Superdiffusive Behavior Research Articles

Anomalous diffusion of E. coli under microfluidic confinement and chemical gradient.

We report a two-layer microfluidic device to study the combined effect of confinement and chemical gradient on the motility of wild-type E. coli. We track individual E. coli in 50 μm and 10 μm wide microchannels, with a channel height of 2 μm, to generate quasi-2D conditions. We find that contrary to expectations, bacterial trajectories are superdiffusive even in the absence of a chemical (glucose) gradient. The superdiffusive behaviour becomes more pronounced upon introducing a chemical gradient or strengthening the lateral confinement. Run length distributions for weak lateral confinement in the absence of chemical gradients follow an exponential distribution. Both confinement and chemoattraction induce deviations from this behaviour, with the run length distributions approaching a power-law form under these conditions. Both confinement and chemoattraction suppress large-angle tumbles as well. Our results suggest that wild-type E. coli modulates both its runs and tumbles in a similar manner under physical confinement and chemical gradient. Our findings have implications for understanding how bacteria modulate their motility behaviour in natural habitats.

Structure and dynamics of an active polymer chain inside a nanochannel grafted with polymers.

We use computer simulations to investigate the complex dynamics of a polymer, made of active Brownian particles, inside a channel grafted internally with passive polymer chains. Our simulations reveal that this probe-polymer, if passive, exhibits a compact structure when its interaction is repulsive with the grafted chains as it tends to stay within the hollow space created along the axis of the channel. On increasing the attractive interaction, the passive probe-polymer is pulled towards the grafted polymeric region and adopts an extended structure. By contrast, switching on the activity helps the probe-polymer to escape from the local traps caused by the sticky grafted chains. The interplay between the activity of the probe-polymer and its sticky interaction with the grafted chains results in shrinking, followed by swelling as the activity is increased. To elucidate the dynamics we compute the mean square displacement (MSD) of the center of mass of the probe-polymer, which increases monotonically with activity and displays superdiffusive behavior at an intermediate time and enhanced diffusion at a long time period. In addition, compared with the attractive interaction, the active probe-polymer shows faster dynamics when the interaction is repulsive to the grafted polymers. We believe that our current study will provide insights into the structural changes and dynamics of active polymers in heterogeneous media and will be useful in designing polymer-based drug delivery vehicles.

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.

Survival in two-species reaction-diffusion system with Lévy flights: renormalization group treatment and numerical simulations

We analyze the two-species reaction-diffusion system including trapping reaction as well as coagulation/annihilation reactions where particles of both species are performing Lévy flights with control parameter , known to lead to superdiffusive behavior. The density as well as the correlation function for target particles B in such systems are known to scale with nontrivial universal exponents at space dimension . Applying the renormalization group formalism we calculate these exponents in a case of superdiffusion below the critical dimension . The numerical simulations in one-dimensional case are performed as well. The quantitative estimates for the decay exponent of the density of survived particles B are in a good agreement with our analytical results. In particular, it is found that the surviving probability of the target particles in superdiffusive regime is higher than that in a system with ordinary diffusion.

Logt-Superdiffusivity for a Brownian particle in the curl of the 2D GFF

The present work is devoted to the study of the large time behaviour of a critical Brownian diffusion in two dimensions, whose drift is divergence-free, ergodic and given by the curl of the 2-dimensional Gaussian Free Field. We prove the conjecture, made in [B. T\'oth, B. Valk\'o, J. Stat. Phys., 2012], according to which the diffusion coefficient $D(t)$ diverges as $\sqrt{\log t}$ for $t\to\infty$. Starting from the fundamental work by Alder and Wainwright [B. Alder, T. Wainright, Phys. Rev. Lett. 1967], logarithmically superdiffusive behaviour has been predicted to occur for a wide variety of out-of-equilibrium systems in the critical spatial dimension $d=2$. Examples include the diffusion of a tracer particle in a fluid, self-repelling polymers and random walks, Brownian particles in divergence-free random environments, and, more recently, the 2-dimensional critical Anisotropic KPZ equation. Even if in all of these cases it is expected that $D(t)\sim\sqrt{\log t}$, to the best of the authors' knowledge, this is the first instance in which such precise asymptotics is rigorously established.

