Memory-induced anomalous dynamics: Emergence of diffusion, subdiffusion, and superdiffusion from a single random walk model

Memory-induced absolute negative mobility

Non-Markovian systems form a broad area of physics that remains greatly unexplored despite years of intensive investigations. The spotlight is on memory as a source of effects that are absent in their Markovian counterparts. In this work, we dive into this problem and analyze a driven Brownian particle moving in a spatially periodic potential and exposed to correlated thermal noise. We show that the absolute negative mobility effect, in which the net movement of the particle is in the direction opposite to the average force acting on it, may be induced by the memory of the setup. To explain the origin of this phenomenon, we resort to the recently developed effective mass approach to dynamics of non-Markovian systems.

Stochastic dynamics of a non-Markovian random walk in the presence of resetting

The discrete stochastic dynamics of a random walker in the presence of resetting and memory is analyzed. Resetting and memory effects may compete in certain parameter regimes, and lead to significant changes in the long-time dynamics of the walker. Analytic exact results are obtained for a model memory where the walker remembers all the past events equally. In most cases, resetting effects dominate at long times and dictate the asymptotic dynamics. We discuss the full phase diagram of the asymptotic dynamics and the resulting changes due to the resetting and the memory effects.

Population heterogeneity in the fractional master equation, ensemble self-reinforcement, and strong memory effects

We formulate a fractional master equation in continuous time with random transition probabilities across the population of random walkers such that the effective underlying random walk exhibits ensemble self-reinforcement. The population heterogeneity generates a random walk with conditional transition probabilities that increase with the number of steps taken previously (self-reinforcement). Through this, we establish the connection between random walks with a heterogeneous ensemble and those with strong memory where the transition probability depends on the entire history of steps. We find the ensemble-averaged solution of the fractional master equation through subordination involving the fractional Poisson process counting the number of steps at a given time and the underlying discrete random walk with self-reinforcement. We also find the exact solution for the variance which exhibits superdiffusion even as the fractional exponent tends to 1.

Random walk with multiple memory channels

A class of one-dimensional, discrete-time random walk models with memory, termed "random walk with n memory channels" (RWnMC), is proposed. In these models the information of n (n∈Z) previous steps from the walker's entire history is needed to decide a future step. Exact calculation of the mean and variance of position of the RW2MC (n=2) has been done, which shows that it can lead to asymptotic diffusive and superdiffusive behavior in different parameter regimes. A connection between RWnMC and a Pólya-type urn model evolving by drawing n balls at a time has also been reported. This connection for the RW2MC is discussed in detail and suggests the applicability of RW2MC in many population dynamics models with multiple competing species.

Sublinear diffusion in the generalized triangle map

Diffusion of the orbits in a nonchaotic area-preserving map called a generalized triangle map (GTM) is numerically and analytically investigated. We provide accurate empirical evidence that the mean-squared displacement of the momentum for generic perturbation parameter settings increases sublinearly in time, and that the distribution of the momentum obeys a time-fractional diffusion equation. We show that the diffusion properties in the GTM do not follow any of the known stochastic processes generating sublinear diffusion since two seemingly incompatible features, non-Markovian yet stationary, coexist in the dynamics.

Anomalous Stochastic Transport of Particles with Self-Reinforcement and Mittag–Leffler Distributed Rest Times

We introduce a persistent random walk model for the stochastic transport of particles involving self-reinforcement and a rest state with Mittag–Leffler distributed residence times. The model involves a system of hyperbolic partial differential equations with a non-local switching term described by the Riemann–Liouville derivative. From Monte Carlo simulations, we found that this model generates superdiffusion at intermediate times but reverts to subdiffusion in the long time asymptotic limit. To confirm this result, we derived the equation for the second moment and find that it is subdiffusive in the long time limit. Analyses of two simpler models are also included, which demonstrate the dominance of the Mittag–Leffler rest state leading to subdiffusion. The observation that transient superdiffusion occurs in an eventually subdiffusive system is a useful feature for applications in stochastic biological transport.

