of the random walk is itself a random variable, its distribution gets clearly reflected in the asymptotic density of random walkers. These features are in contrast with more standard non-ballistic random walks.
... In this case the particle position PDF has a Lorentzian form, independently of the specific time subordinator η(s). Magdziarz et al. (2012) suggested a different system of Langevin equations to describe the Lévy walk model, which, in fact, corresponds to the wait-first coupled model; although similar in the spirit, it is not identical to the Lévy walk, as we discussed in Section III.C. ...
... By appropriate modifications, the above equation can be simplified to give the equations of the random walk with random velocity model and of the Lévy walk model (Eule et al., 2008). The genetic link between Lévy walks, Langevin equations, and fractional Fokker-Planck equations certainly needs to be investigated further (Lubashevsky et al., 2009a,b;Magdziarz et al., 2012;Turgeman et al., 2009). ...
Article
Random walk is a fundamental concept with applications ranging from quantum physics to econometrics. Remarkably, one specific model of random walks appears to be ubiquitous across many research fields as a tool to analyze non-Brownian dynamics exhibited by different systems. The L\'evy walk model combines two key features: a finite velocity of a random walker and the ability to generate anomalously fast diffusion. Recent results in optics, Hamiltonian many-particle chaos, cold atom dynamics, bio-physics, and behavioral science, demonstrate that this particular type of random walks provides significant insight into complex transport phenomena. This review provides a self-consistent introduction into the theory of L\'evy walks, surveys its existing applications, including latest advances, and outlines its further perspectives.
... The original work of Montroll and Weiss [30] assumed there were no correlations between step size χ and and the waiting time τ , corresponding to a situation called a decoupled CTRW. The diffusion of atoms in optical lattices corresponds to a coupled spatial-temporal random walk theory first considered by Scher and Lax [63] (see [64][65][66] for recent developments). Define η s (x, t)dtdx as the probability that the particle crossed the momentum state p = 0 for the sth time in the time interval (t, t + dt) and that the particle's position was in the interval (x, x + dx). ...
... Given the formidable structure of the scaling function B(v 3/2 ), we do not describe here [67] the direct method to obtain non-integer moments like Eq. (64). Instead, we present here a method which gives v 3/2 ν indirectly. ...
Article
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Levy flights are random walks in which the probability distribution of the step sizes is fat-tailed. Levy spatial diffusion has been observed for a collection of ultra-cold Rb atoms and single Mg+ ions in an optical lattice. Using the semiclassical theory of Sisyphus cooling, we treat the problem as a coupled Levy walk, with correlations between the length and duration of the excursions. The problem is related to the area under Bessel excursions, overdamped Langevin motions that start and end at the origin, constrained to remain positive, in the presence of an external logarithmic potential. In the limit of a weak potential, the Airy distribution describing the areal distribution of the Brownian excursion is found. Three distinct phases of the dynamics are studied: normal diffusion, Levy diffusion and, below a certain critical depth of the optical potential, x~ t^{3/2} scaling. The focus of the paper is the analytical calculation of the joint probability density function from a newly developed theory of the area under the Bessel excursion. The latter describes the spatiotemporal correlations in the problem and is the microscopic input needed to characterize the spatial diffusion of the atomic cloud. A modified Montroll-Weiss (MW) equation for the density is obtained, which depends on the statistics of velocity excursions and meanders. The meander, a random walk in velocity space which starts at the origin and does not cross it, describes the last jump event in the sequence. In the anomalous phases, the statistics of meanders and excursions are essential for the calculation of the mean square displacement, showing that our correction to the MW equation is crucial, and points to the sensitivity of the transport on a single jump event. Our work provides relations between the statistics of velocity excursions and meanders and that of the diffusivity.
... (15) and (16), which are obtained from suitably generalising the formalism of PK processes. This formulation neatly results from the definition of a LW process and, in this sense, greatly differs from other phenomenological models published in the literature that rely on subordination techniques [110,111] or fractional derivatives [112], as the statistical characterization involves first-order evolution equations in time and space, whose mathematical structure resembles the linear Boltzmann equation [113]. Owing to this analogy and to the analogy between the evolution equations for the partial densities and the mathematics of radiative transfer [114], the mathematical approaches developed in these two fields can be consistently transferred to the study of EPK processes [113,114]. ...
Article
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Stochastic processes play a key role for modeling a huge variety of transport problems out of equilibrium, with manifold applications throughout the natural and social sciences. To formulate models of stochastic dynamics, the conventional approach consists in superimposing random fluctuations on a suitable deterministic evolution. These fluctuations are sampled from probability distributions that are prescribed a priori, most commonly as Gaussian or Lévy. While these distributions are motivated by (generalized) central limit theorems, they are nevertheless unbounded, meaning that arbitrarily large fluctuations can be obtained with finite probability. This property implies the violation of fundamental physical principles such as special relativity and may yield divergencies for basic physical quantities like energy. Here, we solve the fundamental problem of unbounded random fluctuations by constructing a comprehensive theoretical framework of stochastic processes possessing physically realistic finite propagation velocity. Our approach is motivated by the theory of Lévy walks, which we embed into an extension of conventional Poisson-Kac processes. The resulting extended theory employs generalized transition rates to model subtle microscopic dynamics, which reproduces nontrivial spatiotemporal correlations on macroscopic scales. It thus enables the modeling of many different kinds of dynamical features, as we demonstrate by three physically and biologically motivated examples. The corresponding stochastic models capture the whole spectrum of diffusive dynamics from normal to anomalous diffusion, including the striking “Brownian yet non-Gaussian” diffusion, and more sophisticated phenomena such as senescence. Extended Poisson-Kac theory can, therefore, be used to model a wide range of finite-velocity dynamical phenomena that are observed experimentally.
... The weakly nonergodic behavior of the standard Lévy walk (ν = 1, η = 1) with respect to the squared displacements was investigated in [54][55][56]. Furthermore, it was shown that the standard Lévy walk can also be described by a set of coupled Langevin equations using a subordination technique [62,63]. From that, the time-lag de- pendence of the ensemble-averaged squared displacement and the ensemble average of the time-averaged squared displacement was recovered [64]. ...
