Superdiffusive Behavior Research Articles

Gaussian Fluctuation for Superdiffusive Elephant Random Walks

Elephant random walk is a kind of one-dimensional discrete-time random walk with infinite memory: For each step, with probability $\alpha$ the walker adopts one of his/her previous steps uniformly chosen at random, and otherwise he/she performs like a simple random walk (possibly with bias). It admits phase transition from diffusive to superdiffusive behavior at the critical value $\alpha_c=1/2$. For $\alpha \in (\alpha_c, 1)$, there is a scaling factor $a_n$ of order $n^{\alpha}$ such that the position $S_n$ of the walker at time $n$ scaled by $a_n$ converges to a nondegenerate random variable $W$, whose distribution is not Gaussian. Our main result shows that the fluctuation of $S_n$ around $W \cdot a_n$ is still Gaussian. We also give a description of phase transition induced by bias decaying polynomially in time.

Particle transport in hybrid PIC shock simulations: A comparison of diagnostics

ABSTRACT Recent in situ and remote observations suggest that the transport regime associated with shock-accelerated particles may be anomalous i.e. the mean square displacement (MSD) of such particles scales non-linearly with time. We use self-consistent hybrid particle-in-cell plasma simulations to simulate a quasi-parallel shock with parameters compatible with heliospheric shocks, and gain insights about the particle transport in such a system. For suprathermal particles interacting with the shock we compute the MSD separately in the upstream and downstream regions. Tracking suprathermal particles for sufficiently long times up and/or downstream of the shock poses problems in particle plasma simulations, such as statistically poor particle ensembles and trajectory fragments of variable length in time. Therefore, we introduce the use of time-averaged mean square displacement (TAMSD), which is based on single-particle trajectories, as an additional technique to address the transport regime for the upstream and the downstream regions. MSD and TAMSD are in agreement for the upstream energetic particle population, and both give a strong indication of superdiffusive transport, consistent with interplanetary shock observations. MSD and TAMSD are also in reasonable agreement downstream, where indications of anomalous transport are also found. TAMSD shows evidence of heterogeneity in the diffusion properties of the downstream particle population, ranging from subdiffusive behaviour of particles trapped in the strong magnetic field fluctuations generated at the shock to superdiffusive behaviour of particles transmitted and moving away from the shock.

Experimental Investigation of Superdiffusion via Coherent Disordered Quantum Walks.

Many disordered systems show a superdiffusive dynamics, intermediate between the diffusive one, typical of a classical stochastic process, and the so-called ballistic behavior, which is generally expected for the spreading in a quantum process. We have experimentally investigated the superdiffusive behavior of a quantum walk, whose dynamics can be related to energy transport phenomena, with a resolution which is high enough to clearly distinguish between different disorder regimes. By our experimental setup, the region between ballistic and diffusive spreading can be effectively scanned by suitably setting few degrees of freedom and without applying any decoherence to the quantum walk evolution.

Limit theorems for a minimal random walk model

We study the minimal random walk introduced by Kumar et al (2014 Phys. Rev. E 90 022136). It is a random process on with unbounded memory which exhibits subdiffusive, diffusive and superdiffusive regimes. We prove the law of large numbers for the whole parameter set. Then we prove the central limit theorem and the law of the iterated logarithm for the minimal random walk under diffusive and marginally superdiffusive behaviors. More interestingly, we establish a result for the minimal random walk when it possesses the three regimes; we show the convergence of its rescaled version to a non-normal random variable.

Dynamical Collective Memory in Fluidized Granular Materials

Recent experiments with rotational diffusion of a probe in a vibrated granular media revealed a rich scenario, ranging from a dilute gas to a dense liquid with cage effects and an unexpected superdiffusive behavior at large times. Here we set up a simulation that reproduces quantitatively the experimental observations and allows us to investigate the properties of the host granular medium, a task not feasible in the experiment. We discover a persistent collective rotational mode which emerges at a high density and a low granular temperature: a macroscopic fraction of the medium slowly rotates, randomly switching direction after very long times. Such a rotational mode of the host medium is the origin of the probe's superdiffusion. Collective motion is accompanied by a kind of dynamical heterogeneity at intermediate times (in the cage stage) followed by a strong reduction of fluctuations at late times, when superdiffusion sets in.

