Superdiffusive Regime Research Articles

Shear-driven diffusion with stochastic resetting

External flows, such as shear flow, add directional biases to particle motion, introducing anisotropic behavior into the system. Here, we explore the non-equilibrium dynamics that emerge from the interplay between linear shear flow and stochastic resetting. The particle diffuses with a constant diffusion coefficient while simultaneously experiencing linear shear and being stochastically returned to its initial position at a constant rate. We perturbatively derive the steady-state probability distribution that captures the effects of shear-induced anisotropy on the spatial structure of the distribution. We show that the dynamics, which initially spread diffusively, will at late times reach a steady state due to resetting. At intermediate timescales, the system approaches this steady state either by passing through a superdiffusive regime (in the shear-dominated case) or by exhibiting purely sub-diffusive behavior (in the resetting-dominated case). The steady state also gains cross correlations, a feature absent in simpler resetting systems. We also show that the skewness has a non-monotonic behavior when one passes from the shear-dominated to the resetting-dominated regime. We demonstrate that at small resetting rates, the energetic cost of maintaining the steady state becomes significantly higher due to the displacement caused by shear, a unique scaling not seen without shear. Surprisingly, if only the x-position is reset, the system can maintain a Brownian yet non-Gaussian diffusion pattern with non-trivial tails in the distribution.

Universality of ecological memory for local and global net ecosystem exchange, atmospheric CO2, and sea surface temperature

Modeling global net ecosystem exchange is essential to understanding and quantifying the complex interactions between the Earth’s terrestrial ecosystems and the atmosphere. Emphasizing the inter-relatedness between the global net ecosystem exchange, global sea surface temperature, and atmospheric CO2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ ext {CO}}_{2}$$\\end{document} levels, intuitively suggests that all three systems may exhibit collective environmental memory. Motivated by this, we explicitly identified a collective memory function and showed a similar non-Markovian stochastic behavior for these systems exhibiting superdiffusive behavior in short time intervals. We obtained the values of the memory parameter, μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu$$\\end{document}, and the characteristic frequencies, ν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ u$$\\end{document}, for global net ecosystem exchange (GNEE) (μ=0.94±0.03,ν=0.67±0.08/mo.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu = 0.94\\pm 0.03, \ u = 0.67\\pm 0.08/mo.$$\\end{document}), global sea surface temperature (GSST) (μ=0.68±0.11,ν=0.30±0.18/mo.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu = 0.68\\pm 0.11, \ u = 0.30\\pm 0.18/mo.$$\\end{document}), and atmospheric CO2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ ext {CO}}_{2}$$\\end{document} (μ=0.78±0.08,ν=0.66±0.13/wk.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu = 0.78\\pm 0.08, \ u =0.66\\pm 0.13/wk.$$\\end{document}). The values of the memory parameter are within the range, 0<μ<1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0< \\mu < 1$$\\end{document}, and thus all three systems are in the superdiffusive regime. We emphasize, further, that these results were consistent with our previous analyses at the ecosystem level (i.e. Great Barrier Reef) suggesting scale invariance for these phenomena. Thus, the observed superdiffusive behavior operating at different scales suggests universality of the collective memory function for these systems.

Stochastic walker with variable long jumps.

Motivated by recent interest in the stochastic resetting of a random walker, we propose a generalized model where the random walker takes stochastic jumps of lengths proportional to its present position with certain probability, otherwise it makes forward and backward jumps of fixed (unit) length with given rates. The model exhibits a rich stochastic dynamic behavior. We obtain exact analytic results for the first two moments of the walker's displacement and show that a phase transition from a diffusive to superdiffusive regime occurs if the stochastic jumps of lengths that are twice (or more) of its present positions are allowed. This phase transition is accompanied by a reentrant diffusive behavior.

Superdiffusive Energy Transport in Kinetically Constrained Models

Universal nonequilibrium properties of isolated quantum systems are typically probed by studying transport of conserved quantities, such as charge or spin, while transport of energy has received considerably less attention. Here, we study infinite-temperature energy transport in the kinetically-constrained PXP model describing Rydberg atom quantum simulators. Our state-of-the-art numerical simulations, including exact diagonalization and time-evolving block decimation methods, reveal the existence of two distinct transport regimes. At moderate times, the energy-energy correlation function displays periodic oscillations due to families of eigenstates forming different su(2) representations hidden within the spectrum. These families of eigenstates generalize the quantum many-body scarred states found in previous works and leave an imprint on the infinite-temperature energy transport. At later times, we observe a broad superdiffusive transport regime that we attribute to the proximity of a nearby integrable point. Intriguingly, strong deformations of the PXP model by the chemical potential do not restore diffusion, but instead lead to a stable superdiffusive exponent $z\approx3/2$. Our results suggest constrained models to be potential hosts of novel transport regimes and call for developing an analytic understanding of their energy transport.

