Langevin description of superdiffusive Lévy processes

Lévy-walk-like Langevin dynamics with random parameters

Anomalous diffusion phenomena have been widely found in systems within an inhomogeneous complex environment. For Lévy walk in an inhomogeneous complex environment, we characterize the particle's trajectory through an underdamped Langevin system coupled with a subordinator. The influence of the inhomogeneous environment on the particle's motion is parameterized by the random system parameters, relaxation timescale τ, and velocity diffusivity σ. We find that the two random parameters make different effects on the original superdiffusion behavior of the Lévy walk. The random σ contributes to a trivial result after an ensemble average, which is independent of the specific distribution of σ. By contrast, we find that a specific distribution of τ, a modified Lévy distribution with a finite mean, slows down the decaying rate of the velocity correlation function with respect to the lag time. However, the random τ does not promote the diffusion behavior in a direct way, but plays a competition role to the superdiffusion of the original Lévy walk. In addition, the effect of the random τ is also related to the specific subordinator in the coupled Langevin model, which corresponds to the distribution of the flight time of the Lévy walk. The random system parameters are capable of leading to novel dynamics, which needs detailed analyses, rather than an intuitive judgment, especially in complex systems.

Modelling the eye movements of dyslexic children during reading as a continuous time random walk

The study of eye movements during reading is considered a valuable tool for understanding the underlying cognitive processes and for its ability to detect alterations that could be associated with neurocognitive deficiencies or visual conditions. During reading, the gaze moves from one position to the next on the text performing a saccade-fixation sequence. This dynamics resembles processes usually described as continuous time random walk, where the jumps are the saccadic movements and waiting times are the duration of fixations. The time between jumps (intersaccadic time) consists of stochastic waiting time and flight time, which is a function of the jump length (the amplitude of the saccade). This motivates the present proposal of a model of eye movements during reading in the framework of the intermittent random walk but considering the time between jumps as a combined stochastic-deterministic process. The parameters used in this model were obtained from records of eye movements of children with dyslexia and typically developed for children performing a reading task. The jump lengths arise from the characteristics of the selected text. The time required for the flights was obtained based on a previously proposed model. Synthetic signals were generated and compared with actual eye movement signals in a complexity-entropy plane.

Langevin picture of anomalous diffusion processes in expanding medium

The 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 a particle's motion in an expanding medium has only been investigated in the framework of a continuous-time random walk. To focus on more diffusion processes and physical observables, we build the Langevin picture of anomalous diffusion in an expanding medium, and conduct detailed analyses in the framework of the Langevin equation. With the help of a subordinator, both subdiffusion process and superdiffusion process in the expanding medium are discussed. We find that the expanding medium with different changing rate (exponential form and power-law form) leads to quite different diffusion phenomena. The particle's intrinsic diffusion behavior also plays an important role. Our detailed theoretical analyses and simulations present a panoramic view of investigating anomalous diffusion in an expanding medium under the framework of the Langevin equation.

Different effects of external force fields on aging Lévy walk

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 the Lévy walk and the generalized Green-Kubo formula, we investigate the quantities that include the ensemble- and time-averaged mean-squared displacements in both weak aging t_a≪t and strong aging t_a≫t cases and compare them to the ones in the absence of any force field. Two typical force fields, constant force F and time-dependent periodic force F(t)=f₀sin⁡(ωt), are considered for comparison. The generalized Einstein relation is also discussed in the case with the constant force. We find that the constant force is the key to causing the aging phenomena and enhancing the diffusion behavior of the aging Lévy walk, while the time-dependent periodic force is not. The different effects of the two kinds of forces on the aging Lévy walk are verified by both theoretical analyses and numerical simulations.

Random diffusivity processes in an external force field

Brownian yet non-Gaussian processes have recently been observed in numerous biological systems, and corresponding theories have been constructed based on random diffusivity models. Considering the particularity of random diffusivity, this paper studies the effect of an external force acting on two kinds of random diffusivity models whose difference is embodied in whether the fluctuation-dissipation theorem is valid. Based on the two random diffusivity models, we derive the Fokker-Planck equations with an arbitrary external force, and we analyze various observables in the case with a constant force, including the Einstein relation, the moments, the kurtosis, and the asymptotic behaviors of the probability density function of particle displacement at different timescales. Both the theoretical results and numerical simulations of these observables show a significant difference between the two kinds of random diffusivity models, which implies the important role of the fluctuation-dissipation theorem in random diffusivity systems.

