Non-normalizable densities in strong anomalous diffusion: beyond the central limit theorem

Generalized two-state random walk model: Nontrivial anomalous diffusion, aging, and ergodicity breaking

The intermittent stochastic motion is a dichotomous process that alternates between two distinct states. This phenomenon, observed across various physical and biological systems, is attracting increasing interest and highlighting the need for comprehensive theories to describe it. In this paper, we introduce a generalized intermittent random walk model based on a renewal process that alternates between the continuous time random walk (CTRW) state and the generalized Lévy walk (gLW) state. Notably, the nonlinear space-time coupling inherent in the gLW state allows this generalized model to encompass a variety of random walk models and makes it applicable to diverse systems. By deriving the velocity correlation function and utilizing the scaling Green-Kubo relation, the ensemble-averaged and time-averaged mean-squared displacement (MSD) is calculated, and the anomalous diffusive behavior, aging effect, and ergodic property of the model are further analyzed and discussed. The results reveal that, due to the intermittent nature, there are two diffusive terms in the expression of the MSD, and the diffusion can be intermediately characterized by the diffusive term with the largest diffusion coefficient instead of the diffusive term with the largest diffusion exponent, which is significantly different from single-state stochastic process. We demonstrate that, due to the power-law distribution of sojourn times, nonlinear space-time coupling, and intermittent characteristics, both ergodicity and nonergodicity can coexist in intermittent stochastic processes.

Random telegraph processes with nonlocal memory

We study two-state (dichotomous, telegraph) random ergodic continuous-time processes with dynamics depending on their past. We take into account the history of the process in an explicit form by introducing integral nonlocal memory term into conditional probability function. We start from an expression for the conditional transition probability function describing additive multistep binary random chain and show that the telegraph processes can be considered as continuous-time interpolations of discrete-time dichotomous random sequences. An equation involving the memory function and the two-point correlation function of the telegraph process is analytically obtained. This integral equation defines the correlation properties of the processes with given memory functions. It also serves as a tool for solving the inverse problem, namely for generation of a telegraph process with a prescribed pair correlation function. We obtain analytically the correlation functions of the telegraph processes with two exactly solvable examples of memory functions and support these results by numerical simulations of the corresponding telegraph processes.

Anomalous diffusion in external-force-affected deterministic systems

This study investigates the impact of external forces on the movement of particles, specifically focusing on a type of box piecewise linear map that generates normal diffusion akin to Brownian motion. Through numerical methods, the research delves into the effects of two distinct external forces: linear forces linked to the particle's current position and periodic sinusoidal forces related to time. The results uncover anomalous dynamical behavior characterized by nonlinear growth in the ensemble-averaged mean-squared displacement (EAMSD), aging, and ergodicity breaking. Notably, the diffusion pattern of particles under linear external forces resembles an Ehrenfest double urn model, with its asymptotic EAMSD coinciding with the Langevin equation under linear potential. Meanwhile, particle movement influenced by periodic sinusoidal forces corresponds to an inhomogeneous Markov chain, with its external force amplitude and diffusion coefficient function exhibiting a "multipeak" fractal structure. The study also provides insights into the formation of this structure through the turnstiles dynamics.

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.

Statistics of long-range force fields in random environments: Beyond Holtsmark

Since the times of Holtsmark (1911), statistics of fields in random environments have been widely studied, for example in astrophysics, active matter, and line-shape broadening. The power-law decay of the two-body interaction of the form 1/|r|^{δ}, and assuming spatial uniformity of the medium particles exerting the forces, imply that the fields are fat-tailed distributed, and in general are described by stable Lévy distributions. With this widely used framework, the variance of the field diverges, which is nonphysical, due to finite size cutoffs. We find a complementary statistical law to the Lévy-Holtsmark distribution describing the large fields in the problem, which is related to the finite size of the tracer particle. We discover biscaling with a sharp statistical transition of the force moments taking place when the order of the moment is d/δ, where d is the dimension. The high-order moments, including the variance, are described by the framework presented in this paper, which is expected to hold for many systems. The new scaling solution found here is nonnormalized similar to infinite invariant densities found in dynamical systems.