Preprint
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We investigate the nonergodicity of the generalized L\'evy walk introduced by Shlesinger et al. [Phys. Rev. Lett. 58, 1100 (1987)] with respect to the squared displacements. We present detailed analytical derivations of our previous findings outlined in a recent Letter [Phys. Rev. Lett. 120, 104501 (2018)], give profound interpretations, and especially emphasize three surprising results: First, we find that the mean-squared displacements can diverge for a certain range of parameter values. Second, we show that an ensemble of trajectories can spread subdiffusively, whereas individual time-averaged squared displacements show superdiffusion. Third, we recognize that the fluctuations of the time-averaged squared displacements can become so large that the ergodicity breaking parameter diverges, what we call infinitely strong ergodicity breaking. The latter phenomenon can also occur for paramter values where the lag-time dependence of the mean-squared displacements is linear indicating normal diffusion. In order to numerically determine the full distribution of time-averaged squared displacements, we use importance sampling. For an embedding of our new findings into existing results in the literature, we define a more general model which we call variable speed generalized L\'evy walk and which includes well known models from the literature as special cases such as the space-time coupled L\'evy flight or the anomalous Drude model. We discuss and interpret our findings regarding the generalized L\'evy walk in detail and compare them with the nonergodicity of the other space-time coupled models following from the more general model.
... As a result, the W-like asymptotic distribution of particles is formed. These solutions are not new and were obtained earlier when considering random walks with constant velocity [49,52,53], and when considering similar models of random walks [37,42,48,59,60] (see also the overview work [50]). In the case of the finite mathematical expectation (1 < α < 2), the process of random walks of a particle with a constant velocity falls under the action of the generalized central limit theorem. ...
Article
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The process of Levy random walks is considered in view of the constant velocity of a particle. A kinetic equation is obtained that describes the process of walks, and fractional differential equations are obtained that describe the asymptotic behavior of the process. It is shown that, in the case of finite and infinite mathematical expectation of paths, these equations have a completely different form. To solve the obtained equations, the method of local estimation of the Monte Carlo method is described. The solution algorithm is described and the advantages and disadvantages of the considered method are indicated.
... Commonly, the underlying random process that drives Langevin dynamics is Brownian motion. However, Langevin dynamics with non-Brownian drivers attracted substantial interest [89][90][91][92][93][94][95][96]. For Langevin dynamics that are driven by a selfsimilar process with Hurst exponent h we obtain that: the only selfsimilar process that the Langevin dynamics can produce is with Hurst exponent H = h. ...
... (18) and (19), which are obtained from suitably generalising the formalism of PK processes. This formulation neatly results from the definition of a LW process and, in this sense, greatly differs from other phenomenological models published in the literature that rely on subordination techniques [103,104] or fractional derivatives [105], whose physical interpretation is often hidden by the mathematical technicalities. With a reverse-engineering approach, we then used the cross-link between these processes to formulate a very general theoretical framework for stochastic models with finite propagation speed, which we called EPK theory. ...
Preprint
Full-text available
Stochastic processes play a key role for mathematically modeling a huge variety of transport problems out of equilibrium. To formulate models of stochastic dynamics the mainstream approach consists in superimposing random fluctuations on a suitable deterministic evolution. These fluctuations are sampled from probability distributions that are prescribed a priori, most commonly as Gaussian or Levy. While these distributions are motivated by (generalised) central limit theorems they are nevertheless unbounded. This property implies the violation of fundamental physical principles such as special relativity and may yield divergencies for basic physical quantities like energy. It is thus clearly never valid in real-world systems by rendering all these stochastic models ontologically unphysical. Here we solve the fundamental problem of unbounded random fluctuations by constructing a comprehensive theoretical framework of stochastic processes possessing finite propagation velocity. Our approach is motivated by the theory of Levy walks, which we embed into an extension of conventional Poisson-Kac processes. Our new theory possesses an intrinsic flexibility that enables the modelling of many different kinds of dynamical features, as we demonstrate by three examples. The corresponding stochastic models capture the whole spectrum of diffusive dynamics from normal to anomalous diffusion, including the striking Brownian yet non Gaussian diffusion, and more sophisticated phenomena such as senescence. Extended Poisson-Kac theory thus not only ensures by construction a mathematical representation of physical reality that is ontologically valid at all time and length scales. It also provides a toolbox of stochastic processes that can be used to model potentially any kind of finite velocity dynamical phenomena observed experimentally.
... In [31], the second moment á ñ µ ( ) x t t 2 2 was obtained, which is consistent to the ballistic regime of Lévy walk. Another way to characterise Lévy walk from overdamped Langevin equation is to assume that jump sizes are some functions of waiting times in [78]. ...
Article
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Continuous-time random walks and Langevin equations are two classes of stochastic models used to describe 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 diffusive behavior of the stochastic process described by this coupled Langevin system, and find the super-ballistic diffusion phenomenon for the system with an unconfined potential on velocity but sub-ballistic superdiffusion phenomenon with a confined potential, which is like Lévy walk for long times. One can further note that the two-point distribution of inverse subordinator affects mean square displacement of this coupled weakly damped Langevin system in essential. © 2019 The Author(s). Published by IOP Publishing Ltd on behalf of the Institute of Physics and Deutsche Physikalische Gesellschaft.
... In this paper we generally assume that waiting times and jump lengths are uncorrelated. For a discussion of the correlated case, we refer to [64][65][66][67][68][69][70]. A natural parametrisation of such a random walk is obtained in terms of the number n of jumps performed. ...
Article
Functionals of a stochastic process Y(t) model many physical time-extensive observables, e.g. particle positions, local and occupation times or accumulated mechanical work. When Y(t) is a normal diffusive process, their statistics are obtained as the solution of the Feynman-Kac equation. This equation provides the crucial link between the expected values of diffusion processes and the solutions of deterministic second-order partial differential equations. When Y(t) is an anomalous diffusive process, generalizations of the Feynman-Kac equation that incorporate power-law or more general waiting time distributions of the underlying random walk have recently been derived. A general representation of such waiting times is provided in terms of a L\'evy process whose Laplace exponent is related to the memory kernel appearing in the generalized Feynman-Kac equation. The corresponding anomalous processes have been shown to capture nonlinear mean square displacements exhibiting crossovers between different scaling regimes, which have been observed in biological systems like migrating cells or diffusing macromolecules in intracellular environments. However, the case where both space- and time-dependent forces drive the dynamics of the generalized anomalous process has not been solved yet. Here, we present the missing derivation of the Feynman-Kac equation in such general case by using the subordination technique. Furthermore, we discuss its extension to functionals explicitly depending on time, which are relevant for the stochastic thermodynamics of anomalous diffusive systems. Exact results on the work fluctuations of a simple non-equilibrium model are obtained. In this paper we also provide a pedagogical introduction to L\'evy processes, semimartingales and their associated stochastic calculus, which underlie the mathematical formulation of anomalous diffusion as a subordinated process.