Infinite density and relaxation for Lévy walks in an external potential: Hermite polynomial approach

Lévy walks are continuous-time random-walk processes with a spatiotemporal coupling of jump lengths and waiting times. We here apply the Hermite polynomial method to study the behavior of LWs with power-law walking time density for four different cases. First we show that the known result for the infinite density of an unconfined, unbiased LW is consistently recovered. We then derive the asymptotic behavior of the probability density function (PDF) for LWs in a constant force field, and we obtain the corresponding qth-order moments. In a harmonic external potential we derive the relaxation dynamic of the LW. For the case of a Poissonian walking time an exponential relaxation behavior is shown to emerge. Conversely, a power-law decay is obtained when the mean walking time diverges. Finally, we consider the case of an unconfined, unbiased LW with decaying speed v(τ)=v_{0}/sqrt[τ]. When the mean walking time is finite, a universal Gaussian law for the position-PDF of the walker is obtained explicitly.

Nonergodicity of d-dimensional generalized Lévy walks and their relation to other space-time coupled models

We investigate the nonergodicity of the generalized Lévy walk introduced by Shlesinger et al. [Phys. Rev. Lett. 58, 1100 (1987)PRLTAO0031-900710.1103/PhysRevLett.58.1100] 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)PRLTAO0031-900710.1103/PhysRevLett.120.104501], 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.

Anomalous diffusion of molecules with rotating polar groups: The joint role played by inertia and dipole coupling in microwave and far-infrared absorption

Budó's generalization [A. Budó, J. Chem. Phys. 17, 686 (1949)10.1063/1.1747370] of the Debye rotational diffusion model of dielectric relaxation of polar molecules to an assembly with internal interacting polar groups is extended to inertial anomalous diffusion. Thus, the theory can be applied both in the GHz and the THz regions, accounting for anomalous behavior as well as the necessary return to optical transparency at very high frequencies. The linking of both dispersion regions in a single model including anomalous effects is accomplished via a fractional Fokker-Planck equation in phase space based on the continuous time random walk ansatz. The latter is written via the Langevin equations for the stochastic dynamics of pairs of interacting heavy polar groups embedded in the frame of reference of a particular molecule or molecular dimer rotating about a space-fixed axis. The fractional Fokker-Planck equation is then converted to a three-term matrix differential recurrence equation for the statistical moments. This is solved in the frequency domain for the linear dielectric response using matrix continued fractions. Thus, one has the complex susceptibility χ(ω) for extensive ranges of damping, group dipole moment ratio, and friction. The susceptibility, as inferred from the small oscillation limit, inherently comprises a low frequency (GHz) band with width depending on the anomalous parameter and a far-infrared (THz) or Poley peak of resonant character with a comblike structure of harmonic peaks. This behavior is due to the double transcendental nature of the after-effect function.

Generalization to anomalous diffusion of Budó's treatment of polar molecules containing interacting rotating groups

A fractional Smoluchowski equation for the orientational distribution of dipoles incorporating interactions with continuous time random walk Ansatz for the collision term is obtained. This equation is written via the non-inertial Langevin equations for the evolution of the Eulerian angles and their associated Smoluchowski equation. These equations govern the normal rotational diffusion of an assembly of non-interacting dipolar molecules with similar internal interacting polar groups hindering their rotation owing to their mutual potential energy. The resulting fractional Smoluchowski equation is then solved in the frequency domain using scalar continued fractions yielding the linear dielectric response as a function of the fractional parameter for extensive ranges of the interaction parameter and friction ratios. The complex susceptibility comprises a multimode Cole-Cole-like low frequency band with width dependent on the fractional parameter and is analogous to the discrete set of Debye mechanisms of the normal diffusion. The results, in general, comprise an extension of Budó's treatment [A. Budó, J. Chem. Phys. 17, 686 (1949)] of the dynamics of complex molecules with internal hindered rotation to anomalous diffusion.

Theory of relaxation dynamics for anomalous diffusion processes in harmonic potential

An important physical property for a stochastic process is how it responds to an external force or spatial confinement. This paper aims to study the relaxation dynamics of a generic process confined in a harmonic potential. We find the dependence of ensemble- and time-averaged mean squared displacements of the confined process on the velocity correlation function C(t,t+τ) of the original process without any external force. Combining two kinds of scaling forms of C(t,t+τ) for small τ and large τ, the stationary value and the relaxation behaviors can be immediately obtained. Our results are valid for a large amount of anomalous diffusion processes, including the ones with single-scaled velocity correlation function (such as fractional Brownian motion and scaled Brownian motion) and the multiscaled ones (like Lévy walk with a broad range of power law exponents of flight time distribution). Note that the latter includes a special case with telegraphic active noise, which could take up athermal energy from the environment.