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.

Infinite ergodic theory for three heterogeneous stochastic models with application to subrecoil laser cooling

We compare ergodic properties of the kinetic energy for three stochastic models of subrecoil-laser-cooled gases. One model is based on a heterogeneous random walk (HRW), another is an HRW with long-range jumps (the exponential model), and the other is a mean-field-like approximation of the exponential model (the deterministic model). All the models show an accumulation of the momentum at zero in the long-time limit, and a formal steady state cannot be normalized, i.e., there exists an infinite invariant density. We obtain the exact form of the infinite invariant density and the scaling function for the exponential and deterministic models, and we devise a useful approximation for the momentum distribution in the HRW model. While the models are kinetically nonidentical, it is natural to wonder whether their ergodic properties share common traits, given that they are all described by an infinite invariant density. We show that the answer to this question depends on the type of observable under study. If the observable is integrable, the ergodic properties, such as the statistical behavior of the time averages, are universal as they are described by the Darling-Kac theorem. In contrast, for nonintegrable observables, the models in general exhibit nonidentical statistical laws. This implies that focusing on nonintegrable observables, we discover nonuniversal features of the cooling process, which hopefully can lead to a better understanding of the particular model most suitable for a statistical description of the process. This result is expected to hold true for many other systems, beyond laser cooling.

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.

Asymptotic densities of planar Lévy walks: A nonisotropic case

Lévy walks are a particular type of continuous-time random walks which results in a super-diffusive spreading of an initially localized packet. The original one-dimensional model has a simple schematization that is based on starting a new unidirectional motion event either in the positive or in the negative direction. We consider two-dimensional generalization of Lévy walks in the form of the so-called XY model. It describes a particle moving with a constant velocity along one of the four basic directions and randomly switching between them when starting a new motion event. We address the ballistic regime and derive solutions for the asymptotic density profiles. The solutions have a form of first-order integrals which can be evaluated numerically. For specific values of parameters we derive an exact expression. The analytic results are in agreement with the results of finite-time numerical samplings.

Nonequilibrium Phase Transition to Anomalous Diffusion and Transport in a Basic Model of Nonlinear Brownian Motion

We investigate a basic model of nonlinear Brownian motion in a thermal environment, where nonlinear friction interpolates between viscous Stokes and dry Coulomb friction. We show that superdiffusion and supertransport emerge as a nonequilibrium critical phenomenon when such a Brownian motion is driven out of thermal equilibrium by a constant force. Precisely at the edge of a phase transition, velocity fluctuations diverge asymptotically and diffusion becomes superballistic. The autocorrelation function of velocity fluctuations in this nonergodic regime exhibits a striking aging behavior.

Lévy walk dynamics in mixed potentials from the perspective of random walk theory

Lévy walk process is one of the most effective models to describe superdiffusion, which underlies some important movement patterns and has been widely observed in micro- and macrodynamics. From the perspective of random walk theory, here we investigate the dynamics of Lévy walks under the influences of the constant force field and the one combined with harmonic potential. Utilizing Hermite polynomial approximation to deal with the spatiotemporally coupled analysis challenges, some striking features are detected, including non-Gaussian stationary distribution, faster diffusion, still strongly anomalous diffusion, etc.

Memory-dependent noise-induced resonance and diffusion in non-Markovian systems

We study random processes with nonlocal memory and obtain solutions of the Mori-Zwanzig equation describing non-Markovian systems. We analyze the system dynamics depending on the amplitudes ν and μ_{0} of the local and nonlocal memory and pay attention to the line in the (ν, μ_{0}) plane separating the regions with asymptotically stationary and nonstationary behavior. We obtain general equations for such boundaries and consider them for three examples of nonlocal memory functions. We show that there exist two types of boundaries with fundamentally different system dynamics. On the boundaries of the first type, diffusion with memory takes place, whereas on borderlines of the second type the phenomenon of noise-induced resonance can be observed. A distinctive feature of noise-induced resonance in the systems under consideration is that it occurs in the absence of an external regular periodic force. It takes place due to the presence of frequencies in the noise spectrum, which are close to the self-frequency of the system. We analyze also the variance of the process and compare its behavior for regions of asymptotic stationarity and nonstationarity, as well as for diffusive and noise-induced-resonance borderlines between them.

