Ratcheting by Stochastic Resetting With Fat-Tailed Time Distributions.

We investigated both numerically and analytically the drift of a Brownian particle in a ratchet potential under stochastic resetting with fat-tailed distributions. As a study case we chose a Pareto time distribution with tail index ß. We observed that for 1 / 2 < ß < 1 ${1/2\char60 \beta \char60 1}$ rectification occurs even if for ß < 1 ${\beta \char60 1}$ the mean resetting time is infinite. However, for ß ≤ 1 / 2 ${\beta \le 1/2}$ rectification is completely suppressed. For low noise levels, the drift speed attains a maximum for ß immediately above 1, that is for finite but large mean resetting times. In correspondence with such an optimal drift the particle diffusion over the ratchet potential turns from normal to superdiffusive, a property also related to the fat tails of the resetting time distribution.

Superdiffusion of quantized vortices uncovering scaling laws in quantum turbulence.

Generic scaling laws, such as Kolmogorov's 5/3 law, are milestone achievements of turbulence research in classical fluids. For quantum fluids such as atomic Bose-Einstein condensates, superfluid helium, and superfluid neutron stars, turbulence can also exist in the presence of a chaotic tangle of evolving quantized vortex lines. However, due to the lack of suitable experimental tools to directly probe the vortex-tangle motion, so far little is known about possible scaling laws that characterize the velocity correlations and trajectory statistics of the vortices in quantum-fluid turbulence, i.e., quantum turbulence (QT). Acquiring such knowledge could greatly benefit the development of advanced statistical models of QT. Here we report an experiment where a tangle of vortices in superfluid 4He are decorated with solidified deuterium tracer particles. Under experimental conditions where these tracers follow the motion of the vortices, we observed an apparent superdiffusion of the vortices. Our analysis shows that this superdiffusion is not due to Lévy flights, i.e., long-distance hops that are known to be responsible for superdiffusion of random walkers. Instead, a previously unknown power-law scaling of the vortex-velocity temporal correlation is uncovered as the cause. This finding may motivate future research on hidden scaling laws in QT.

Superdiffusive transport of energy in one-dimensional metals.

Metals in one spatial dimension are described at the lowest energy scales by the Luttinger liquid theory. It is well understood that this free theory, and even interacting integrable models, can support ballistic transport of conserved quantities including energy. In contrast, realistic one-dimensional metals, even without disorder, contain integrability-breaking interactions that are expected to lead to thermalization and conventional diffusive linear response. We argue that the expansion of energy when such a nonintegrable Luttinger liquid is locally heated above its ground state shows superdiffusive behavior (i.e., spreading of energy that is intermediate between diffusion and ballistic propagation), by combining an analytical anomalous diffusion model with numerical matrix-product-state calculations on a specific perturbed spinless fermion chain. Different metals will have different scaling exponents and shapes in their energy spreading, but the superdiffusive behavior is stable and should be visible in time-resolved experiments.

Dynamics of a cross-superdiffusive SIRS model with delay effects in transmission and treatment.

We investigate the dynamics of a SIRS epidemiological model taking into account cross-superdiffusion and delays in transmission, Beddington-DeAngelis incidence rate and Holling type II treatment. The superdiffusion is induced by inter-country and inter-urban exchange. The linear stability analysis for the steady-state solutions is performed, and the basic reproductive number is calculated. The sensitivity analysis of the basic reproductive number is presented, and we show that some parameters strongly influence the dynamics of the system. A bifurcation analysis to determine the direction and stability of the model is carried out using the normal form and center manifold theorem. The results reveal a proportionality between the transmission delay and the diffusion rate. The numerical results show the formation of patterns in the model, and their epidemiological implications are discussed.

Generalisation of continuous time random walk to anomalous diffusion MRI models with an age-related evaluation of human corpus callosum.