Multidimensional elephant random walk with coupled memory

The elephant random walk (ERW) is a microscopic, one-dimensional, discrete-time, non-Markovian random walk, which can lead to anomalous diffusion due to memory effects. In this study, I propose a multidimensional generalization in which the probability of taking a step in a certain direction depends on the previous steps in other directions. The original model is generalized in a straightforward manner by introducing coefficients that couple the probability of moving in one direction with the previous steps in all directions. I motivate the model by first introducing a two-elephant system and then elucidating it with a specific coupling. With the explicit calculation of the first moments, I show the existence of two newsworthy relative movement behaviors: one in which one elephant follows the other and another in which they go in opposite directions. With the aid of a Fokker-Planck equation, the second moment is evaluated and two superdiffusion regimes appear, not found in other ERWs. Then, I reinterpret the equations as a bidimensional elephant random walk model, and further generalize it to N dimensions. I argue that the introduction of coupling coefficients is a way of extending any one-dimensional ERW to many dimensions.

A Novel Bulk-Optics Scheme for Quantum Walk with High Phase Stability

A novel bulk optics scheme for quantum walks is presented. It consists of a one-dimensional lattice built on two concatenated displaced Sagnac interferometers that make it possible to reproduce all the possible trajectories of an optical quantum walk. Because of the closed loop configuration, the interferometric structure is intrinsically stable in phase. Moreover, the lattice structure is highly configurable, as any phase component perceived by the walker is accessible, and finally, all output modes can be measured at any step of the quantum walk evolution. We report here on the experimental implementation of ordered and disordered quantum walks.

Narrow log-periodic modulations in non-Markovian random walks

What are the necessary ingredients for log-periodicity to appear in the dynamics of a random walk model? Can they be subtle enough to be overlooked? Previous studies suggest that long-range damaged memory and negative feedback together are necessary conditions for the emergence of log-periodic oscillations. The role of negative feedback would then be crucial, forcing the system to change direction. In this paper we show that small-amplitude log-periodic oscillations can emerge when the system is driven by positive feedback. Due to their very small amplitude, these oscillations can easily be mistaken for numerical finite-size effects. The models we use consist of discrete-time random walks with strong memory correlations where the decision process is taken from memory profiles based either on a binomial distribution or on a delta distribution. Anomalous superdiffusive behavior and log-periodic modulations are shown to arise in the large time limit for convenient choices of the models parameters.

Nonstationary Markovian replication process causing diverse diffusions

We introduce a single generative mechanism that can be used to describe diverse nonstationary diffusions. A nonstationary Markovian replication process for steps is considered for which we derive analytically the time evolution of the probability distribution of the walker's displacement and the generalized telegrapher equation with time-varying coefficients, and we find that diffusivity can be determined by temporal changes of replication of an immediate step. By controlling the replications, we realize diverse diffusions such as alternating diffusion, superdiffusion, subdiffusion, and marginal diffusion, which originate from oscillating, increasing, decreasing, and slowly increasing or decreasing replications with time, respectively.

Limited capacity of working memory in unihemispheric random walks implies conceivable slow dispersal

Phenomenologically inspired by dolphins' unihemispheric sleep, we introduce a minimal model for random walks with physiological memory. The physiological memory consists of long-term memory which includes unconscious implicit memory and conscious explicit memory, and working memory which serves as a multi-component system for integrating, manipulating and managing short-term storage. The model assumes that the sleeping state allows retrievals of episodic objects merely from the episodic buffer where these memory objects are invoked corresponding to the ambient objects and are thus object-oriented, together with intermittent but increasing use of implicit memory in which decisions are unconsciously picked up from historical time series. The process of memory decay and forgetting is constructed in the episodic buffer. The walker's risk attitude, as a product of physiological heuristics according to the performance of objected-oriented decisions, is imposed on implicit memory. The analytical results of unihemispheric random walks with the mixture of object-oriented and time-oriented memory, as well as the long-time behavior which tends to the use of implicit memory, are provided, indicating the common sense that a conservative risk attitude is inclinable to slow movement.