Lévy-walk-like Langevin dynamics affected by a time-dependent force

The Lévy walk is a popular and more 'physical' model to describe the phenomena of superdiffusion, because of its finite velocity. The movements of particles are under the influence of external potentials at almost any time and anywhere. In this paper, we establish a Langevin system coupled with a subordinator to describe the Lévy walk in a time-dependent periodic force field. The effects of external force are detected and carefully analyzed, including the nonzero first moment (even though the force is periodic), adding an additional dispersion on the particle position, a consistent influence on the ensemble- and time-averaged mean-squared displacement, etc. Besides, the generalized Klein-Kramers equation is obtained, not only for the time-dependent force but also for the space-dependent one.

Correlated continuous-time random walk in a velocity field: Anomalous bifractional crossover

The diffusion of space-time correlated continuous-time random walk moving in the velocity field, which includes the fluid flowing freely and the fluid flowing through porous media, is investigated in this paper. Results reveal that it presents anomalous diffusion merely caused by space-time correlation in the freely flowing fluid, and the bias from the velocity field only supplies a standard advection, which is verified by the corresponding generalized diffusion equation which includes a standard advection term. However, the diffusion in the fluid flowing through porous media, i.e., the mean squared displacement, can display a bifractional form of which one originates from space-time correlation and the other one originates from dispersive bias caused by sticking of the porous media. The fractional advection term emerging in the corresponding generalized diffusion equation confirms the results. Moreover, the coexistence of correlation and dispersive bias result in crossover phenomenon in-between the diffusive process at an intermediate timescale, but just as the definition of diffusion, the one owning the largest diffusion exponent always prevails at large timescales. However, since the two fractional diffusions originate from a different mechanism, even if it owns the smaller diffusion exponent, that one can dominate the diffusion if it fluctuates much stronger than the other one, which no longer obeys the previous conclusion.

Non-Normalizable Quasi-Equilibrium Solution of the Fokker-Planck Equation for Nonconfining Fields

We investigate the overdamped Langevin motion for particles in a potential well that is asymptotically flat. When the potential well is deep as compared to the temperature, physical observables, like the mean square displacement, are essentially time-independent over a long time interval, the stagnation epoch. However, the standard Boltzmann-Gibbs (BG) distribution is non-normalizable, given that the usual partition function is divergent. For this regime, we have previously shown that a regularization of BG statistics allows for the prediction of the values of dynamical and thermodynamical observables in the non-normalizable quasi-equilibrium state. In this work, based on the eigenfunction expansion of the time-dependent solution of the associated Fokker–Planck equation with free boundary conditions, we obtain an approximate time-independent solution of the BG form, being valid for times that are long, but still short as compared to the exponentially large escape time. The escaped particles follow a general free-particle statistics, where the solution is an error function, which is shifted due to the initial struggle to overcome the potential well. With the eigenfunction solution of the Fokker–Planck equation in hand, we show the validity of the regularized BG statistics and how it perfectly describes the time-independent regime though the quasi-stationary state is non-normalizable.

A least microenvironmental uncertainty principle (LEUP) as a generative model of collective cell migration mechanisms

Collective migration is commonly observed in groups of migrating cells, in the form of swarms or aggregates. Mechanistic models have proven very useful in understanding collective cell migration. Such models, either explicitly consider the forces involved in the interaction and movement of individuals or phenomenologically define rules which mimic the observed behavior of cells. However, mechanisms leading to collective migration are varied and specific to the type of cells involved. Additionally, the precise and complete dynamics of many important chemomechanical factors influencing cell movement, from signalling pathways to substrate sensing, are typically either too complex or largely unknown. The question is how to make quantitative/qualitative predictions of collective behavior without exact mechanistic knowledge. Here we propose the least microenvironmental uncertainty principle (LEUP) that may serve as a generative model of collective migration without precise incorporation of full mechanistic details. Using statistical physics tools, we show that the famous Vicsek model is a special case of LEUP. Finally, to test the biological applicability of our theory, we apply LEUP to construct a model of the collective behavior of spherical Serratia marcescens bacteria, where the underlying migration mechanisms remain elusive.