... where B(·) is a standard Brownian motion and S −1 α (t) is inverse α-stable subordinator defined in (6). Proof of this statement is an consequence of the Theorem 1 in [39]. ...
Article
The continuous time random walk model plays an important role in modeling of so called anomalous diffusion behaviour. One of the specific property of such model are constant time periods visible in trajectory. In the continuous time random walk approach they are realizations of the sequence called waiting times. The main attention of the paper is paid on the analysis of waiting times distribution. We introduce here novel methods of estimation and statistical investigation of such distribution. The methods are based on the modified cumulative distribution function. In this paper we consider three special cases of waiting time distributions, namely $\alpha$-stable, tempered stable and gamma. However the proposed methodology can be applied to broad set of distributions - in general it may serve as a method of fitting any distribution function if the observations are rounded. The new statistical techniques we apply to the simulated data as well as to the real data describing $CO_2$ concentration in indoor air.
... The derivation of this classical result is provided in an Appendix. Note that in the original work of Montroll and Weiss a decoupled random walk was considered and the origin of Eq. (11) can be traced to the work of Scher and Lax [47] and that of Shlesinger, West and Klafter [27] (see also [28,30,45,[48][49][50]). Examples [5,24,29] for physical processes described by the Lévy walk are certain non-linear dynamical systems [2,6,27,51], polymer dynamics [52], blinking quantum dots [53][54][55][56], cold atoms diffusing in optical lattices [57,58], intermittent search strategies [59], and dynamics generated by many body Hamiltonian systems [60,61]. ...
Article
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Motion of particles in many systems exhibits a mixture between periods of random diffusive like events and ballistic like motion. These systems exhibit strong anomalous diffusion, where low order moments $\langle |x(t)|^q \rangle$ with $q$ below a critical value $q_c$ exhibit diffusive scaling while for $q>q_c$ a ballistic scaling emerges. Such mixed dynamics exhibits a theoretical challenge since it does not fall into a unique category of motion, e.g., all known diffusion equations and central limit theorems fail to describe both aspects of the motion. In this paper we investigate this problem using the widely applicable L\'evy walk model, which is a basic and well known random walk. We recently showed by examining a few special cases that an infinite density describes the system. The infinite density is a measurable non-normalized density emerging from the norm conserving dynamics. We find a general formula for this non-normalized density showing that it is fully determined by the particles velocity distribution, the anomalous diffusion exponent $\alpha$ and the diffusion coefficient $K_\alpha$. A distinction between observables integrable (ballistic observables) and non-integrable (diffusive observables) with respect to the infinite density is crucial for the statistical description of the motion. The infinite density is complementary to the central limit theorem as it captures the ballistic elements of the transport, while the latter describes the diffusive elements of the problem. We show how infinite densities play a central role in the description of dynamics of a large class of physical processes and explain how it can be evaluated from experimental or numerical data.
... There is an extensive literature in this field: general results for the scaling limits of CTRWs on the stochastic process level can be found in [6,7,29,35,37,44,45]. Governing equations for the densities of the CTRW limits and the related fractional Cauchy problems were analyzed in [2,4,20,23,33,37,45]. Some recent results for particular classes of correlated and coupled CTRWs can be found in [17,24,30,31,47] The trajectories of CTRW are step functions, thus they are discontinuous. However, the usual physical requirement for a mathematical model is to have continuous realizations. ...
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The Levy Walk is the process with continuous sample paths which arises from consecutive linear motions of i.i.d. lengths with i.i.d. directions. Assuming speed 1 and motions in the domain of beta-stable attraction, we prove functional limit theorems and derive governing pseudo-differential equations for the law of the walker's position. Both Levy Walk and its limit process are continuous and ballistic in the case beta in (0,1). In the case beta in (1,2), the scaling limit of the process is beta-stable and hence discontinuous. This case exhibits an interesting situation in which scaling exponent 1/beta on the process level is seemingly unrelated to the scaling exponent 3-beta of the second moment. For beta = 2, the scaling limit is Brownian motion.
... Recent progress in the field of CTRWs with correlated temporal and spatial structure can be found in Meerschaert et al. [14] and Magdziarz et al. [15,16]. The Langevin description of some classes of anomalous diffusions has been recently studied in Magdziarz [17], Magdziarz et al. [18] and Teuerle et al. [19]. We would like to emphasize that the word 'correlated' is used here in the broad sense and should be understood as 'dependent'. ...
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We introduce a continuous-time random walk process with correlated temporal structure. The dependence between consecutive waiting times is generated by weighted sums of independent random variables combined with a reflecting boundary condition. The weights are determined by the memory kernel, which belongs to the broad class of regularly varying functions. We derive the corresponding diffusion limit and prove its subdiffusive character. Analysing the set of corresponding coupled Langevin equations, we verify the speed of relaxation, Einstein relations, equilibrium distributions, ageing and ergodicity breaking.
... Space-and time-dependent Lévy Walks. Lévy Walks were introduced in [34], and have only recently been studied on the stochastic process level [35]. Their main feature is that waiting times and jumps are strongly coupled: A waiting time of length W k is accompanied by a jump of size |J k | = v W k , which achieves a finite travel velocity v of the random walker; note however that we assume each jump to be instantaneous, and that the term "velocity" is only to be interpreted in an averaged sense. ...
Article
Continuous Time Random Walks (CTRWs) are jump processes with random waiting times between jumps. We study scaling limits for CTRWs where the distribution of jumps and waiting times is coupled and varies in space and time. Such processes model e.g.\ anomalous diffusion processes in a space- and time-dependent potential. Conditions for the process-convergence of CTRWs are given, and the limits are characterised by four coefficients. Kolmogorov forwards and backwards equations with non-local time operators are derived, and three models for anomalous diffusion are presented: i) Subdiffusion in a time-dependent potential, ii) subdiffusion with spatially varying waiting times and iii) L\'evy walks with space- and time-dependent drift.