Diffusion MRI measures of the human brain provide key insight into microstructural variations across individuals and into the impact of central nervous system diseases and disorders. One approach to extract information from diffusion signals has been to use biologically relevant analytical models to link millimetre scale diffusion MRI measures with microscale influences. The other approach has been to represent diffusion as an anomalous transport process and infer microstructural information from the different anomalous diffusion equation parameters. In this study, we investigated how parameters of various anomalous diffusion models vary with age in the human brain white matter, particularly focusing on the corpus callosum. We first unified several established anomalous diffusion models (the super-diffusion, sub-diffusion, quasi-diffusion and fractional Bloch-Torrey models) under the continuous time random walk modelling framework. This unification allows a consistent parameter fitting strategy to be applied from which meaningful model parameter comparisons can be made. We then provided a novel way to derive the diffusional kurtosis imaging (DKI) model, which is shown to be a degree two approximation of the sub-diffusion model. This link between the DKI and sub-diffusion models led to a new robust technique for generating maps of kurtosis and diffusivity using the sub-diffusion parameters ßSUB and DSUB. Superior tissue contrast is achieved in kurtosis maps based on the sub-diffusion model. 7T diffusion weighted MRI data for 65 healthy participants in the age range 19-78 years was used in this study. Results revealed that anomalous diffusion model parameters α and ß have shown consistent positive correlation with age in the corpus callosum, indicating α and ß are sensitive to tissue microstructural changes in ageing.

Parsimonious modeling of skeletal muscle perfusion: Connecting the stretched exponential and fractional Fickian diffusion.

PURPOSE: To develop an anomalous (non-Gaussian) diffusion model for characterizing skeletal muscle perfusion using multi-b-value DWI. THEORY AND METHODS: Fick's first law was extended for describing tissue perfusion as anomalous superdiffusion, which is non-Gaussian diffusion exhibiting greater particle spread than that of the Gaussian case. This was accomplished using a space-fractional derivative that gives rise to a power-law relationship between mean squared displacement and time, and produces a stretched exponential signal decay as a function of b-value. Numerical simulations were used to estimate parameter errors under in vivo conditions, and examine the effect of limited SNR and residual fat signal. Stretched exponential DWI parameters, α and D , were measured in thigh muscles of 4 healthy volunteers at rest and following in-magnet exercise. These parameters were related to a stable distribution of jump-length probabilities and used to estimate microvascular volume fractions. RESULTS: Numerical simulations showed low dispersion in parameter estimates within 1.5% and 1%, and bias errors within 3% and 10%, for α and D , respectively. Superdiffusion was observed in resting muscle, and to a greater degree following exercise. Resting microvascular volume fraction was between 0.0067 and 0.0139 and increased between 2.2-fold and 4.7-fold following exercise. CONCLUSIONS: This model captures superdiffusive molecular motions consistent with perfusion, using a parsimonious representation of the DWI signal, providing approximations of microvascular volume fraction comparable with histological estimates. This signal model demonstrates low parameter-estimation errors, and therefore holds potential for a wide range of applications in skeletal muscle and elsewhere in the body.

Hierarchical random walks in trace fossils and the origin of optimal search behavior.

Efficient searching is crucial for timely location of food and other resources. Recent studies show that diverse living animals use a theoretically optimal scale-free random search for sparse resources known as a Lévy walk, but little is known of the origins and evolution of foraging behavior and the search strategies of extinct organisms. Here, using simulations of self-avoiding trace fossil trails, we show that randomly introduced strophotaxis (U-turns)--initiated by obstructions such as self-trail avoidance or innate cueing--leads to random looping patterns with clustering across increasing scales that is consistent with the presence of Lévy walks. This predicts that optimal Lévy searches may emerge from simple behaviors observed in fossil trails. We then analyzed fossilized trails of benthic marine organisms by using a novel path analysis technique and find the first evidence, to our knowledge, of Lévy-like search strategies in extinct animals. Our results show that simple search behaviors of extinct animals in heterogeneous environments give rise to hierarchically nested Brownian walk clusters that converge to optimal Lévy patterns. Primary productivity collapse and large-scale food scarcity characterizing mass extinctions evident in the fossil record may have triggered adaptation of optimal Lévy-like searches. The findings suggest that Lévy-like behavior has been used by foragers since at least the Eocene but may have a more ancient origin, which might explain recent widespread observations of such patterns among modern taxa.

Evidence of Levy walk foraging patterns in human hunter-gatherers.