Memory-induced diffusive-superdiffusive transition: Ensemble and time-averaged observables

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. The diffusion process is nonstationary, and its probability develops the phenomenon of aging. 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 nonergodic. In addition, and similarly to Lévy walks [Godec and Metzler, Phys. Rev. Lett. 110, 020603 (2013)PRLTAO0031-900710.1103/PhysRevLett.110.020603], 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 nonstationary power-law decay of the transition probabilities to their stationary values. Time-averaged response to a bias is also calculated. In contrast with Lévy walks [Froemberg and Barkai, Phys. Rev. E 87, 030104(R) (2013)PLEEE81539-375510.1103/PhysRevE.87.030104], the response always vanishes asymptotically.

Elephant random walks and their connection to Pólya-type urns

In this paper, we explain the connection between the elephant random walk (ERW) and an urn model à la Pólya and derive functional limit theorems for the former. The ERW model was introduced in [Phys. Rev. E 70, 045101 (2004)10.1103/PhysRevE.70.045101] to study memory effects in a highly non-Markovian setting. More specifically, the ERW is a one-dimensional discrete-time random walk with a complete memory of its past. The influence of the memory is measured in terms of a memory parameter p between zero and one. In the past years, a considerable effort has been undertaken to understand the large-scale behavior of the ERW, depending on the choice of p. Here, we use known results on urns to explicitly solve the ERW in all memory regimes. The method works as well for ERWs in higher dimensions and is widely applicable to related models.

Inhomogeneous diffusion and ergodicity breaking induced by global memory effects

We introduce a class of discrete random-walk model driven by global memory effects. At any time, the right-left transitions depend on the whole previous history of the walker, being defined by an urnlike memory mechanism. The characteristic function is calculated in an exact way, which allows us to demonstrate that the ensemble of realizations is ballistic. Asymptotically, each realization is equivalent to that of a biased Markovian diffusion process with transition rates that strongly differs from one trajectory to another. Using this "inhomogeneous diffusion" feature, the ergodic properties of the dynamics are analytically studied through the time-averaged moments. Even in the long-time regime, they remain random objects. While their average over realizations recovers the corresponding ensemble averages, departure between time and ensemble averages is explicitly shown through their probability densities. For the density of the second time-averaged moment, an ergodic limit and the limit of infinite lag times do not commutate. All these effects are induced by the memory effects. A generalized Einstein fluctuation-dissipation relation is also obtained for the time-averaged moments.

Weak ergodicity breaking induced by global memory effects

We study the phenomenon of weak ergodicity breaking for a class of globally correlated random walk dynamics defined over a finite set of states. The persistence in a given state or the transition to another one depends on the whole previous temporal history of the system. A set of waiting time distributions, associated to each state, sets the random times between consecutive steps. Their mean value is finite for all states. The probability density of time-averaged observables is obtained for different memory mechanisms. This statistical object explicitly shows departures between time and ensemble averages. While the residence time in each state may have a divergent mean value, we demonstrate that this condition is in general not necessary for breaking ergodicity. Hence, we conclude that global memory effects are an alternative mechanism able to induce ergodicity breaking without involving power-law statistics. Analytical and numerical calculations support these results.

Island nucleation and growth with anomalous diffusion

While most studies of submonolayer island nucleation and growth have been based on the assumption of ordinary monomer diffusion corresponding to diffusion exponent μ=1, in some cases either subdiffusive (μ<1) or superdiffusive (μ>1) behavior may occur. Here we present general expressions for the exponents describing the flux dependence of the island and monomer densities as a function of the critical island size i, substrate dimension d, island fractal dimension d_{f}, and diffusion exponent μ, where 0≤μ≤2. Our results are compared with kinetic Monte Carlo simulations for the case of irreversible island growth (i=1) with 0≤μ≤2 and d=2 as well as simulation results for d=1, 3, and 4, and excellent agreement is found.