Universal and Non-Universal Features in the Random Shear Model

The stochastic transport of particles in a disordered two-dimensional layered medium, driven by correlated y-dependent random velocity fields is usually referred to as random shear model. This model exhibits a superdiffusive behavior in the x direction ascribable to the statistical properties of the disorder advection field. By introducing layered random amplitude with a power-law discrete spectrum, the analytical expressions for the space and time velocity correlation functions, together with those of the position moments, are derived by means of two distinct averaging procedures. In the case of quenched disorder, the average is performed over an ensemble of uniformly spaced initial conditions: albeit the strong sample-to-sample fluctuations, and universality appears in the time scaling of the even moments. Such universality is exhibited in the scaling of the moments averaged over the disorder configurations. The non-universal scaling form of the no-disorder symmetric or asymmetric advection fields is also derived.

Universality aspects of quantum corrections to transverse momentum broadening in QCD media

We study non-linear quantum corrections to transverse momentum broadening (TMB) of a fast parton propagating in dense QCD matter in the leading logarithmic approximation. These non-local corrections yield an anomalous super-diffusive behavior characterized by a heavy tailed distribution which is associated with Lévy random walks. Using a formal analogy with the physics of traveling waves, we show that at late times the transverse momentum distribution tends to a universal scaling regime. We derive analytic solutions in terms of an asymptotic expansion around the scaling limit for both fixed and running coupling. We note that our analytic approach yields a good agreement with the exact numerical solutions down to realistic values of medium length. Finally, we discuss the interplay between system size and energy dependence of the diffusion coefficient hat{q} and its connection with the gluon distribution function that is manifest at large transverse momentum transfer.

Emergent dynamics of light-induced active colloids probed by XPCS

Self-propelled particulate systems manifest certain collective behavior of living matter, which have been the subject of intense research over the past decades. One of the elegant methods for realizing such active motions is by means of custom synthesized Janus particles suspended in a catalytic medium that can be triggered upon illumination by ultraviolet light. In this work, the evolution of the particle dynamics from passive diffusive to active ballistic behavior upon light illumination was probed by multispeckle x-ray photon correlation spectroscopy (XPCS). This technique enables not only studying the emergence of active motions in three dimensions (3D) but also deciphering different contributions to the overall dynamics. Using a combination of homodyne and heterodyne analysis, the ensemble averaged mean velocity, velocity fluctuations and diffusion coefficient of particles were determined in the thermodynamic limit. Results revealed a gradual transition from diffusive to ballistic dynamics with systematic increase of the catalytic activity. At the intermediate region, the dynamics is dominated by Gaussian velocity fluctuations and an enhanced relaxation rate with a weaker wave vector dependence similar to superdiffusive behavior. For the highest activity, the dynamics became purely ballistic with Lorentzian-like distribution of velocity fluctuations. Presented results demonstrate that different aspects of active dynamics can be investigated in 3D over a broad range of Péclet numbers and other control parameters by means of multispeckle XPCS.

Universal critical behavior of transition to chaos: Intermittency route

Abstract The robustness of the universality class concept of the chaotic transition was investigated by analytically obtaining its critical exponent for a wide class of maps. In particular, we extended the existing one-dimensional chaotic maps, thereby generalizing the invariant density function from the Cauchy distribution by adding one parameter. This generalization enables the adjustment of the power exponents of the density function and superdiffusive behavior. We proved that these generalized one-dimensional chaotic maps are exact (a stronger condition than ergodicity) to obtain the critical exponent of the Lyapunov exponent from the phase average. Furthermore, we proved that the critical exponent of the Lyapunov exponent is $\frac{1}{2}$ regardless of the power exponent of the density function and is thus universal. This result can be considered as rigorous proof of the universality of the critical exponent of the Lyapunov exponent for a countably infinite number of maps.

Probability distribution functions of sub- and superdiffusive systems

We study the anomalous transport in systems of random walks (RW's) on comb-like lattices with fractal sidebranches, showing subdiffusion, and in a system of Brownian particles driven by a random shear along the x-direction, showing a superdiffusive behavior. In particular, we discuss whether scaling and universality are present or not in the shapes of the particle distribution along the preferential transport direction (x-axis).