Particle dynamics due to interaction between a breaking-induced vortex and a nearbed vortex

One single regular wave, traveling over a submerged abrupt discontinuity, can generate a pair of counter-rotating vortices. A nearbed vortex is generated by the flow separation that occurs at the bed, while a surface vortex can be generated by either a direct (co-rotating vortex) or a backward (counter-rotating vortex) breaking. Starting from recent laboratory test results, which highlighted the role of wave nonlinearity on the interaction between the counter-rotating vortices and led to the identification of three different regimes [Brocchini et al., “Interaction between breaking-induced vortices and near-bed structures: Part 1—Experimental and theoretical investigation,” J. Fluid Mech. 940, A44 (2022)], the present work illustrates the main findings obtained from the optical analysis of the flow field induced by three different wave conditions, each belonging to a specific nonlinear regime. For each test, the measured domain has been seeded with virtual particles to obtain long-lasting trajectories driven by the Eulerian flow field recovered through the particle tracking velocimetry analysis, to be studied by means of single-particle and multi-particle statistics. Both absolute and relative statistics confirm that a ballistic regime exists just after particle release (t≲TL) at each location of the domain. At times larger than the Lagrangian timescale (t≳TL), the absolute statistics suggest a sub-diffusive regime both within the vortices and between such areas (i.e., in correspondence of the breaking-induced jet), followed by a superdiffusive regime, dominated by rotation and particle release. Differently, the relative diffusivity suggests the occurrence of a superdiffusive regime at t≳TL, corresponding to an enstrophy cascade and exponential growth, followed by a Richardson regime and then by an oscillatory behavior, during which particles are periodically trapped and released by vortices.

Structural and dynamic heterogeneity in associative networks formed by artificially engineered protein polymers.

This work investigates static gel structure and cooperative multi-chain motion in associative networks using a well-defined model system composed of artificial coiled-coil proteins. The combination of small-angle and ultra-small-angle neutron scattering provides evidence for three static length scales irrespective of protein gel design which are attributed to correlations arising from the blob length, inter-junction spacing, and multi-chain density fluctuations. Self-diffusion measurements using forced Rayleigh scattering demonstrate an apparent superdiffusive regime in all gels studied, reflecting a transition between distinct "slow" and "fast" diffusive species. The interconversion between the two diffusive modes occurs on a length scale on the order of the largest correlation length observed by neutron scattering, suggesting a possible caging effect. Comparison of the self-diffusive behavior with characteristic molecular length scales and the single-sticker dissociation time inferred from tracer diffusion measurements supports the primarily single-chain mechanisms of self-diffusion as previously conceptualized. The step size of the slow mode is comparable to the root-mean-square length of the midblock strands, consistent with a single-chain walking mode rather than collective motion of multi-chain aggregates. The transition to the fast mode occurs on a timescale 10-1000 times the single-sticker dissociation time, which is consistent with the onset of single-molecule hopping. Finally, the terminal diffusivity depends exponentially on the number of stickers per chain, further suggesting that long-range diffusion occurs by molecular hopping rather than sticky Rouse motion of larger assemblies. Collectively, the results suggest that diffusion of multi-chain clusters is dominated by the single-chain pictures proposed in previous coarse-grained modeling.

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.

How to estimate the memory of the elephant random walk

We introduce an original way to estimate the memory parameter of the elephant random walk, a fascinating discrete time random walk on integers having a complete memory of its entire history. Our estimator is nothing more than a quasi-maximum likelihood estimator, based on a second order Taylor approximation of the log-likelihood function. We show the almost sure convergence of our estimate in the diffusive, critical and superdiffusive regimes. The local asymptotic normality of our statistical procedure is established in the diffusive regime, while the local asymptotic mixed normality is proven in the superdiffusive regime. Asymptotic and exact confidence intervals as well as statistical tests are also provided. All our analysis relies on asymptotic results for martingales and the quadratic variations associated.

Further results on the minimal random walk

The aim of this paper is to go further in the analysis of the asymptotic behavior of the so-called minimal random walk (MRW) using a new martingale approach. The MRW is a discrete-time random walk with infinite memory that has three regimes depending on the location of its two parameters. In the diffusive and critical regimes, we establish new results on the asymptotic behavior of the MRW such as quadratic strong laws and functional central limit theorems. In the superdiffusive regime, we prove the almost sure convergence of the MRW, properly normalized, to a nondegenerate random variable. Moreover, we show that the fluctuation of the MRW around its limiting random variable is still Gaussian. Finally, several results on the center of mass of the MRW are also provided.