Large deviations of the ballistic Lévy walk model

We study the ballistic Lévy walk stemming from an infinite mean traveling time between collision events. Our study focuses on the density of spreading particles all starting from a common origin, which is limited by a "light" cone -v_{0}t<x<v_{0}t. In particular we study this density close to its maximum in the vicinity of the light cone. The spreading density follows the Lamperti-arcsine law describing typical fluctuations. However, this law blows up in the vicinity of the spreading horizon, which is nonphysical in the sense that any finite-time observation will never diverge. We claim that one can find two laws for the spatial density: The first one is the mentioned Lamperti-arcsine law describing the central part of the distribution, and the second is an infinite density illustrating the dynamics for x≃v_{0}t. We identify the relationship between a large position and the longest traveling time describing the single big jump principle. From the renewal theory we find that the distribution of rare events of the position is related to the derivative of the average of the number of renewals at a short "time" using a rate formalism.

Extreme value theory for constrained physical systems

We investigate extreme value theory for physical systems with a global conservation law which describes renewal processes, mass transport models, and long-range interacting spin models. As shown previously, a special feature is that the distribution of the extreme value exhibits a nonanalytical point in the middle of the support. We expose exact relationships between constrained extreme value theory and well-known quantities of the underlying stochastic dynamics, all valid beyond the midpoint in general, i.e., even far from the thermodynamic limit. For example, for renewal processes the distribution of the maximum time between two renewal events is exactly related to the mean number of these events. In the thermodynamic limit, we show how our theory is suitable to describe typical and rare events which deviate from classical extreme value theory. For example, for the renewal process we unravel dual scaling of the extreme value distribution, pointing out two types of limiting laws: a normalizable scaling function for the typical statistics and a non-normalized state describing the rare events.

Lévy walk dynamics in an external harmonic potential

Lévy walks (LWs) are spatiotemporally coupled random-walk processes describing superdiffusive heat conduction in solids, propagation of light in disordered optical materials, motion of molecular motors in living cells, or motion of animals, humans, robots, and viruses. We here investigate a key feature of LWs-their response to an external harmonic potential. In this generic setting for confined motion we demonstrate that LWs equilibrate exponentially and may assume a bimodal stationary distribution. We also show that the stationary distribution has a horizontal slope next to a reflecting boundary placed at the origin, in contrast to correlated superdiffusive processes. Our results generalize LWs to confining forces and settle some longstanding puzzles around LWs.

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.

Infinite invariant density in a semi-Markov process with continuous state variables

We report on a fundamental role of a non-normalized formal steady state, i.e., an infinite invariant density, in a semi-Markov process where the state is determined by the interevent time of successive renewals. The state describes certain observables found in models of anomalous diffusion, e.g., the velocity in the generalized Lévy walk model and the energy of a particle in the trap model. In our model, the interevent-time distribution follows a fat-tailed distribution, which makes the state value more likely to be zero because long interevent times imply small state values. We find two scaling laws describing the density for the state value, which accumulates in the vicinity of zero in the long-time limit. These laws provide universal behaviors in the accumulation process and give the exact expression of the infinite invariant density. Moreover, we provide two distributional limit theorems for time-averaged observables in these nonstationary processes. We show that the infinite invariant density plays an important role in determining the distribution of time averages.

Universality in the Onset of Superdiffusion in Lévy Walks

Anomalous dynamics in which local perturbations spread faster than diffusion are ubiquitously observed in the long-time behavior of a wide variety of systems. Here, the manner by which such systems evolve towards their asymptotic superdiffusive behavior is explored using the 1D Lévy walk of order 1<β<2. The approach towards superdiffusion, as captured by the leading correction to the asymptotic behavior, is shown to remarkably undergo a transition as β crosses the critical value β_{c}=3/2. Above β_{c}, this correction scales as |x|∼t^{1/2}, describing simple diffusion. However, below β_{c} it is instead found to remain superdiffusive, scaling as |x|∼t^{1/(2β-1)}. This transition is shown to be independent of the precise model details and is thus argued to be universal.