Effects of active fluctuations on energetics of a colloidal particle: Superdiffusion, dissipation and entropy production

We consider a colloidal particle immersed in an active bath and derive a Smoluchowski equation that governs the dynamics of the colloidal particle. We address this as active Smoluchowski equation. Our analysis based on this active Smoluchowski equation shows a short time superdiffusive behavior that strongly depends on the activity. Our model also predicts a non-monotonic dependence of the mean energy dissipation against time, a signature of activity-induced dynamics. By introducing a frequency-dependent effective temperature, we show that the mean rate of entropy production is time-dependent, unlike in a thermal system. The main reason for these anomalies is the absence of any fluctuation–dissipation theorem for the active noise. We also comment on how microscopic details of activity can reverse the trends for the mean energy dissipation and mean rate of entropy production.

Anomalous Diffusion of Motile Colloids Dispersed in Liquid Crystals.

We study the superdiffusion of driven colloidal particles dispersed in a nematic liquid crystal. While motion is ballistic in the driving direction, our experiments show that transversal fluctuations become superdiffusive depending on the topological defect pattern around the inclusions. The phenomenon can be reproduced with different driving methods and propulsion speeds, while it is strongly dependent on particle size and temperature. We propose a mechanism based on the geometry of the liquid crystal backflow around the inclusions to justify the persistence of thermal fluctuations and to explain the observed temperature and particle size dependence of the superdiffusive behavior based on material and geometrical parameters.

Numerical calculation of overlap distribution of (2 + 1)-dimensional directed polymer in random media

We investigate -dimensional discretized directed polymers in Gaussian random media. By numerically calculating the probability distribution function of overlap between two independent and identical systems on a common random potential, we show that there is no replica symmetry breaking. We also show that while the mean-squared displacement of one polymer end exhibits superdiffusive behavior, the mean-squared relative distance of two polymer ends exhibits subdiffusive behavior.

The microscopic role of deformation in the dynamics of soft colloids

Soft colloids enable the exploration of states with densities exceeding that of random close packing, but it remains unclear whether softness controls the dynamics under these dense conditions. Experimental studies have reported conflicting results, and numerical studies have so far focused primarily on simple models that allow particles to overlap, but neglect particle deformations. This makes the concept of softness in simulations and experiments difficult to compare. Here, we propose a model system consisting of polymer rings with internal elasticity. At high packing fractions, the system displays compressed exponential decay of the intermediate scattering functions and super-diffusive behaviour of the mean-squared displacements. These features are explained in terms of the complex interplay between particle deformations and dynamic heterogeneities, which gives rise to persistent motion of ballistic particles. We also observe a striking variation of the relaxation times with increasing particle softness, clearly demonstrating the crucial role of deformation in the dynamics of realistic soft colloids. Simulations of a system comprising polymer rings with internal elasticity reveal a key role for deformation in controlling the microscopic dynamics of soft colloids.

Time-resolved x-ray phase-contrast tomography of sedimenting micro-spheres

We have implemented a time-dependent (dynamic) x-ray tomography of sedimenting micro-spheres suspended in water. To achieve phase contrast at high magnification we use the divergent and highly coherent beam emitted from an x-ray waveguide. Holograms are recorded with 5 ms acquisition time while the sample is rotated at 1 Hz, over a run of 40 s. We show that under these conditions, more than 20 000 individual particle trajectories can be tracked. The analysis of the trajectories shows apparent super-diffusive behavior due to collective flow patterns, as also further evidenced by plotting the temporal averaged spatial distribution of particle densities and velocities.

Kinetic modeling of ELM-induced tungsten transport in a tokamak plasma

Impurity accumulation in the core plasma leads to fuel dilution and higher radiative losses that can lead to loss of the H-mode, to thermal collapse of the plasma, and eventually even to a disruption in tokamaks. In present experiments, it has been shown that Edge Localized Modes (ELMs) at sufficiently high frequency are required to prevent W accumulation in the core, by expelling impurities from the edge plasma region, thus preventing their penetration into the plasma core. We present a full-orbit particle extension of the MHD code JOREK suitable for simulating impurity transport during ELMs. This model has been applied to the simulation of an ELM crash in ASDEX Upgrade, where we have quantified the displacement of W particles across flux surfaces. The transport mechanism is shown to be the particle E × B-drifts due to the electric field created by the MHD instability underlying the ELM. In- and outward transport is observed as a series of interchange motions, leading to a superdiffusive behavior. This causes not only the particles near the plasma pedestal to move outwards but also the particles outside of the pedestal to move inwards. This has important consequences for operation with W in ITER, where it is expected to be screened in the pedestal, and ELMs are shown here to increase the core W density. A comparison with existing diffusive modeling is made, showing a qualitative agreement and the limitations of this simplified modeling approach.