Self-attracting walk on heterogeneous networks

Understanding human mobility in cyberspace becomes increasingly important in this information era. While human mobility, memory-dependent and subdiffusive, is well understood in Euclidean space, it remains elusive in random heterogeneous networks like the World Wide Web. Here we study the diffusion characteristics of self-attracting walks, in which a walker is more likely to move to the locations visited previously than to unvisited ones, on scale-free networks. Under strong attraction, the number of distinct visited nodes grows linearly in time with larger coefficients in more heterogeneous networks. More interestingly, crossovers to sublinear growths occur in strongly heterogeneous networks. To understand these phenomena, we investigate the characteristic volumes and topology of the cluster of visited nodes and find that the reinforced attraction to hubs results in expediting exploration first but delaying later, as characterized by the scaling exponents that we derive. Our findings and analysis method can be useful for understanding various diffusion processes mediated by human.

Slow Lévy flights

Among Markovian processes, the hallmark of Lévy flights is superdiffusion, or faster-than-Brownian dynamics. Here we show that Lévy laws, as well as Gaussian distributions, can also be the limit distributions of processes with long-range memory that exhibit very slow diffusion, logarithmic in time. These processes are path dependent and anomalous motion emerges from frequent relocations to already visited sites. We show how the central limit theorem is modified in this context, keeping the usual distinction between analytic and nonanalytic characteristic functions. A fluctuation-dissipation relation is also derived. Our results may have important applications in the study of animal and human displacements.

Random recursive trees and the elephant random walk

One class of random walks with infinite memory, so-called elephant random walks, are simple models describing anomalous diffusion. We present a surprising connection between these models and bond percolation on random recursive trees. We use a coupling between the two models to translate results from elephant random walks to the percolation process. We calculate, besides other quantities, exact expressions for the first and the second moment of the root cluster size and of the number of nodes in child clusters of the first generation. We further introduce another model, the skew elephant random walk, and calculate the first and second moment of this process.

An analytical correlated random walk model and its application to understand subdiffusion in crowded environment

Subdiffusion in crowded environment such as movement of macromolecule in a living cell has often been observed experimentally. The primary reason for subdiffusion is volume exclusion by the crowder molecules. However, other effects such as hydrodynamic interaction may also play an important role. Although there are a large number of computer simulation studies on understanding molecular crowding, there is a lack of theoretical models that can be connected to both experiment and simulation. In the current work, we have formulated a one-dimensional correlated random walk model by connecting this to the motion in a crowded environment. We have found the exact solution of the probability distribution function of the model by solving it analytically. The parameters of our model can be obtained either from simulation or experiment. It has been shown that this analytical model captures some of the general features of diffusion in crowded environment as given in the previous literature and its prediction for transient subdiffusion closely matches the observations of a previous study of computer simulation of Escherichia coli cytoplasm. It is likely that this model will open up more development of theoretical models in this area.

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.

Memory-induced anomalous dynamics in a minimal random walk model

A discrete-time dynamics of a non-Markovian random walker is analyzed using a minimal model where memory of the past drives the present dynamics. In recent work [N. Kumar et al., Phys. Rev. E 82, 021101 (2010)] we proposed a model that exhibits asymptotic superdiffusion, normal diffusion, and subdiffusion with the sweep of a single parameter. Here we propose an even simpler model, with minimal options for the walker: either move forward or stay at rest. We show that this model can also give rise to diffusive, subdiffusive, and superdiffusive dynamics at long times as a single parameter is varied. We show that in order to have subdiffusive dynamics, the memory of the rest states must be perfectly correlated with the present dynamics. We show explicitly that if this condition is not satisfied in a unidirectional walk, the dynamics is only either diffusive or superdiffusive (but not subdiffusive) at long times.