... The Lévy walk model is a generalization of the classical Drude model describing a particle moving with constant velocity and changing its direction randomly. While in the Drude model exponential waiting times between turning events due to strong collisions result in a Markov process, the Lévy walk model postulates power law distributed waiting times between randomization events resulting in long flights [17][18][19]. The Lévy walk [18] describes enhanced transport phenomena in many systems, ranging from chaotic diffusion to animal foraging patterns [16,[20][21][22][23][24][25][26]. ...
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The L\'evy walk model is a stochastic framework of enhanced diffusion with many applications in physics and biology. Here we investigate the time averaged mean squared displacement $\bar{\delta^2}$ often used to analyze single particle tracking experiments. The ballistic phase of the motion is non-ergodic and we obtain analytical expressions for the fluctuations of $\bar{\delta^2}$. For enhanced sub-ballistic diffusion we observe numerically apparent ergodicity breaking on long time scales. As observed by Akimoto \textit{Phys. Rev. Lett.} \textbf{108}, 164101 (2012) deviations of temporal averages $\bar{\delta^2}$ from the ensemble average $< x^2 >$ depend on the initial preparation of the system, and here we quantify this discrepancy from normal diffusive behavior. Time averaged response to a bias is considered and the resultant generalized Einstein relations are discussed.
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In this paper we analyze the asymptotic behavior of Lévy walks with rests. Applying recent results in the field of functional convergence of continuous-time random walks we find the corresponding limiting processes. Depending on the parameters of the model, we show that in the limit we can obtain standard Lévy walk or the process describing competition between subdiffusion and Lévy flights. Some other more complicated limit forms are also possible to obtain. Finally we present some numerical results, which confirm our findings.
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We investigate the nonergodicity of the generalized Lévy walk introduced by Shlesinger et al. [Phys. Rev. Lett. 58, 1100 (1987)] with respect to the squared displacements. We present detailed analytical derivations of our previous findings outlined in a recent letter [Phys. Rev. Lett. 120, 104501 (2018)], give detailed interpretations, and in particular emphasize three surprising results. First, we find that the mean-squared displacements can diverge for a certain range of parameter values. Second, we show that an ensemble of trajectories can spread subdiffusively, whereas individual time-averaged squared displacements show superdiffusion. Third, we recognize that the fluctuations of the time-averaged squared displacements can become so large that the ergodicity breaking parameter diverges, what we call infinitely strong ergodicity breaking. This phenomenon can also occur for paramter values where the lag-time dependence of the mean-squared displacements is linear indicating normal diffusion. In order to numerically determine the full distribution of time-averaged squared displacements, we use importance sampling. For an embedding of our findings into existing results in the literature, we define a more general model which we call variable speed generalized Lévy walk and which includes well-known models from the literature as special cases such as the space-time coupled Lévy flight or the anomalous Drude model. We discuss and interpret our findings regarding the generalized Lévy walk in detail and compare them with the nonergodicity of the other space-time coupled models following from the more general model.
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A Brownian yet non-Gaussian phenomenon has recently been observed in many biological and active matter systems. The main idea of explaining this phenomenon is to introduce a random diffusivity for particles moving in inhomogeneous environment. This paper considers a Langevin system containing a random diffusivity and an α-stable subordinator with α<1. This model describes the particle's motion in complex media where both the long trapping events and random diffusivity exist. We derive the general expressions of ensemble- and time-averaged mean-squared displacements which only contain the values of the inverse subordinator and diffusivity. Further taking specific time-dependent diffusivity, we obtain the analytic expressions of ergodicity breaking parameter and probability density function of the time-averaged mean-squared displacement. The results imply the nonergodicity of the random diffusivity model with any kind of diffusivity, including the critical case where the model presents normal diffusion.
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Continuous-time random walks (CTRWs) are an elementary model for particle motion subject to randomized waiting times. In this paper, we consider the case where the distribution of waiting times depends on the location of the particle. In particular, we analyze the case where the medium exhibits a bounded trapping region in which the particle is subject to CTRW with power-law waiting times and regular diffusion elsewhere. We derive a diffusion limit for this inhomogeneous CTRW. We show that depending on the index of the power-law distribution, we can observe either nonlinear subdiffusive or standard diffusive motion.
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A generalization of the model of Lévy walks with traps is considered. The main difference between the model under consideration and the already existing models is the introduction of multiplicative particle acceleration at collisions. The introduction of acceleration transfers the consideration of walks to coordinate–momentum phase space, which allows both the spatial distribution of particles and their spectrum to be obtained. The kinetic equations in coordinate–momentum phase space have been derived for the case of walks with two possible states. This system of equations in a special case is shown to be reduced to ordinary Lévy walks. This system of kinetic equations admits of integration over the spatial variable, which transfers the consideration only to momentum space and allows the spectrum to be calculated. An exact solution of the kinetic equations can be obtained in terms of the Laplace–Mellin transform. The inverse transform can be performed only for the asymptotic solutions. The calculated spectra are compared with the results of Monte Carlo simulations, which confirm the validity of the derived asymptotics.
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We introduce a coupled continuous-time random walk with coupling which is characteristic for Lévy walks. Additionally we assume that the walker moves in a quenched random environment, i.e. the site disorder at each lattice point is fixed in time. We analyze the scaling limit of such a random walk. We show that for large times the behaviour of the analyzed process is exactly the same as in the case of uncoupled quenched trap model for Lévy flights.
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Aging can be observed for numerous physical systems. In such systems statistical properties [like probability distribution, mean square displacement (MSD), first-passage time] depend on a time span ta between the initialization and the beginning of observations. In this paper we study aging properties of ballistic Lévy walks and two closely related jump models: wait-first and jump-first. We calculate explicitly their probability distributions and MSDs. It turns out that despite similarities these models react very differently to the delay ta. Aging weakly affects the shape of probability density function and MSD of standard Lévy walks. For the jump models the shape of the probability density function is changed drastically. Moreover for the wait-first jump model we observe a different behavior of MSD when ta≪t and ta≫t.