Surface Waves Enhance Particle Dispersion

We study the horizontal dispersion of passive tracer particles on the free surface of gravity waves in deep water. For random linear waves with the JONSWAP spectrum, the Lagrangian particle trajectories are computed using an exact nonlinear model known as the John–Sclavounos equation. We show that the single-particle dispersion exhibits an unusual super-diffusive behavior. In particular, for large times t, the variance of the tracer ⟨ | X ( t ) | 2 ⟩ increases as a quadratic function of time, i.e., ⟨ | X ( t ) | 2 ⟩ ∼ t 2 . This dispersion is markedly faster than Taylor’s single-particle dispersion theory which predicts that the variance of passive tracers grows linearly with time for large t. Our results imply that the wave motion significantly enhances the dispersion of fluid particles. We show that this super-diffusive behavior is a result of the long-term correlation of the Lagrangian velocities of fluid parcels on the free surface.

Enhanced diffusion, swelling, and slow reconfiguration of a single chain in non-Gaussian active bath.

A prime example of a non-equilibrium or active environment is a biological cell. In order to understand in vivo functioning of biomolecules such as proteins and chromatins, a description beyond equilibrium is absolutely necessary. In this context, biomolecules have been modeled as Rouse chains in a Gaussian active bath. However, these non-equilibrium fluctuations in biological cells are non-Gaussian. This motivates us to take a Rouse chain subjected to a series of pulses of force with a finite duration, mimicking the run and tumble motion of a class of microorganisms. Thus by construction, this active force is non-Gaussian. Our analytical calculations show that the mean square displacement (MSD) of the center of mass grows faster and even shows superdiffusive behavior at higher activity. The MSD of a tagged monomer in an active bath also shows superdiffusion at an intermediate time unlike a monomer of a Rouse chain. In the case of a short chain length, reconfiguration is slower and the reconfiguration time of a chain with N monomers scales as Nσ, with σ ≈ 1.6 - 2. In addition, the chain swells. We compare this activity-induced swelling with that of a Rouse chain in a Gaussian active bath. In principle, our predictions can be verified by future single molecule experiments.

Origin of superdiffusive behavior in a class of nonequilibrium systems.

Experiments and simulations have established that dynamics in a class of living and abiotic systems that are far from equilibrium exhibit superdiffusive behavior at long times, which in some cases (for example, an evolving tumor) is preceded by slow glass-like dynamics. By using the evolution of a collection of tumor cells, driven by mechanical forces and subject to cell birth and apoptosis, as a case study we show theoretically that on short timescales the mean-square displacement is subdiffusive due to jamming, whereas at long times it is superdiffusive. The results obtained by using a stochastic quantization method, which is needed because of the absence of the fluctuation-dissipation theorem, show that the superdiffusive behavior is universal and impervious to the nature of cell-cell interactions. Surprisingly, the theory also quantitatively accounts for the nontrivial dynamics observed in simulations of a model soap foam characterized by creation and destruction of spherical bubbles, which suggests that the two nonequilibrium systems belong to the same universality class. The theoretical prediction for the superdiffusion exponent is in excellent agreement with simulations for collective motion of tumor cells and dynamics associated with soap bubbles.

Transient superdiffusive motion on a disordered ratchet potential

The relationship between anomalous superdiffusive behavior and particle trapping probability is analyzed on a rocking ratchet potential with spatially correlated weak disorder. The trapping probability density is shown, analytically and numerically, to have an exponential form as a function of space. The trapping processes with a low or no thermal noise are only transient, but they can last much longer than the characteristic time scale of the system and therefore might be detected experimentally. Using the result for the trapping probability we obtain an analytical expression for the number of wells where a given number of particles are trapped. We have also obtained an analytical approximation for the second-moment of the particle distribution function C2 as a function of time, when trapped particles coexist with constant velocity untrapped particles. We also use the expression for C2 to characterize the anomalous superdiffusive motion in the absence of thermal noise for the transient time.

Probing Fluctuations and Avalanches in the Cytoskeleton with Active Micropost Arrays