Anomalous diffusion induced by enhancement of memory

We introduced simple microscopic non-Markovian walk models which describe the underlying mechanism of anomalous diffusions. In the models, we considered the competitions between randomness and memory effects of previous history by introducing the probability parameters. The memory effects were considered in two aspects: one is the perfect memory of whole history and the other is the latest memory enhanced with time. In the perfect memory model superdiffusion was induced with the relation of the Hurst exponent H to the controlling parameter p as H = p for p>1/2, while in the latest memory enhancement models, anomalous diffusions involving both superdiffusion and subdiffusion were induced with the relations H = (1+α)/2 and H = (1-α)/2 for 0 ≤ α ≤ 1, where α is the parameter controlling the degree of the latest memory enhancement. Also we found that, although the latest memory was only considered, the memory improved with time results in the long-range correlations between steps and the correlations increase as time goes on. Thus we suggest the memory enhancement as a key origin describing anomalous diffusions.

Ultraslow diffusion in an exactly solvable non-Markovian random walk

We study a one-dimensional discrete-time non-Markovian random walk with strong memory correlations subjected to pauses. Unlike the Scher-Montroll continuous-time random walk, which can be made Markovian by defining an operational time equal to the random-walk step number, the model we study keeps a record of the entire history of the walk. This new model is closely related to the one proposed recently by Kumar, Harbola, and Lindenberg [Phys. Rev. E 82, 021101 (2010)], with the difference that in our model the stochastic dynamics does not stop even in the extreme limit of subdiffusion. Surprisingly, this small difference leads to large consequences. The main results we report here are exact results showing ultraslow diffusion and a stationary diffusion regime (i.e., localization). Specifically, the equations of motion are solved analytically for the first two moments, allowing the determination of the Hurst exponent. Several anomalous diffusion regimes are apparent, ranging from superdiffusion to subdiffusion, as well as ultraslow and stationary regimes. We present the complete phase diffusion diagram, along with a study of the persistence and the statistics in the regions of interest.

Microscopic theory of anomalous diffusion based on particle interactions

We present a master equation formulation based on a Markovian random walk model that exhibits subdiffusion, classical diffusion, and superdiffusion as a function of a single parameter. The nonclassical diffusive behavior is generated by allowing for interactions between a population of walkers. At the macroscopic level, this gives rise to a nonlinear Fokker-Planck equation. The diffusive behavior is reflected not only in the mean squared displacement [<r(2)(t)>~t(γ) with 0<γ≤1.5] but also in the existence of self-similar scaling solutions of the Fokker-Planck equation. We give a physical interpretation of sub- and superdiffusion in terms of the attractive and repulsive interactions between the diffusing particles and we discuss analytically the limiting values of the exponent γ. Simulations based on the master equation are shown to be in agreement with the analytical solutions of the nonlinear Fokker-Planck equation in all three diffusion regimes.

Non-Gaussian propagator for elephant random walks

For almost a decade the consensus has held that the random walk propagator for the elephant random walk (ERW) model is a Gaussian. Here we present strong numerical evidence that the propagator is, in general, non-Gaussian and, in fact, non-Lévy. Motivated by this surprising finding, we seek a second, non-Gaussian solution to the associated Fokker-Planck equation. We prove mathematically, by calculating the skewness, that the ERW Fokker-Planck equation has a non-Gaussian propagator for the superdiffusive regime. Finally, we discuss some unusual aspects of the propagator in the context of higher order terms needed in the Fokker-Planck equation.