Anomalous diffusion of a dipole interacting with its surroundings

A fractional Fokker-Planck equation based on the continuous time random walk Ansatz is written via the Langevin equations for the dynamics of a dipole interacting with its surroundings, as represented by a cage of dipolar molecules. This equation is solved in the frequency domain using matrix continued fractions, thus yielding the linear dielectric response for extensive ranges of damping, dipole moment ratio, and cage-dipole inertia ratio, and hence the complex susceptibility. The latter comprises a low frequency band with width depending on the anomalous parameter and a far infrared (THz) band with a comb-like structure of peaks. Several physical consequences of the model relevant to anomalous diffusion in the presence of interactions are discussed. The entire calculation may be regarded as an extension of the cage model interpretation of the dynamics of polar molecules to anomalous diffusion, taking into account inertial effects.

Langevin picture of Lévy walk in a constant force field

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∈(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, as it is 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 moments of velocity processes for these two models are consistent. In particular, based on the velocity correlation function derived from our subordinated Langevin equation, we investigate more interesting statistical quantities, such as the ensemble- and time-averaged mean-squared displacements. Under the influence of constant force, the diffusion of particles becomes faster. Finally, the superballistic diffusion and the nonergodic behavior are verified by the simulations with different α.

Langevin dynamics for a Lévy walk with memory

Memory effects, sometimes, cannot 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 correlated waiting time process, as well as the one of its inverse process, and present the Langevin description of Lévy walk with memory. We call this model a Lévy-walk-type model with correlated waiting times. Based on the built Langevin picture, the properties of aging and nonstationary are discussed. This Langevin system exhibits sub-ballistic superdiffusion 〈x^{2}(t)〉∝t^{2-α^{2}β/αβ+1} if the friction force is involved, while it displays superballistic diffusion or hyperdiffusion 〈x^{2}(t)〉∝t^{2+α/αβ+1} if there is no friction. The parameter 0<α<1 is for the white α-stable Lévy noise, while 0≤β≤1 is to characterize the strength of the correlation of waiting times; β=0 corresponds to uncorrelated case and β=1 the strongest correlation. It is discovered that the correlation of waiting times suppresses the diffusion behavior whether a friction is involved or not. The stronger the correlation of waiting times becomes, the slower the diffusion is. In particular, the correlation function, correlation coefficient, ergodicity, and scaling property of the corresponding stochastic process are also investigated.

Spectral decomposition of nonlinear systems with memory

We present an alternative approach to the analysis of nonlinear systems with long-term memory that is based on the Koopman operator and a Lévy transformation in time. Memory effects are considered to be the result of interactions between a system and its surrounding environment. The analysis leads to the decomposition of a nonlinear system with memory into modes whose temporal behavior is anomalous and lacks a characteristic scale. On average, the time evolution of a mode follows a Mittag-Leffler function, and the system can be described using the fractional calculus. The general theory is demonstrated on the fractional linear harmonic oscillator and the fractional nonlinear logistic equation. When analyzing data from an ill-defined (black-box) system, the spectral decomposition in terms of Mittag-Leffler functions that we propose may uncover inherent memory effects through identification of a small set of dynamically relevant structures that would otherwise be obscured by conventional spectral methods. Consequently, the theoretical concepts we present may be useful for developing more general methods for numerical modeling that are able to determine whether observables of a dynamical system are better represented by memoryless operators, or operators with long-term memory in time, when model details are unknown.

Swarming bacteria migrate by Lévy Walk

Individual swimming bacteria are known to bias their random trajectories in search of food and to optimize survival. The motion of bacteria within a swarm, wherein they migrate as a collective group over a solid surface, is fundamentally different as typical bacterial swarms show large-scale swirling and streaming motions involving millions to billions of cells. Here by tracking trajectories of fluorescently labelled individuals within such dense swarms, we find that the bacteria are performing super-diffusion, consistent with Lévy walks. Lévy walks are characterized by trajectories that have straight stretches for extended lengths whose variance is infinite. The evidence of super-diffusion consistent with Lévy walks in bacteria suggests that this strategy may have evolved considerably earlier than previously thought.

Langevin formulation of a subdiffusive continuous-time random walk in physical time

Systems living in complex nonequilibrated 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 in terms of the subordination of a normal diffusive process by an inverse Lévy-stable process. Here, we propose an equivalent Langevin formulation of a force-free CTRW without subordination. By introducing a different type of non-Gaussian noise, we are able to express the CTRW dynamics in terms of a single Langevin equation in physical time with additive noise. We derive the full multipoint statistics of this noise and compare it with the scaled Brownian motion (SBM), an alternative stochastic model describing subdiffusive dynamics. Interestingly, these two noises are identical up to the second order correlation functions, but different in the higher order statistics. We extend our formalism to general waiting time distributions and force fields and compare our results with those of the SBM. In the presence of external forces, our proposed noise generates a different class of stochastic processes, resembling a CTRW but with forces acting at all times.