The cellular actomyosin network plays an important role in a wide range of cell behavior, including motility, morphology and mechanotransduction. Although the building blocks of actomyosin networks are well understood at the molecular level, the picture connecting these microscale units to cell-scale behavior remains incomplete. Here we present results using active micropost array detectors (AMPADS) to characterize the dynamical fluctuations and local rheology of cellular actomyosin networks in detail. AMPADS are poly(dimethylsiloxane) (PDMS) micropillar arrays with embedded magnetic nanowires that enable mechanical actuation of an adherent cell. We classified the cell-associated microposts based on their traction force, and identified two distinct populations, one containing posts coupled to stress fibers and the other containg posts coupled to the actomyosin cortex. Both groups show weak power law rheology, but we find that the fluctuations of stress fiber-associated posts are more active and highly anisotropic in comparison to the cortically-associated posts. Notably, the fluctuations of both populations are highly heterogenous and resemble a Lévy process. Specifically, the mean-squared displacements of both types of posts display broadly distributed amplitudes and super-diffusive behavior. We find that the amplitude distribution is fat-tailed, and is dominated by a corresponding distribution of intermittent, large step-like displacements, resembling avalanches and earthquakes in physical systems. Our regular array of detectors also allows us to determine the spatial extent and average symmetry of the largest rearrangements in the cortex, which are both spatially and temporally complex, resembling those in plastic solids. These results suggest that the dynamics of the cellular actomyosin network resembles what is seen in soft matter systems such as sandpiles and foams, where constituent components self-organize into marginally stable, plastic networks, with significant implications for future models of the cortex.

Self-diffusion in two-dimensional quasimagnetized rotating dusty plasmas.

The self-diffusion phenomenon in a two-dimensional dusty plasma at extremely strong (effective) magnetic fields is studied experimentally and by means of molecular dynamics simulations. In the experiment the high magnetic field is introduced by rotating the particle cloud and observing the particle trajectories in a corotating frame, which allows reaching effective magnetic fields up to 3000 T. The experimental results confirm the predictions of the simulations: (i) superdiffusive behavior is found at intermediate timescales and (ii) the dependence of the self-diffusion coefficient on the magnetic field is well reproduced.

Active microrheology in two-dimensional magnetic networks.

We study active microrheology in two-dimensional (2D) magnetic networks. To this end, we use Langevin dynamics computer simulations where single non-magnetic or magnetic tracer particles are pulled through the network structures via a constant force f. Structural changes in the network around the pulled tracer particle are characterized in terms of pair correlation functions. These functions indicate that the non-magnetic tracer particles tend to strongly affect the network structure leading to the formation of channels at sufficiently high forces, while the magnetic tracer particles modify the network structure only slightly. At zero pulling force, f = 0, both non-magnetic and magnetic tracer particles are localized, i.e. they do not show diffusive behavior in the long-time limit. Nevertheless, the friction coefficient, as obtained from the steady-state velocity of the tracer particles, seems to indicate a linear-response regime at small values of f. Beyond the latter linear response regime, the diffusion dynamics of the tracer particles are anisotropic with superdiffusive behavior in force direction. This transport anomaly is investigated via van Hove correlation functions and residence time distributions.

Non-Markovian intracellular transport with sub-diffusion and run-length dependent detachment rate.

Intracellular transport of organelles is fundamental to cell function and health. The mounting evidence suggests that this transport is in fact anomalous. However, the reasons for the anomaly is still under debate. We examined experimental trajectories of organelles inside a living cell and propose a mathematical model that describes the previously reported transition from sub-diffusive to super-diffusive motion. In order to explain super-diffusive behaviour at long times, we introduce non-Markovian detachment kinetics of the cargo: the rate of detachment is inversely proportional to the time since the last attachment. Recently, we observed the non-Markovian detachment rate experimentally in eukaryotic cells. Here we further discuss different scenarios of how this effective non-Markovian detachment rate could arise. The non-Markovian model is successful in simultaneously describing the time averaged variance (the time averaged mean squared displacement corrected for directed motion), the mean first passage time of trajectories and the multiple peaks observed in the distributions of cargo velocities. We argue that non-Markovian kinetics could be biologically beneficial compared to the Markovian kinetics commonly used for modelling, by increasing the average distance the cargoes travel when a microtubule is blocked by other filaments. In turn, sub-diffusion allows cargoes to reach neighbouring filaments with higher probability, which promotes active motion along the microtubules.

Hygrothermoelastic response of a hollow cylinder based on a coupled time-fractional heat and moisture transfer model

In this paper, the time-fractional diffusion-wave equations are introduced to describe the coupled heat conduction and moisture diffusion in a hollow pipe. The outer surface of the hollow pipe is assumed to be insulated and damp-proof, while the inner surface is subjected to thermal and moisture shocks. Using the method of separation of variables and the Laplace transform, analytic expressions for transient hygrothermoelastic response of temperature change, moisture distribution, displacement and stress components in the hollow cylinder are obtained in terms of Mittag–Leffler functions, respectively. Numerical results are calculated, and the effects of fractional order on the dynamic response of temperature, moisture and hygrothermal stresses are illustrated graphically. Sub-diffusive and super-diffusive transport coupling behaviors as well as wave-like behaviors are shown. The results in the case of the fractional order $$\alpha =1$$ reduce to those for the classical Fourier heat conduction and the Fick’s moisture diffusion coupling model.