Rare events in generalized Lévy Walks and the Big Jump principle

The prediction and control of rare events is an important task in disciplines that range from physics and biology, to economics and social science. The Big Jump principle deals with a peculiar aspect of the mechanism that drives rare events. According to the principle, in heavy-tailed processes a rare huge fluctuation is caused by a single event and not by the usual coherent accumulation of small deviations. We consider generalized Lévy walks, a class of stochastic processes with power law distributed step durations and with complex microscopic dynamics in the single stretch. We derive the bulk of the probability distribution and using the big jump principle, the exact form of the tails that describes rare events. We show that the tails of the distribution present non-universal and non-analytic behaviors, which depend crucially on the dynamics of the single step. The big jump estimate also provides a physical explanation of the processes driving the rare events, opening new possibilities for their correct prediction.

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.

Infinite horizon billiards: Transport at the border between Gauss and Lévy universality classes

We consider transport in two billiard models, the infinite horizon Lorentz gas and the stadium channel, presenting analytical results for the spreading packet of particles. We first obtain the cumulative distribution function of traveling times between collisions, which exhibits nonanalytical behavior. Using a renewal assumption and the Lévy walk model, we obtain the particles' probability density. For the Lorentz gas, it shows a distinguished difference when compared with the known Gaussian propagator, as the latter is valid only for extremely long times. In particular, we show plumes of particles spreading along the infinite corridors, creating power-law tails of the density. We demonstrate the slow convergence rate via summation of independent and identically distributed random variables on the border between Lévy and Gauss laws. The renewal assumption works well for the Lorentz gas with intermediate-size scattering centers, but fails for the stadium channel due to strong temporal correlations. Our analytical results are supported with numerical samplings.

Single-big-jump principle in physical modeling

The big-jump principle is a well-established mathematical result for sums of independent and identically distributed random variables extracted from a fat-tailed distribution. It states that the tail of the distribution of the sum is the same as the distribution of the largest summand. In practice, it means that when in a stochastic process the relevant quantity is a sum of variables, the mechanism leading to rare events is peculiar: Instead of being caused by a set of many small deviations all in the same direction, one jump, the biggest of the lot, provides the main contribution to the rare large fluctuation. We reformulate and elevate the big-jump principle beyond its current status to allow it to deal with correlations, finite cutoffs, continuous paths, memory, and quenched disorder. Doing so we are able to predict rare events using the extended big-jump principle in Lévy walks, in a model of laser cooling, in a scattering process on a heterogeneous structure, and in a class of Lévy walks with memory. We argue that the generalized big-jump principle can serve as an excellent guideline for reliable estimates of risk and probabilities of rare events in many complex processes featuring heavy-tailed distributions, ranging from contamination spreading to active transport in the cell.

Collectively enhanced thermalization via multiqubit collisions

We investigate the evolution of a target qubit caused by its multiple random collisions with N-qubit clusters. Depending on the cluster state, the evolution of the target qubit may correspond to its effective interaction with a thermal bath, a coherent (laser) drive, or a squeezed bath. In cases where the target qubit relaxes to a thermal state, its dynamics can exhibit a quantum advantage, whereby the target-qubit temperature can be scaled up proportionally to N^{2} and the thermalization time can be shortened by a similar factor, provided the appropriate coherence in the cluster is prepared by nonthermal means. We dub these effects quantum superthermalization because of the analogies to superradiance. Experimental realizations of these effects are suggested.

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.

Dispersion of particles in an infinite-horizon Lorentz gas

We consider a two-dimensional Lorentz gas with infinite horizon. This paradigmatic model consists of pointlike particles undergoing elastic collisions with fixed scatterers arranged on a periodic lattice. It was rigorously shown that when t→∞, the distribution of particles is Gaussian. However, the convergence to this limit is ultraslow, hence it is practically unattainable. Here, we obtain an analytical solution for the Lorentz gas' kinetics on physically relevant timescales, and find that the density in its far tails decays as a universal power law of exponent -3. We also show that the arrangement of scatterers is imprinted in the shape of the distribution.