When searching for food, many organisms adopt a superdiffusive, scale-free movement pattern called a Lévy walk, which is considered optimal when foraging for heterogeneously located resources with little prior knowledge of distribution patterns [Viswanathan GM, da Luz MGE, Raposo EP, Stanley HE (2011) The Physics of Foraging: An Introduction to Random Searches and Biological Encounters]. Although memory of food locations and higher cognition may limit the benefits of random walk strategies, no studies to date have fully explored search patterns in human foraging. Here, we show that human hunter-gatherers, the Hadza of northern Tanzania, perform Lévy walks in nearly one-half of all foraging bouts. Lévy walks occur when searching for a wide variety of foods from animal prey to underground tubers, suggesting that, even in the most cognitively complex forager on Earth, such patterns are essential to understanding elementary foraging mechanisms. This movement pattern may be fundamental to how humans experience and interact with the world across a wide range of ecological contexts, and it may be adaptive to food distribution patterns on the landscape, which previous studies suggested for organisms with more limited cognition. Additionally, Lévy walks may have become common early in our genus when hunting and gathering arose as a major foraging strategy, playing an important role in the evolution of human mobility.

How animals move along? Exactly solvable model of superdiffusive spread resulting from animal's decision making.

Patterns of individual animal movement have been a focus of considerable attention recently. Of particular interest is a question how different macroscopic properties of animal dispersal result from the stochastic processes occurring on the microscale of the individual behavior. In this paper, we perform a comprehensive analytical study of a model where the animal changes the movement velocity as a result of its behavioral response to environmental stochasticity. The stochasticity is assumed to manifest itself through certain signals, and the animal modifies its velocity as a response to the signals. We consider two different cases, i.e. where the change in the velocity is or is not correlated to its current value. We show that in both cases the early, transient stage of the animal movement is super-diffusive, i.e. ballistic. The large-time asymptotic behavior appears to be diffusive in the uncorrelated case but super-ballistic in the correlated case. We also calculate analytically the dispersal kernel of the movement and show that, whilst it converge to a normal distribution in the large-time limit, it possesses a fatter tail during the transient stage, i.e. at early and intermediate time. Since the transients are known to be highly relevant in ecology, our findings may indicate that the fat tails and superdiffusive spread that are sometimes observed in the movement data may be a feature of the transitional dynamics rather than an inherent property of the animal movement.

Turing pattern dynamics and adaptive discretization for a super-diffusive Lotka-Volterra model.

In this paper we analyze the effects of introducing the fractional-in-space operator into a Lotka-Volterra competitive model describing population super-diffusion. First, we study how cross super-diffusion influences the formation of spatial patterns: a linear stability analysis is carried out, showing that cross super-diffusion triggers Turing instabilities, whereas classical (self) super-diffusion does not. In addition we perform a weakly nonlinear analysis yielding a system of amplitude equations, whose study shows the stability of Turing steady states. A second goal of this contribution is to propose a fully adaptive multiresolution finite volume method that employs shifted Grünwald gradient approximations, and which is tailored for a larger class of systems involving fractional diffusion operators. The scheme is aimed at efficient dynamic mesh adaptation and substantial savings in computational burden. A numerical simulation of the model was performed near the instability boundaries, confirming the behavior predicted by our analysis.

Fractal Lévy Heat Transport in Nanoparticle Embedded Semiconductor Alloys.

Materials with embedded nanoparticles are of considerable interest for thermoelectric applications. Here, we experimentally characterize the effect of nanoparticles on the recently discovered Lévy phonon transport in semiconductor alloys. The fractal space dimension α ≈ 1.55 of quasiballistic (superdiffusive) heat conduction in (ErAs)x:InGaAlAs is virtually independent of the Er content 0.001 < x < 0.1 but instead controlled by alloy scattering of the host matrix. The increased nanoparticle concentration does reduce the diffusive recovery length by an order of magnitude. The bulk conductivity drops by 3-fold, in close agreement with a Callaway model. Our results may provide helpful hints toward engineering superdiffusive heat transport similar to what has been achieved with light in Lévy glasses.

Turing Pattern Formation in a Semiarid Vegetation Model with Fractional-in-Space Diffusion.