Kinetic lattice Monte Carlo simulation of viscoelastic subdiffusion

We propose a kinetic Monte Carlo method for the simulation of subdiffusive random walks on a cartesian lattice. The random walkers are subject to viscoelastic forces which we compute from their individual trajectories via the fractional Langevin equation. At every step the walkers move by one lattice unit, which makes them differ essentially from continuous time random walks, where the subdiffusive behavior is induced by random waiting. To enable computationally inexpensive simulations with n-step memories, we use an approximation of the memory and the memory kernel functions with a complexity O(log n). Eventual discretization and approximation artifacts are compensated with numerical adjustments of the memory kernel functions. We verify with a number of analyses that this new method provides binary fractional random walks that are fully consistent with the theory of fractional brownian motion.

Random walk in chemical space of Cantor dust as a paradigm of superdiffusion

We point out that the chemical space of a totally disconnected Cantor dust K(n) [Symbol: see text E(n) is a compact metric space C(n) with the spectral dimension d(s) = d(ℓ) = n > D, where D and d(ℓ) = n are the fractal and chemical dimensions of K(n), respectively. Hence, we can define a random walk in the chemical space as a Markovian Gaussian process. The mapping of a random walk in C(n) into K(n) [Symbol: see text] E(n) defines the quenched Lévy flight on the Cantor dust with a single step duration independent of the step length. The equations, describing the superdiffusion and diffusion-reaction front propagation ruled by the local quenched Lévy flight on K(n) [Symbol: see text] E(n), are derived. The use of these equations to model superdiffusive phenomena, observed in some physical systems in which propagators decay faster than algebraically, is discussed.

Age-related changes in speed and mechanism of adult skeletal muscle stem cell migration

Skeletal muscle undergoes a progressive age-related loss in mass and function. Preservation of muscle mass depends in part on satellite cells, the resident stem cells of skeletal muscle. Reduced satellite cell function may contribute to the age-associated decrease in muscle mass. Here, we focused on characterizing the effect of age on satellite cell migration. We report that aged satellite cells migrate at less than half the speed of young cells. In addition, aged cells show abnormal membrane extension and retraction characteristics required for amoeboid-based cell migration. Aged satellite cells displayed low levels of integrin expression. By deploying a mathematical model approach to investigate mechanism of migration, we have found that young satellite cells move in a random "memoryless" manner, whereas old cells demonstrate superdiffusive tendencies. Most importantly, we show that nitric oxide, a key regulator of cell migration, reversed the loss in migration speed and reinstated the unbiased mechanism of movement in aged satellite cells. Finally, we found that although hepatocyte growth factor increased the rate of aged satellite cell movement, it did not restore the memoryless migration characteristics displayed in young cells. Our study shows that satellite cell migration, a key component of skeletal muscle regeneration, is compromised during aging. However, we propose clinically approved drugs could be used to overcome these detrimental changes.

Target search on a dynamic DNA molecule

We study a protein-DNA target search model with explicit DNA dynamics applicable to in vitro experiments. We show that the DNA dynamics plays a crucial role for the effectiveness of protein "jumps" between sites distant along the DNA contour but close in three-dimensional space. A strongly binding protein that searches by one-dimensional sliding and jumping alone explores the search space less redundantly when the DNA dynamics is fast on the time scale of protein jumps than in the opposite "frozen DNA" limit. We characterize the crossover between these limits using simulations and scaling theory. We also rationalize the slow exploration in the frozen limit as a subtle interplay between long jumps and long trapping times of the protein in "islands" within random DNA configurations in solution.

Transport in periodic potentials induced by fractional Gaussian noise

Directed transport of overdamped Brownian particles driven by fractional Gaussian noises is investigated in asymmetrically periodic potentials. By using Langevin dynamics simulations, we find that rectified currents occur in the absence of any external driving forces. Unlike white Gaussian noises, fractional Gaussian noises can break thermodynamical equilibrium and induce directed transport. Remarkably, the average velocity for persistent fractional noise is opposite to that for antipersistent fractional noise. The velocity increases monotonically with Hurst exponent for the persistent case, whereas there exists an optimal value of Hurst exponent at which the velocity takes its maximal value for the antipersistent case.