Large deviations in the presence of cooperativity and slow dynamics

We study simple models of intermittency, involving switching between two states, within the dynamical large-deviation formalism. Singularities appear in the formalism when switching is cooperative or when its basic time scale diverges. In the first case the unbiased trajectory distribution undergoes a symmetry breaking, leading to a change in shape of the large-deviation rate function for a particular dynamical observable. In the second case the symmetry of the unbiased trajectory distribution remains unbroken. Comparison of these models suggests that singularities of the dynamical large-deviation formalism can signal the dynamical equivalent of an equilibrium phase transition but do not necessarily do so.

Infinite invariant densities due to intermittency in a nonlinear oscillator

Dynamical intermittency is known to generate anomalous statistical behavior of dynamical systems, a prominent example being the Pomeau-Manneville map. We present a nonlinear oscillator, i.e., a physical model in continuous time, whose properties in terms of weak ergodity breaking and aging have a one-to-one correspondence to the properties of the Pomeau-Manneville map. So for both systems in a wide range of parameters no physical invariant density exists. We show how this regime can be characterized quantitatively using the techniques of infinite invariant densities and the Thaler-Dynkin limit theorem. We see how expectation values exhibit aging in terms of scaling in time.

Large Fluctuations for Spatial Diffusion of Cold Atoms

We use a new approach to study the large fluctuations of a heavy-tailed system, where the standard large-deviations principle does not apply. Large-deviations theory deals with tails of probability distributions and the rare events of random processes, for example, spreading packets of particles. Mathematically, it concerns the exponential falloff of the density of thin-tailed systems. Here we investigate the spatial density P_{t}(x) of laser-cooled atoms, where at intermediate length scales the shape is fat tailed. We focus on the rare events beyond this range, which dominate important statistical properties of the system. Through a novel friction mechanism induced by the laser fields, the density is explored with the recently proposed non-normalized infinite-covariant density approach. The small and large fluctuations give rise to a bifractal nature of the spreading packet. We derive general relations which extend our theory to a class of systems with multifractal moments.

Heat perturbation spreading in the Fermi-Pasta-Ulam-β system with next-nearest-neighbor coupling: Competition between phonon dispersion and nonlinearity

We employ the heat perturbation correlation function to study thermal transport in the one-dimensional Fermi-Pasta-Ulam-β lattice with both nearest-neighbor and next-nearest-neighbor couplings. We find that such a system bears a peculiar phonon dispersion relation, and thus there exists a competition between phonon dispersion and nonlinearity that can strongly affect the heat correlation function's shape and scaling property. Specifically, for small and large anharmoncities, the scaling laws are ballistic and superdiffusive types, respectively, which are in good agreement with the recent theoretical predictions; whereas in the intermediate range of the nonlinearity, we observe an unusual multiscaling property characterized by a nonmonotonic delocalization process of the central peak of the heat correlation function. To understand these multiscaling laws, we also examine the momentum perturbation correlation function and find a transition process with the same turning point of the anharmonicity as that shown in the heat correlation function. This suggests coupling between the momentum transport and the heat transport, in agreement with the theoretical arguments of mode cascade theory.

Anomalous temperature-dependent heat transport in one-dimensional momentum-conserving systems with soft-type interparticle interaction

We numerically investigate the heat transport problem in a one-dimensional momentum-conserving lattice with a soft-type (ST) anharmonic interparticle interaction. It is found that with the increase of the system's temperature, while the introduction of ST anharmonicity softens phonons and decreases their velocities, this type of nonlinearity like its hard type (HT) counterpart, can still not be able to fully damp the longest wavelength phonons. Therefore, a usual anomalous temperature dependence of heat transport with certain scaling properties similarly to those shown in the Fermi-Pasta-Ulam-β-like systems with HT interactions can be seen. Our detailed examination from simulations verifies this temperature-dependent behavior well.