Characterization of anomalous diffusion in one-dimensional quantum walks

Quantum walks are known to propagate quadratically faster than their classical counterparts and are used to model dynamics in various quantum systems. The spread of the quantum walk in position space shows anomalous diffusion behavior. By controlling the action of quantum coin operation on the corresponding coin degree of freedom of the walker, one can demonstrate control over the diffusion behavior. In this work, we report different forms of coin operations on quantum walks exhibiting anomalous diffusion behavior. Homogeneous and accelerated quantum walks display superdiffusive behavior, whereas uncorrelated static and dynamic disorders in the evolution induce strong and weak localization of the particle indicating subdiffusive and normal diffusive behavior. The role played by the interference effects in the spreading of the walker has remained elusive and our aim in this work is to present the interplay between quantum coherence and mean squared displacement of the walker. We employ two reliable measures of coherence for conclusively establishing the role of quantum interference as the driving force behind the anomalous diffusive behavior in the dynamics of quantum walks.

LFA: A Lévy Walk and Firefly-Based Search Algorithm: Application to Multi-Target Search and Multi-Robot Foraging

In the literature, several exploration algorithms have been proposed so far. Among these, Lévy walk is commonly used since it is proved to be more efficient than the simple random-walk exploration. It is beneficial when targets are sparsely distributed in the search space. However, due to its super-diffusive behavior, some tuning is needed to improve its performance, specifically when targets are clustered. Firefly algorithm is a swarm intelligence-based algorithm useful for intensive search, but its exploration rate is very limited. An efficient and reliable search could be attained by combining the two algorithms since the first one allows exploration space, and the second one encourages its exploitation. In this paper, we propose a swarm intelligence-based search algorithm called Lévy walk and Firefly-based Algorithm (LFA), which is a hybridization of the two aforementioned algorithms. The algorithm is applied to Multi-Target Search and Multi-Robot Foraging. Numerical experiments to test the performances are conducted on the robotic simulator ARGoS. A comparison with the original firefly algorithm proves the goodness of our contribution.

Transient Superdiffusion of Energetic Carriers in Transition Metal Dichalcogenides Visualized by Ultrafast Pump-Probe Microscopy

Because of the strong Coulomb interaction and quantum confinement effect, 2-dimensional transition metal dichalcogenides possess a stable excitonic population. To realize excitonic device applications, such as excitonic circuits, switches, and transistors, it is of paramount importance for understanding the optical properties of transition metal dichalcogenides. Furthermore, the strong quantum confinement in 2-dimensional space introduces exotic properties, such as enhanced phonon bottlenecking effect, many-body interaction of excitons, and ultrafast nonequilibrium exciton–exciton annihilation. Exciton diffusion is the primary energy dissipation process and a working horse in excitonic devices. In this work, we investigated time-resolved exciton propagation in monolayer semiconductors of WSe 2 , MoWSe 2 , and MoSe 2 , with a home-built femtosecond pump-probe microscope. We observed ultrafast exciton expansion behavior with an equivalent diffusivity of up to 502 cm 2 s −1 at the initial delay time, followed by a slow linear diffusive regime (20.9 cm 2 s −1 ) in the monolayer WSe 2 . The fast expansion behavior is attributed to energetic carrier-dominated superdiffusive behavior. We found that in the monolayers MoWSe 2 and MoSe 2 , the energetic carrier-induced exciton expansion is much more effective, with diffusivity up to 668 and 2295 cm 2 s −1 , respectively. However, the “cold” exciton transport is trap limited in MoWSe 2 and MoSe 2 , leading to negative diffusion behavior at later time. Our findings are helpful to better understand the ultrafast nonlinear diffusive behavior in strongly quantum-confined systems. It may be harnessed to break the limit of conventional slow diffusion of excitons for advancing more efficient and ultrafast optoelectronic devices.