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In this paper we derive explicit formulas for the densities of Lévy walks. Our results cover both jump-first and wait-first scenarios. The obtained densities solve certain fractional differential equations involving fractional material derivative operators. In the particular case, when the stability index is rational, the densities can be represented as an integral of Meijer G-function. This allows to efficiently evaluate them numerically. We also compute two-point distribution of wait-first model. Our results show perfect agreement with the Monte Carlo simulations.
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Many real data exhibit behavior adequate to subdiffusion processes. Very often it is manifested by so-called “trapping events”. The visible evidence of subdiffusion we observe not only in financial time series but also in technical data. In this paper we propose a model which can be used for description of such kind of data. The model is based on the continuous time autoregressive time series with stable noise delayed by the infinitely divisible inverse subordinator. The proposed system can be applied to real datasets with short-time dependence, visible jumps and mentioned periods of stagnation. In this paper we extend the theoretical considerations in analysis of subordinated processes and propose a new model that exhibits mentioned properties. We concentrate on the main characteristics of the examined subordinated process expressed mainly in the language of the measures of dependence which are main tools used in statistical investigation of real data. We present also the simulation procedure of the considered system and indicate how to estimate its parameters. The theoretical results we illustrate by the analysis of real technical data.
In this paper we analyze multidimensional Lévy walks with power-law dependence between waiting times and jumps. We obtain the detailed structure of the scaling limits of such multidimensional processes for all positive values of the power-law exponent. It appears that the scaling limit strongly depends on the value of the power-law exponent and has two possible scenarios: an αα-stable Lévy motion subordinated to a strongly dependent inverse subordinator, or a Brownian motion subordinated to an independent inverse subordinator. Moreover, we derive the mean-squared displacement for the scaling limit processes. Based on these results we conclude that the resulting limiting processes belong to sub-, quasi- and superdiffusion regimes. The corresponding fractional diffusion equation and Langevin picture of considered models are also derived. Theoretical results are illustrated using the proposed numerical scheme for simulation of considered processes.
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Lévy walks (LWs) are a popular stochastic tool to model anomalous diffusion and have recently been used to describe a variety of phenomena. We study the linear response behavior of this generic model of superdiffusive LWs in finite systems to an external force field under both stationary and nonstationary conditions. These finite-size LWs are based on power-law waiting time distributions with a finite-time regularization at τ_{c}, such that the physical requirements are met to apply linear response theory and derive the power spectrum with the correct short frequency limit, without the introduction of artificial cutoffs. We obtain the generalized Einstein relation for both ensemble and time averages over the entire process time and determine the turnover to normal Brownian motion when the full system is explored. In particular, we obtain an exact expression for the long time diffusion constant as a function of the scaling exponent of the waiting time density and the characteristic time scale τ_{c}.
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In this paper we obtain the scaling limit of a multidimensional Lévy walk and describe the detailed structure of the limiting process. The scaling limit is a subordinated α-stable Lévy motion with the parent process and subordinator being strongly dependent processes. The corresponding Langevin picture is derived. We also introduce a useful method of simulating Lévy walks with a predefined spectral measure, which controls the direction of each jump. Our approach can be applied in the analysis of real-life data—we are able to recover the spectral measure from the data and obtain the full characterization of a Lévy walk. We also give examples of some useful spectral measures, which cover a large class of possible scenarios in the modeling of real-life phenomena.
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We study a class of random walk, the stored-energy-driven Lévy flight (SEDLF), whose jump length is determined by a stored energy during a trapped state. The SEDLF is a continuous-time random walk with jump lengths being coupled with the trapping times. It is analytically shown that the ensemble-averaged mean-square displacements exhibit subdiffusion as well as superdiffusion, depending on the coupling parameter. We find that time-averaged mean-square displacements increase linearly with time and the diffusion coefficients are intrinsically random, a manifestation of distributional ergodicity. The diffusion coefficient shows aging in subdiffusive regime, whereas it increases with the measurement time in superdiffusive regime.
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The description of diffusion processes is possible in different frameworks such as random walks or Fokker-Planck or Langevin equations. Whereas for classical diffusion the equivalence of these methods is well established, in the case of anomalous diffusion it often remains an open problem. In this paper we aim to bring three approaches describing anomalous superdiffusive behavior to a common footing. While each method clearly has its advantages it is crucial to understand how those methods relate and complement each other. In particular, by using the method of subordination, we show how the Langevin equation can describe anomalous diffusion exhibited by Lévy-walk-type models and further show the equivalence of the random walk models and the generalized Kramers-Fokker-Planck equation. As a result a synergetic and complementary description of anomalous diffusion is obtained which provides a much more flexible tool for applications in real-world systems.
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We study the ergodic properties of superdiffusive, spatiotemporally coupled Levy walk processes. For trajectories of finite duration, we reveal a distinct scatter of the scaling exponents of the time averaged mean squared displacement delta**2 around the ensemble value 3-alpha (1<alpha<2) ranging from ballistic motion to subdiffusion, in strong contrast to the behavior of subdiffusive processes. In addition we find a significant dependence of the trajectory-to-trajectory average of delta**2 as function of the finite measurement time. This so-called finite-time amplitude depression and the scatter of the scaling exponent is vital in the quantitative evaluation of superdiffusive processes. Comparing the long time average of the second moment with the ensemble mean squared displacement, these only differ by a constant factor, an ultraweak ergodicity breaking.
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Many useful descriptions of stochastic models can be obtained from functional limit theorems (invariance principles or weak convergence theorems for probability measures on function spaces). These descriptions typically come from standard functional limit theorems via the continuous mapping theorem. this study facilitates applications of the continuous mapping theorem by determining when several important functions and sequences of functions preserve convergence. the functions considered are composition, addition, composition plus addition, multiplication, supremum, reflecting barrier, first passage time and time reversal. These functions provide means for proving new functional limit theorems from previous ones.
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We introduce a fractional Langevin equation with fi-stable noise and show that its so- lution fY•(t); t 2 Rg is the stationary fi-stable Ornstein-Uhlenbeck-type process recently studied in (14). We examine the asymptotic dependence structure of Y•(t) via the measure of its codependence r(µ1;µ2;t) being the difierence between the joint characteristic function of (Y•(t);Y•(0)) and the product of the characteristic functions of Y•(t) and Y•(0). We prove that Y•(t) is not a long-memory process in the sense of codependence r(µ1;µ2;t). Moreover, we have found a natural continuous-time analogue of fractional ARIMA time series.