Emergence of Lévy walks in systems of interacting individuals

We propose a model of superdiffusive Lévy walk as an emergent nonlinear phenomenon in systems of interacting individuals. The aim is to provide a qualitative explanation of recent experiments [G. Ariel et al., Nat. Commun. 6, 8396 (2015)2041-172310.1038/ncomms9396] revealing an intriguing behavior: swarming bacteria fundamentally change their collective motion from simple diffusion into a superdiffusive Lévy walk dynamics. We introduce microscopic mean-field kinetic equations in which we combine two key ingredients: (1) alignment interactions between individuals and (2) non-Markovian effects. Our interacting run-and-tumble model leads to the superdiffusive growth of the mean-squared displacement and the power-law distribution of run length with infinite variance. The main result is that the superdiffusive behavior emerges as a cooperative effect without using the standard assumption of the power-law distribution of run distances from the inception. At the same time, we find that the collision and repulsion interactions lead to the density-dependent exponential tempering of power-law distributions. This qualitatively explains the experimentally observed transition from superdiffusion to the diffusion of mussels as their density increases [M. de Jager et al., Proc. R. Soc. B 281, 20132605 (2014)PRLBA40962-845210.1098/rspb.2013.2605].

Transient anomalous diffusion in periodic systems: ergodicity, symmetry breaking and velocity relaxation

We study far from equilibrium transport of a periodically driven inertial Brownian particle moving in a periodic potential. As detected for a SQUID ratchet dynamics, the mean square deviation of the particle position from its average may involve three distinct intermediate, although extended diffusive regimes: initially as superdiffusion, followed by subdiffusion and finally, normal diffusion in the asymptotic long time limit. Even though these anomalies are transient effects, their lifetime can be many, many orders of magnitude longer than the characteristic time scale of the setup and turns out to be extraordinarily sensitive to the system parameters like temperature or the potential asymmetry. In the paper we reveal mechanisms of diffusion anomalies related to ergodicity of the system, symmetry breaking of the periodic potential and ultraslow relaxation of the particle velocity towards its steady state. Similar sequences of the diffusive behaviours could be detected in various systems including, among others, colloidal particles in random potentials, glass forming liquids and granular gases.

Strange kinetics of bulk-mediated diffusion on lipid bilayers

Diffusion at solid-liquid interfaces is crucial in many technological and biophysical processes. Although its behavior seems to be deceivingly simple, recent studies showing passive superdiffusive transport suggest that diffusion on surfaces may hide rich complexities. In particular, bulk-mediated diffusion occurs when molecules are transiently released from the surface to perform three-dimensional excursions into the liquid bulk. This phenomenon bears the dichotomy where a molecule always return to the surface but the mean jump length is infinite. Such behavior is associated with a breakdown of the central limit theorem and weak ergodicity breaking. Here, we use single-particle tracking to study the statistics of bulk-mediated diffusion on a supported lipid bilayer. We find that the time-averaged mean square displacement (MSD) of individual trajectories, the archetypal measure in diffusion processes, does not converge to the ensemble MSD but it remains a random variable, even in the long observation-time limit. The distribution of time averages is shown to agree with a Lévy flight model. Our results also unravel intriguing anomalies in the statistics of displacements. The time-averaged MSD is shown to depend on experimental time and investigations of fractional moments show a scaling 〈|r(t)|(q)〉∼t(qν(q)) with non-linear exponents, i.e. ν(q) ≠ const. This type of behavior is termed strong anomalous diffusion and is rare among experimental observations.

Single integrodifferential wave equation for a Lévy walk

We derive the single integrodifferential wave equation for the probability density function of the position of a classical one-dimensional Lévy walk with continuous sample paths. This equation involves a classical wave operator together with memory integrals describing the spatiotemporal coupling of the Lévy walk. It is valid at all times, not only in the long time limit, and it does not involve any large-scale approximations. It generalizes the well-known telegraph or Cattaneo equation for the persistent random walk with the exponential switching time distribution. Several non-Markovian cases are considered when the particle's velocity alternates at the gamma and power-law distributed random times. In the strong anomalous case we obtain the asymptotic solution to the integrodifferential wave equation. We implement the nonlinear reaction term of Kolmogorov-Petrovsky-Piskounov type into our equation and develop the theory of wave propagation in reaction-transport systems involving Lévy diffusion.