Magnetic Energy Release, Plasma Dynamics, and Particle Acceleration in Relativistic Turbulent Magnetic Reconnection

Abstract In strongly magnetized astrophysical plasma systems, magnetic reconnection is believed to be the primary process during which explosive energy release and particle acceleration occur, leading to significant high-energy emission. Past years have witnessed active development of kinetic modeling of relativistic magnetic reconnection, supporting this magnetically dominated scenario. A much less explored issue in studies of relativistic reconnection is the consequence of three-dimensional dynamics, where turbulent structures are naturally generated as various types of instabilities develop. This paper presents a series of three-dimensional, fully kinetic simulations of relativistic turbulent magnetic reconnection (RTMR) in positron–electron plasmas with system domains much larger than kinetic scales. Our simulations start from a force-free current sheet with several different modes of long-wavelength magnetic field perturbations, which drive additional turbulence in the reconnection region. Because of this, the current layer breaks up and the reconnection region quickly evolves into a turbulent layer filled with coherent structures such as flux ropes and current sheets. We find that plasma dynamics in RTMR is vastly different from its 2D counterpart in many aspects. The flux ropes evolve rapidly after their generation, and can be completely disrupted by the secondary kink instability. This turbulent evolution leads to superdiffusive behavior of magnetic field lines as seen in MHD studies of turbulent reconnection. Meanwhile, nonthermal particle acceleration and the timescale for energy release can be very fast and do not depend strongly on the turbulence amplitude. The main acceleration mechanism is a Fermi-like acceleration process supported by the motional electric field, whereas the nonideal electric field acceleration plays a subdominant role. We also discuss possible observational implications of three-dimensional RTMR in high-energy astrophysics.

Self-adaptive behavior of nunchakus-like tracer induced by active Brownian particles

We design a nunchakus-like tracer and investigate its self-adaptive behavior in an active Brownian particle (ABP) bath via systematically tuning the self-propelled capability and density of ABPs. Specifically, the nunchakus-like tracer will have a stable wedge-like shape in the ABP bath when the self-propelled force is high enough. We analyze the angle between the two arms of the tracer and the velocity of the joint point of the tracer. The angle exhibits a non-monotonic phenomenon as a function of active force. However, it increases with density of ABPs increasing monotonically. A simple linear relationship between the velocity and the self-propelled force is found under the highly active force. In other words, the joint points of the tracer diffuse and the super-diffusive behavior can make the relation between the self-propelled force and the density of ABPs persist longer. In addition, we find that the tracer can flip at high density of ABPs. Our results also suggest the new self-adaptive model research of the transport properties in a non-equilibrium medium.

Single molecule/particle tracking analysis program SMTracker 2.0 reveals different dynamics of proteins within the RNA degradosome complex in Bacillus subtilis.

Single-molecule (particle) tracking is a powerful method to study dynamic processes in cells at highest possible spatial and temporal resolution. We have developed SMTracker, a graphical user interface for automatic quantifying, visualizing and managing of data. Version 2.0 determines distributions of positional displacements in x- and y-direction using multi-state diffusion models, discriminates between Brownian, sub- or superdiffusive behaviour, and locates slow or fast diffusing populations in a standardized cell. Using SMTracker, we show that the Bacillus subtilis RNA degradosome consists of a highly dynamic complex of RNase Y and binding partners. We found similar changes in molecule dynamics for RNase Y, CshA, PNPase and enolase, but not for phosphofructokinase, RNase J1 and J2, to inhibition of transcription. However, the absence of PfkA or of RNase J2 affected molecule dynamics of RNase Y-mVenus, indicating that these two proteins are indeed part of the degradosome. Molecule counting suggests that RNase Y is present as a dimer in cells, at an average copy number of about 500, of which 46% are present in a slow-diffusive state and thus likely engaged within degradosomes. Thus, RNase Y, CshA, PNPase and enolase likely play central roles, and RNase J1, J2 and PfkA more peripheral roles, in degradosome architecture.

Magnetized wake driven anomalous diffusion in complex plasma

Diffusion of a three-dimensional dust ensemble embedded in a flowing plasma in the presence of a moderate magnetic field is investigated using Langevin dynamics simulation. It is found that the asymmetric wake potential created due to streaming ions drive the dusty plasma from the subdiffusive to the superdiffusive regime. A novel wake dominant regime is identified that shows superdiffusive behavior for suitable adjustment of the magnetic field. The dependence of the cross field diffusion coefficient on the magnetic field exhibits three distinct behaviors characterized by (a) B2−3 for an ultra-low magnetic field and strongly correlated state, (b) B≃0.1 for a moderate magnetic field, and (c) classical B−2 for a relatively large magnetic field. Our analysis shows that the magnetic field is mediated via the streaming ions in regimes (a) and (b) of relatively low magnetic field where the dependence of the diffusion coefficient on the magnetic field is faster than the usual classical regime.