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In statistical physics, subdiffusion processes constitute one of the most relevant subclasses of the family of anomalous diffusion models. These processes are characterized by certain power-law deviations from the classical Brownian linear time dependence of the mean-squared displacement. In this article we study sample path properties of subdiffusion. We propose a martingale approach to the stochastic analysis of subdiffusion models. We verify the martingale property, Hölder continuity of the trajectories, and derive the law of large numbers. The precise asymptotic behavior of subdiffusion is obtained in the law of the iterated logarithm. The presented results may be applied to identify the type of subdiffusive dynamics in experimental data.
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Subdiffusion in the presence of an external force field has been recently described in phase space by the fractional Klein–Kramers equation. In this paper using a subordination method, we identify a two-dimensional stochastic process (position, velocity) whose probability density function is a solution of the fractional Klein–Kramers equation. The structure of this process agrees with the two-stage scenario underlying the anomalous diffusion mechanism, in which trapping events are superimposed on the Langevin dynamics. Applying an extension of the celebrated Itô formula for subdiffusion we found that the velocity process can be represented explicitly by a corresponding fractional Ornstein–Uhlenbeck process. A basic feature arising in the context of this stochastic representation is the random change of time of the system made by subordination. For the position and velocity processes we present a computer visualization of their sample paths and we derive an explicit expression for the two-point correlation function of the velocity process. The obtained stochastic representation is crucial in constructing an algorithm to simulate sample paths of the anomalous diffusion, which in turn allows us to detect and examine many relevant properties of the system under consideration.
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The fractional Boltzmann equation for resonance radiation transport in plasma is proposed. We start with the standard Boltzmann equation; averaging over photon frequencies leads to the appearance of a fractional derivative. This fact is in accordance with the conception of latent variables leading to hereditary and non-local dynamics (in particular, fractional dynamics). The presence of a fractional material derivative in the equation is concordant with heavy tailed distribution of photon path lengths and with spatiotemporal coupling peculiar to the process. We discuss some methods of solving the obtained equation and demonstrate numerical results in some simple cases.
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In a continuous time random walk (CTRW), a random waiting time precedes each random jump. The CTRW model is useful in physics, to model diffusing particles. Its scaling limit is a time-changed process, whose densities solve an anomalous diffusion equation. This paper develops limit theory and governing equations for cluster CTRW, in which a random number of jumps cluster together into a single jump. The clustering introduces a dependence between the waiting times and jumps that significantly affects the asymptotic limit. Vector jumps are considered, along with oracle CTRW, where the process anticipates the next jump.
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The continuous-time random walk of Montroll and Weiss has been modified by Scher and Lax to include a coupled spatial-temporal memory. We treat novel cases for the random walk and the corresponding generalized master equation when combinations of both spatial, and temporal moments of the memory are infinite. The asymptotic properties of the probability distribution for being at any lattice site as a function of time and its variance are calculated. The resulting behavior includes localized, diffusive, wavelike, and Levy's stable laws for the appropriate scaled variable. We show that an infinite mean waiting time can lead to long time diffusive behavior, while a finite mean waiting time is not sufficient to ensure the same.
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In this paper we study a Langevin approach to modeling of subdiffusion in the presence of time-dependent external forces. We construct a subordinated Langevin process, whose probability density function solves the subdiffusive fractional Fokker-Planck equation. We generalize the results known for the Lévy-stable waiting times to the case of infinitely divisible waiting-time distributions. Our approach provides a complete mathematical description of subdiffusion with time-dependent forces. Moreover, it allows to study the trajectories of the constructed process both analytically and numerically via Monte-Carlo methodology.
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Fractional kinetic equations of the diffusion, diffusion–advection, and Fokker–Planck type are presented as a useful approach for the description of transport dynamics in complex systems which are governed by anomalous diffusion and non-exponential relaxation patterns. These fractional equations are derived asymptotically from basic random walk models, and from a generalised master equation. Several physical consequences are discussed which are relevant to dynamical processes in complex systems. Methods of solution are introduced and for some special cases exact solutions are calculated. This report demonstrates that fractional equations have come of age as a complementary tool in the description of anomalous transport processes.
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In this paper we extend the subdiffusive Klein-Kramers model, in which the waiting times are modeled by the α-stable laws, to the case of waiting times belonging to the class of tempered α-stable distributions. We introduce a generalized version of the Klein-Kramers equation, in which the fractional Riemman-Liouville derivative is replaced with a more general integro-differential operator. This allows a transition from the initial subdiffusive character of motion to the standard diffusion for long times to be modeled. Taking advantage of the corresponding Langevin equation, we study some properties of the tempered dynamics, in particular, we approximate solutions of the tempered Klein-Kramers equation via Monte Carlo methods. Also, we study the distribution of the escape time from the potential well and compare it to the classical results in the Kramers escape theory. Finally, we derive the analytical formula for the first-passage-time distribution for the case of free particles. We show that the well-known Sparre Andersen scaling holds also for the tempered subdiffusion.
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Most statistical theories of anomalous diffusion rely on ensemble-averaged quantities such as the mean squared displacement. Single molecule tracking measurements require, however, temporal averaging. We contrast the two approaches in the case of continuous-time random walks with a power-law distribution of waiting times psi(t) proportional to t{-1-alpha}, with 0<alpha<1, lacking the mean. We show that, contrary to what is expected, the temporal averaged mean squared displacement leads to a simple diffusive behavior with diffusion coefficients that strongly differ from one trajectory to another. This distribution of diffusion coefficients renders a system inhomogeneous: an ensemble of simple diffusers with different diffusion coefficients. Taking an ensemble average over these diffusion coefficients results in an effective diffusion coefficient K{eff} approximately T{alpha-1} which depends on the length of the trajectory T.