A simple non-chaotic map generating subdiffusive, diffusive, and superdiffusive dynamics

Analytically tractable dynamical systems exhibiting a whole range of normal and anomalous deterministic diffusion are rare. Here, we introduce a simple non-chaotic model in terms of an interval exchange transformation suitably lifted onto the whole real line which preserves distances except at a countable set of points. This property, which leads to vanishing Lyapunov exponents, is designed to mimic diffusion in non-chaotic polygonal billiards that give rise to normal and anomalous diffusion in a fully deterministic setting. As these billiards are typically too complicated to be analyzed from first principles, simplified models are needed to identify the minimal ingredients generating the different transport regimes. For our model, which we call the slicer map, we calculate all its moments in position analytically under variation of a single control parameter. We show that the slicer map exhibits a transition from subdiffusion over normal diffusion to superdiffusion under parameter variation. Our results may help to understand the delicate parameter dependence of the type of diffusion generated by polygonal billiards. We argue that in different parameter regions the transport properties of our simple model match to different classes of known stochastic processes. This may shed light on difficulties to match diffusion in polygonal billiards to a single anomalous stochastic process.

Fluctuations around equilibrium laws in ergodic continuous-time random walks

We study occupation time statistics in ergodic continuous-time random walks. Under thermal detailed balance conditions, the average occupation time is given by the Boltzmann-Gibbs canonical law. But close to the nonergodic phase, the finite-time fluctuations around this mean are large and nontrivial. They exhibit dual time scaling and distribution laws: the infinite density of large fluctuations complements the Lévy-stable density of bulk fluctuations. Neither of the two should be interpreted as a stand-alone limiting law, as each has its own deficiency: the infinite density has an infinite norm (despite particle conservation), while the stable distribution has an infinite variance (although occupation times are bounded). These unphysical divergences are remedied by consistent use and interpretation of both formulas. Interestingly, while the system's canonical equilibrium laws naturally determine the mean occupation time of the ergodic motion, they also control the infinite and Lévy-stable densities of fluctuations. The duality of stable and infinite densities is in fact ubiquitous for these dynamics, as it concerns the time averages of general physical observables.

Asymptotic densities of ballistic Lévy walks

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 nonballistic random walks.

Generalized Lyapunov exponent as a unified characterization of dynamical instabilities

The Lyapunov exponent characterizes an exponential growth rate of the difference of nearby orbits. A positive Lyapunov exponent (exponential dynamical instability) is a manifestation of chaos. Here, we propose the Lyapunov pair, which is based on the generalized Lyapunov exponent, as a unified characterization of nonexponential and exponential dynamical instabilities in one-dimensional maps. Chaos is classified into three different types, i.e., superexponential, exponential, and subexponential chaos. Using one-dimensional maps, we demonstrate superexponential and subexponential chaos and quantify the dynamical instabilities by the Lyapunov pair. In subexponential chaos, we show superweak chaos, which means that the growth of the difference of nearby orbits is slower than a stretched exponential growth. The scaling of the growth is analytically studied by a recently developed theory of a continuous accumulation process, which is related to infinite ergodic theory.

Infinite densities for Lévy walks

Motion of particles in many systems exhibits a mixture between periods of random diffusive-like events and ballistic-like motion. In many cases, such systems exhibit strong anomalous diffusion, where low-order moments 〈|x(t)|(q)〉 with q below a critical value q(c) exhibit diffusive scaling while for q>q(c) a ballistic scaling emerges. The mixed dynamics constitutes a theoretical challenge since it does not fall into a unique category of motion, e.g., the known diffusion equations and central limit theorems fail to describe both aspects. In this paper we resolve this problem by resorting to the concept of infinite density. Using the widely applicable Lévy walk model, we find a general expression for the corresponding non-normalized density which is fully determined by the particles velocity distribution, the anomalous diffusion exponent α, and the diffusion coefficient K(α). We explain how infinite densities play a central role in the description of dynamics of a large class of physical processes and discuss how they can be evaluated from experimental or numerical data.

Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking

This Perspective summarises the properties of a variety of anomalous diffusion processes and provides the necessary tools to analyse and interpret recorded anomalous diffusion data.