A fractional power of the Laplacian is introduced to a reaction-diffusion system to describe water's anomalous diffusion in a semiarid vegetation model. Our linear stability analysis shows that the wavenumber of Turing pattern increases with the superdiffusive exponent. A weakly nonlinear analysis yields a system of amplitude equations, and the analysis of these amplitude equations shows that the spatial patterns are asymptotic stable due to the supercritical Turing bifurcation. Numerical simulations exhibit a bistable regime composed of hexagons and stripes, which confirm our analytical results. Moreover, the characteristic length of the emergent spatial pattern is consistent with the scale of vegetation patterns observed in field studies.

Anisotropic Diffusion of Elongated Particles in Active Coherent Flows.

The study of particle diffusion, a classical conundrum in scientific inquiry, holds manifold implications for various real-world applications. Particularly within the domain of active flows, where the motion of self-propelled particles instigates fluid movement, extensive research has been dedicated to unraveling the dynamics of passive spherical particles. This scrutiny has unearthed intriguing phenomena, such as superdiffusion at brief temporal scales and conventional diffusion at longer intervals. In contrast to the spherical counterparts, anisotropic particles, which manifest directional variations, are prevalent in nature. Although anisotropic behavior in passive fluids has been subject to exploration, enigmatic aspects persist in comprehending the interplay of anisotropic particles within active flows. This research delves into the intricacies of anisotropic passive particle diffusion, exposing a notable escalation in translational and rotational diffusion coefficients, as well as the superdiffusion index, contingent upon bacterial concentration. Through a detailed examination of particle coordinates, the directional preference of particle diffusion is not solely dependent on the particle length, but rather determined by the ratio of the particle length to the associated length scale of the background flow field. These revelations accentuate the paramount importance of unraveling the nuances of anisotropic particle diffusion within the context of active flows. Such insights not only contribute to the fundamental understanding of particle dynamics, but also have potential implications for a spectrum of applications.

Complex Monte Carlo Light-Driven Dynamics of Monomers in Functionalized Bond Fluctuation Model Polymer Chains.

We study Monte Carlo dynamics of the monomers and center of mass of a model polymer chain functionalized with azobenzene molecules in the presence of an inhomogeneous linearly polarized laser light. The simulations use a generalized Bond Fluctuation Model. The mean squared displacements of the monomers and the center of mass are analyzed in a period of Monte Carlo time typical for a build-up of Surface Relief Grating. Approximate scaling laws for mean squared displacements are found and interpreted in terms of sub- and superdiffusive dynamics for the monomers and center of mass. A counterintuitive effect is observed, where the monomers perform subdiffusive motion but the resulting motion of the center of mass is superdiffusive. This result disparages theoretical approaches based on an assumption that the dynamics of single monomers in a chain can be characterized in terms of independent identically distributed random variables.

Activity-Induced Enhancement of Superdiffusive Transport in Bacterial Turbulence.

Superdiffusion processes significantly promote the transport of tiny passive particles within biological fluids. Activity, one of the essential measures for living matter, however, is less examined in terms of how and to what extent it can improve the diffusivity of the moving particles. Here, bacterial suspensions are confined within the microfluidic channel at the state of bacterial turbulence, and are tuned to different activity levels by oxygen consumption in control. Systematic measurements are conducted to determine the superdiffusion exponent, which characterizes the diffusivity strength of tracer particles, depending on the continuously injecting energy converted to motile activity from swimming individuals. Higher activity is quantified to drastically enhance the superdiffusion process of passive tracers in the short-time regime. Moreover, the number density of the swimming bacteria is controlled to contribute to the field activity, and then to strengthen the super-diffusivity of tracers, distinguished by regimes with and without collective motion of interacting bacteria. Finally, the non-slip surfaces of the microfluidic channel lower the superdiffusion of immersed tracers due to the resistance, with the small diffusivity differing from the counterpart in the bulk. The findings here suggest ways of controlled diffusion and transport of substances within the living system with different levels of nutrition and resources and boundary walls, leading to efficient mixing, drug delivery and intracellular communications.

Fully Quantum Modeling of Exciton Diffusion in Mesoscale Light Harvesting Systems.