Asymmetric Tsallis distributions for modeling financial market dynamics

Financial markets are highly non-linear and non-equilibrium systems. Earlier works have suggested that the behavior of market returns can be well described within the framework of non-extensive Tsallis statistics or superstatistics. For small time scales (delays), a good fit to the distributions of stock returns is obtained with q-Gaussian distributions, which can be derived either from Tsallis statistics or superstatistics. These distributions are symmetric. However, as the time lag increases, the distributions become increasingly non-symmetric. In this work, we address this problem by considering the data distribution as a linear combination of two independent normalized distributions — one for negative returns and one for positive returns. Each of these two independent distributions are half q-Gaussians with different non-extensivity parameter q and temperature parameter beta. Using this model, we investigate the behavior of stock market returns over time scales from 1 to 80 days. The data covers both the .com bubble and the 2008 crash periods. These investigations show that for all the time lags, the fits to the data distributions are better using asymmetric distributions than symmetric q-Gaussian distributions. The behaviors of the q parameter are quite different for positive and negative returns. For positive returns, q approaches a constant value of 1 after a certain lag, indicating the distributions have reached equilibrium. On the other hand, for negative returns, the q values do not reach a stationary value over the time scales studied. In the present model, the markets show a transition from normal to superdiffusive behavior (a possible phase transition) during the 2008 crash period. Such behavior is not observed with a symmetric q-Gaussian distribution model with q independent of time lag.

Liver-localized CD8 T cells do not need Levy walks to be efficient killers

Abstract Pathogen-specific CD8 T cells face the problem of finding rare cells that present their cognate antigen either in the lymph node or infected tissue. To optimize the search for rare targets it has been proposed that T cells might perform a random walk with long displacements called Levy walks enabling superdiffusive behavior and shorter search times which may be evolutionary selected. We examined the behavior of activated CD8 T cells in the liver where both the movement of the cells and the underlying structural constraints can be clearly defined. We show that Plasmodium-specific liver-localized CD8 T cells perform short displacement, Brownian-like walks and yet display transiently superdiffusive displacement, the cardinal feature of efficient Levy walks. Because liver-localized CD8 T cells are mainly associated with liver sinusoids, simulations of Brownian or Levy walkers in structures derived from the liver sinusoids illustrate that structure together with preferred forward movement results in superdiffusive movement. Interestingly, Brownian walkers require less time to find a rare target when T cells search for the infection in physiologically-derived liver structures. Our results strongly suggest that observed patterns of movement of CD8 T cells are likely to result from a combination of a cell-intrinsic movement program, physical constraints imposed by the environmental structures, and other environmental cues. Future work needs to focus on quantifying the relative contributions of these factors to the overall observed movement patterns of biological agents.

Plasmon-enhanced fluorescence correlation spectroscopy for super-localized detection of nanoscale subcellular dynamics

In this report, we investigate plasmon-enhanced imaging fluorescence correlation spectroscopy (p-FCS). p-FCS takes advantage of extreme light confinement by localization at nanogap-based plasmonic nanodimer arrays (PNAs) for enhanced signal-to-noise ratio (SNR) and improved precision by registration with surface plasmon microscopy images. Theoretical results corroborate the enhancement by PNAs in the far-field. Near-field scanning optical microscopy was used to confirm near-field localization experimentally. Experimental confirmation was also conducted with fluorescent nanobeads. The concept was further applied to studying the diffusion dynamics of lysosomes in HEK293T cells stimulated by phorbol 12-myristate 13-acetate treatment. It was found that lysosomes demonstrate stronger super-diffusive behavior with relatively weaker sub-diffusion after stimulation. SNR measured of p-FCS was improved by 9.77 times over conventional FCS. This report is expected to serve as the foundation for an enhanced analytical tool to explore subcellular dynamics.