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Diffusion in the plasma membrane of living cells is often found to display anomalous dynamics. However, the mechanism underlying this diffusion pattern remains highly controversial. Here, we study the physical mechanism underlying Kv2.1 potassium channel anomalous dynamics using single-molecule tracking. Our analysis includes both time series of individual trajectories and ensemble averages. We show that an ergodic and a nonergodic process coexist in the plasma membrane. The ergodic process resembles a fractal structure with its origin in macromolecular crowding in the cell membrane. The nonergodic process is found to be regulated by transient binding to the actin cytoskeleton and can be accurately modeled by a continuous-time random walk. When the cell is treated with drugs that inhibit actin polymerization, the diffusion pattern of Kv2.1 channels recovers ergodicity. However, the fractal structure that induces anomalous diffusion remains unaltered. These results have direct implications on the regulation of membrane receptor trafficking and signaling.
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Combining extensive single particle tracking microscopy data of endogenous lipid granules in living fission yeast cells with analytical results we show evidence for anomalous diffusion and weak ergodicity breaking. Namely we demonstrate that at short times the granules perform subdiffusion according to the laws of continuous time random walk theory. The associated violation of ergodicity leads to a characteristic turnover between two scaling regimes of the time averaged mean squared displacement. At longer times the granule motion is consistent with fractional Brownian motion.
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In this paper we present an approach to anomalous diffusion based on subordination of stochastic processes. Application of such a methodology to analysis of the diffusion processes helps better understanding of physical mechanisms underlying the nonexponential relaxation phenomena. In the subordination framework we analyze a coupling between the very large jumps in physical and two different operational times, modeled by under- and overshooting subordinators, respectively. We show that the resulting diffusion processes display features by means of which all experimentally observed two-power-law dielectric relaxation patterns can be explained. We also derive the corresponding fractional equations governing the spatiotemporal evolution of the diffusion front of an excitation mode undergoing diffusion in the system under consideration. The commonly known type of subdiffusion, corresponding to the Mittag-Leffler (or Cole-Cole) relaxation, appears as a special case of the studied anomalous diffusion processes.
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We demonstrate that continuous time random walks in which successive waiting times are correlated by Gaussian statistics lead to anomalous diffusion with mean squared displacement <r^2(t)>~t^{2/3}. Long-ranged correlations of the waiting times with power-law exponent alpha (0<alpha<=2) give rise to subdiffusion of the form <r^2(t)>~t^{alpha/(1+alpha)}. In contrast correlations in the jump lengths are shown to produce superdiffusion. We show that in both cases weak ergodicity breaking occurs. Our results are in excellent agreement with simulations.
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In this paper, we propose a transparent subordination approach to anomalous diffusion processes underlying the nonexponential relaxation. We investigate properties of a coupled continuous-time random walk that follows from modeling the occurrence of jumps with compound counting processes. As a result, two different diffusion processes corresponding to over- and undershooting operational times, respectively, have been found. We show that within the proposed framework, all empirical two-power-law relaxation patterns may be derived. This work is motivated by the so-called "less typical" relaxation behavior observed, e.g., for gallium-doped Cd0.99Mn0.01Te mixed crystals.
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Scaling limits of continuous time random walks are used in physics to model anomalous diffusion, in which a cloud of particles spreads at a different rate than the classical Brownian motion. Governing equations for these limit processes generalize the classical diffusion equation. In this article, we characterize scaling limits in the case where the particle jump sizes and the waiting time between jumps are dependent. This leads to an efficient method of computing the limit, and a surprising connection to fractional derivatives.
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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.
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A century after the celebrated Langevin paper [C.R. Seances Acad. Sci. 146, 530 (1908)] we study a Langevin-type approach to subdiffusion in the presence of time-dependent force fields. Using a subordination technique, we construct rigorously a stochastic Langevin process, whose probability density function is equal to the solution of the fractional Fokker-Planck equation with a time-dependent force. Our model provides physical insight into the nature of the corresponding process through the simulated trajectories. Moreover, the subordinated Langevin equation allows us to study subdiffusive dynamics both analytically and numerically via Monte Carlo methods.
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Single particle tracking of mRNA molecules and lipid granules in living cells shows that the time averaged mean squared displacement delta2[over ] of individual particles remains a random variable while indicating that the particle motion is subdiffusive. We investigate this type of ergodicity breaking within the continuous time random walk model and show that delta2[over ] differs from the corresponding ensemble average. In particular we derive the distribution for the fluctuations of the random variable delta2[over ]. Similarly we quantify the response to a constant external field, revealing a generalization of the Einstein relation. Consequences for the interpretation of single molecule tracking data are discussed.
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We present an appraisal of differential-equation models for anomalous diffusion, in which the time evolution of the mean-square displacement is 〈r2(t)〉∼tγ with γ≠1. By comparison, continuous-time random walks lead via generalized master equations to an integro-differential picture. Using Lévy walks and a kernel which couples time and space, we obtain a generalized picture for anomalous transport, which provides a unified framework both for dispersive (γ<1) and for enhanced diffusion (γ>1).
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Continuous time random walks with jump sizes equal to the corresponding waiting times for jumps are considered. Sufficient conditions for the weak convergence of such processes are established and the limiting processes are identified. Furthermore one-dimensional distributions of the limiting processes are given under an additional assumption.
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The convergence of stochastic processes is defined in terms of the so-called “weak convergence” (w. c.) of probability measures in appropriate functional spaces (c. s. m. s.). Chapter 1. Let $\Re$ be the c.s.m.s. and v a set of all finite measures on $\Re$. The distance $L(\mu _1 ,\mu _2 )$ (that is analogous to the Lévy distance) is introduced, and equivalence of L-convergence and w. c. is proved. It is shown that $V\Re = (v,L)$ is c. s. m. s. Then, the necessary and sufficient conditions for compactness in $V\Re$ are given. In section 1.6 the concept of “characteristic functionals” is applied to the study of w. cc of measures in Hilbert space. Chapter 2. On the basis of the above results the necessary and sufficient compactness conditions for families of probability measures in spaces $C[0,1]$ and $D[0,1]$ (space of functions that are continuous in $[0,1]$ except for jumps) are formulated. Chapter 3. The general form of the “invariance principle” for the sums of independent random variables is developed. Chapter 4. An estimate of the remainder term in the well-known Kolmogorov theorem is given (cf. [3.1]).