It has long been a challenge to accurately and efficiently simulate exciton-phonon dynamics in mesoscale photosynthetic systems with a fully quantum mechanical treatment due to extensive computational resources required. In this work, we tackle this seemingly intractable problem by combining the Dirac-Frenkel time-dependent variational method with Davydov trial states and implementing the algorithm in graphic processing units. The phonons are treated on the same footing as the exciton. Tested with toy models, which are nanoarrays of the B850 pigments from the light harvesting 2 complexes of purple bacteria, the methodology is adopted to describe exciton diffusion in huge systems containing more than 1600 molecules. The superradiance enhancement factor extracted from the simulations indicates an exciton delocalization over two to three pigments, in agreement with measurements of fluorescence quantum yield and lifetime in B850 systems. With fractal analysis of the exciton dynamics, it is found that exciton transfer in B850 nanoarrays exhibits a superdiffusion component for about 500 fs. Treating the B850 ring as an aggregate and modeling the inter-ring exciton transfer as incoherent hopping, we also apply the method of classical master equations to estimate exciton diffusion properties in one-dimensional (1D) and two-dimensional (2D) B850 nanoarrays using derived analytical expressions of time-dependent excitation probabilities. For both coherent and incoherent propagation, faster energy transfer is uncovered in 2D nanoarrays than 1D chains, owing to availability of more numerous propagating channels in the 2D arrangement.

Photoinduced Mass Transport in Azo-Polymers in 2D: Monte Carlo Study of Polarization Effects.

We studied the impact of light polarization on photoinduced dynamics of model azo-polymer chains in two dimensions, using bond-fluctuation Monte Carlo simulations. For two limiting models-sensitive to and independent of light polarization-their dynamics driven by photoisomerization of azo-dyes as well as by thermal effects was studied, including characterization of mass transport and chain reorientations. The corresponding schemes of light-matter interaction promote qualitatively different dynamics of photoinduced motion of azo-polymer chains. In particular, they can inhibit or trigger off a directed mass transport along a gradient of light illumination. The generic dynamics of single chains is superdiffusive and is promoted by breaking a symmetry present in the polarization independent model.

Impact of anomalous transport kinetics on the progress of wound healing.

This work focuses on the transport kinetics of chemical and cellular species during wound healing. Anomalous transport kinetics, coupling sub- and superdiffusion with chemotaxis, and fractional viscoelasticity of soft tissues are analyzed from a modeling point of view. The paper presents a generalization of well stablished mechano-chemical models of wound contraction (Murphy et al., 2012; Valero et al., 2014) to include the previously mentioned anomalous effects by means of partial differential equations of fractional order. Results show the effect that anomalous dynamics have on the contraction rate and extension and on the distribution of biological species, and indicators of fibroproliferative disorders are identified.

Mechanistic analysis of the search behaviour of Caenorhabditis elegans.

A central question in movement research is how animals use information and movement to promote encounter success. Current random search theory identifies reorientation patterns as key to the compromise between optimizing encounters for both nearby and faraway targets, but how the balance between intrinsic motor programmes and previous environmental experience determines the occurrence of these reorientation behaviours remains unknown. We used high-resolution tracking and imaging data to describe the complete motor behaviour of Caenorhabditis elegans when placed in a novel environment (one in which food is absent). Movement in C. elegans is structured around different reorientation behaviours, and we measured how these contributed to changing search strategies as worms became familiar with their new environment. This behavioural transition shows that different reorientation behaviours are governed by two processes: (i) an environmentally informed 'extrinsic' strategy that is influenced by recent experience and that controls for area-restricted search behaviour, and (ii) a time-independent, 'intrinsic' strategy that reduces spatial oversampling and improves random encounter success. Our results show how movement strategies arise from a balance between intrinsic and extrinsic mechanisms, that search behaviour in C. elegans is initially determined by expectations developed from previous environmental experiences, and which reorientation behaviours are modified as information is acquired from new environments.

Emergence of an optimal search strategy from a simple random walk.

In reports addressing animal foraging strategies, it has been stated that Lévy-like algorithms represent an optimal search strategy in an unknown environment, because of their super-diffusion properties and power-law-distributed step lengths. Here, starting with a simple random walk algorithm, which offers the agent a randomly determined direction at each time step with a fixed move length, we investigated how flexible exploration is achieved if an agent alters its randomly determined next step forward and the rule that controls its random movement based on its own directional moving experiences. We showed that our algorithm led to an effective food-searching performance compared with a simple random walk algorithm and exhibited super-diffusion properties, despite the uniform step lengths. Moreover, our algorithm exhibited a power-law distribution independent of uniform step lengths.