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One of several critical issues in the development of optimal disease containment and eradication strategies is the knowledge of underlying contacts between individuals. Here we employ random search strategies to identify all possible links, representing direct or indirect interactions between individuals building up the system. In order to recognize all contacts, the searcher performs symmetric Lévy flights onto the accessible area. We investigate the influence of local and non-local information, the exponent characterizing asymptotic behavior of Lévy flights, boundary conditions, density of links and type of a search strategy on the efficiency of the search process. Monte Carlo examination of the suggested model reveals that the efficiency of the search process is sensitive to the type of boundary conditions. Depending on the assumed type of boundary conditions, efficiency of the search process can be a monotonic or non-monotonic function of the exponents characterizing asymptotic behavior of Lévy flights. Consequently, among the whole spectrum of exponents characterizing the power law behavior of jumps’ length, there exist distinguished values of stability index representing the most efficient search processes. These exponents correspond to extreme (minimal or maximal) or intermediate values of stability index associated with Gaussian, maximally heavy-tailed or Cauchy-like strategies, respectively.
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In a continuous time random walk (CTRW), a random waiting time precedes each random jump. The CTRW is coupled if the waiting time and the subsequent jump are dependent random variables. The CTRW is used in physics to model diffusing particles. Its scaling limit is governed by an anomalous diffusion equation. Some applications require an overshoot continuous time random walk (OCTRW), where the waiting time is coupled to the previous jump. This paper develops stochastic limit theory and governing equations for CTRW and OCTRW. The governing equations involve coupled space–time fractional derivatives. In the case of infinite mean waiting times, the solutions to the CTRW and OCTRW governing equations can be quite different.
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Measurements of the transient photocurrent I(t) in an increasing number of inorganic and organic amorphous materials display anomalous transport properties. The long tail of I(t) indicates a dispersion of carrier transit times. However, the shape invariance of I(t) to electric field and sample thickness (designated as universality for the classes of materials here considered) is incompatible with traditional concepts of statistical spreading, i.e., a Gaussian carrier packet. We have developed a stochastic transport model for I(t) which describes the dynamics of a carrier packet executing a time-dependent random walk in the presence of a field-dependent spatial bias and an absorbing barrier at the sample surface. The time dependence of the random walk is governed by hopping time distribution Ψ(t). A packet, generated with a Ψ(t) characteristic of hopping in a disordered system [e.g., Ψ(t)∼t-(1+α), 0<α<1], is shown to propagate with a number of anomalous non-Gaussian properties. The calculated I(t) associated with this packet not only obeys the property of universality but can account quantitatively for a large variety of experiments. The new method of data analysis advanced by the theory allows one to directly extract the transit time even for a featureless current trace. In particular, we shall analyze both an inorganic (a-As2Se3) and an organic (trinitrofluorenone-polyvinylcarbazole) system. Our function Ψ(t) is related to a first-principles calculation. It is to be emphasized that these Ψ(t)'s characterize a realization of a non-Markoffian transport process. Moreover, the theory shows the limitations of the concept of a mobility in this dispersive type of transport.
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Electronic junctions made from porous, nanocrystalline TiO2 films in contact with an electrolyte are important for applications such as dye-sensitized solar cells. They exhibit anomalous electron transport properties: extremely slow, nonexponential current and charge recombination transients, and intensity-dependent response times. These features are attributed to a high density of intraband-gap trap states. Most available models of the electron transport are based on the diffusion equation and predict transient and intensity-dependent behavior which is not observed. In this paper, a preliminary model of dispersive transport based on the continuous-time random walk is applied to nanocrystalline TiO2 electrodes. Electrons perform a random walk on a lattice of trap states, each electron moving after a waiting time which is determined by the activation energy of the trap currently occupied. An exponential density of trap states g(E)∼eα(EC-E)/kT is used giving rise to a power-law waiting-time distribution, ψ(t)=At-1-α. Occupancy of traps is limited to simulate trap filling. The model predicts photocurrents that vary like t-1-α at long time, and charge recombination transients that are approximately stretched exponential in form. Monte Carlo simulations of photocurrent and charge recombination transients reproduce many of the features that have been observed in practice. Using α=0.37, good quantitative agreement is obtained with measurements of charge recombination kinetics in dye-sensitized TiO2 electrodes under applied bias. The intensity dependence of photocurrent transients can be reproduced. It is also shown that normal diffusive transport, which is represented by ψ(t)=λe-λt fails to explain the observed kinetic behavior. The model is proposed as a starting point for a more refined microscopic treatment in which an experimentally determined density of states can be easily incorporated.
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We show that, similar to the Gaussian case, the fractional Ornstein–Uhlenbeck α-stable process obtained via the Lamperti transformation of the linear fractional stable motion is a different stationary process than the one defined as the solution of the Langevin equation driven by a linear fractional stable noise. We investigate the asymptotic dependence structure of the first process and prove that, in contrast to the second case, it is a short-memory process in the sense of the measure of dependence appropriate for processes with infinite second moment.
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We discuss the problem of deriving Lévy diffusion, in the form of a Lévy walk, from a density approach, namely using a Liouville equation. We address this problem for a case that has already been discussed using the method of the continuous time random walk, and consequently an approach based on trajectory dynamics rather than density time evolution. We show that the use of the Liouville equation requires the knowledge of the correlation functions of the fluctuation that generates the Lévy diffusion. We benefit from the results of earlier work proving that the correlation function is not factorized as in the Poisson case. We show that the Liouville equation generates a long-time diffusion whose probability distribution density keeps a memory of the detailed form of the fluctuation correlation function, and not only of its long-time inverse power law structure. Although the main result of this paper rests on an approximate expression for higher-order correlation functions, it becomes exact in the long-time limit. Thus, we argue that it explains the failure to derive Lévy diffusion from the Liouville equation, thereby supporting the claim that there exists a conflict between trajectory and density approaches in this case.
Chapter
Introduction and MotivationQuantitative Assessments of Human MobilityStatistical Properties and Scaling Laws in Multi-Scale Mobility NetworksSpatially Extended Epidemic ModelsSpatial ModelsReferences
Book
CONTENTS: Preliminary remarks; Brownian motion, poisson process, alpha-stable Levy motion; Computer simulation of alpha-stable random variables; Stochastic integration; Spectral representations of stationary processes; Computer approximations of continuous time processes; Examples of alpha-stable stochastic modelling; Convergence of approximate methods; Chaotic behaviour of stationary processes; Hierarchy of chaos for stable and ID stationary processes. Appendix - A guide to simulation.
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