Dispersion Processes in Weakly Dissipative Tidal Channels

AbstractWe report the results of an extensive experimental campaign dedicated to the analysis of turbulent dispersion owing to the circulations in tidal environments, characterized by a tidal inlet and a channel with lateral tidal flats. We focus on weakly‐convergent and weakly‐dissipative estuaries or tidal embankments, where the internal waters communicate with the open sea through an inlet mouth. Tides are reproduced as single or multiple harmonics waves. Particle Image Velocimetry is employed to measure two‐dimensional surface velocity fields. Large‐scale macro‐vortices, generated by vortex shedding during the flood phase from the inlet barrier, tend to occupy the entire tidal flats width and are completely flushed out during the ebb phase. In all experiments, an intense residual current, with shape influenced by the large‐scale flood vortices, is observed. The presence of large‐scale vortices and of a residual current strongly influences the Lagrangian auto‐correlation functions and the corresponding absolute dispersion time evolution. Looping auto‐correlations are the signature of both periodic forcing and vortices, ultimately, leading to super diffusive regimes. An asymptotic Brownian regime is always found for the investigated range of parameters allowing for an estimate of the horizontal dispersion coefficients which turn out to decrease with the friction parameter and tend to be enhanced when the semi‐diurnal constituents prevail. Finally, multiple particle statistics show multiple regimes depending on particle separations, compared to a typical injection length scale that seems to coincide with the inlet mouth dimension.

Random walks with memory profile of the κ-exponential: a first measurement of diffusive regimes

A new class of random walks, Elephant Random Walks (ERW) in the memory type, non-Markovian, was proposed by Schütz and Trimper characterized by diffusive regimes typical of anomalous diffusion. Following this line, we proposed a model of random walks guided by the principle of kinetic interaction (KIP), where, according to KIP, the temporal evolution of the distribution function of identical particles subject to binary collisions leads us to an ever-increasing functional with time, satisfying the statement of the irreversibility of the second law of Thermodynamics. We derived a distribution of the exponential that is discrete. In the limit , the distribution of the -exponential falls into the ordinary exponential distribution, where we found the typical Brownian diffusion. In the upper limit of anti-equilibrium ( ), we found a transition from the ordinary diffusive regime to the super-diffusive regime. Additionally, moving away from the anti-equilibrium region, we found the ERW’s diffusive behavior. Our results are yet another case of anomalous transport in complex systems, which are associated with particle diffusion for which the variance spreads non-linearly with time.

On the Elephant Random Walk with Stops Playing Hide and Seek with the Mittag–Leffler Distribution

The aim of this paper is to investigate the asymptotic behavior of the so-called elephant random walk with stops (ERWS). In contrast with the standard elephant random walk, the elephant is allowed to be lazy by staying on his own position. We prove that the number of ones of the ERWS, properly normalized, converges almost surely to a Mittag–Leffler distribution. It allows us to carry out a sharp analysis on the asymptotic behavior of the ERWS. In the diffusive and critical regimes, we establish the almost sure convergence of the ERWS. We also show that it is necessary to self-normalized the position of the ERWS by the random number of ones in order to prove the asymptotic normality. In the superdiffusive regime, we establish the almost sure convergence of the ERWS, properly normalized, to a nondegenerate random variable. Moreover, we also show that the fluctuation of the ERWS around its limiting random variable is still Gaussian.

Lévy walk of quasiballistic phonons in nanowires

Phonon transport in square-cross-sectionnanowires is studied using spectral Monte Carlo simulations. Our results show the evolution of the different transport regimes described by Lévy statistics as a function of the surface roughness-to-thermal wavelength ratio σ/λ. More precisely, the relationship between the Lévy index γ describing the mean free path distribution Ψ(Λ) and σ/λ is established for the classical diffusive regime, the superdiffusive regime, and the ballistic regime in the nanowire. Besides the conventional superdiffusive regime that is marked by Ψ(Λ) with a single heavy-tailed peak, we reveal an unconventional superdiffusive subregime featuring Ψ(Λ) with sawtooth oscillations when σ/λ∼0.01. Investigation of the direction of propagation of phonons shows a significant narrowing of the angular distribution around the long axis of the nanowire due to the diffuse scattering at rough boundaries when σ/λ>0.1. These results shed light on the transport mechanisms of quasiballistic phonons and will help in nanowire design for specific applications.

Mechanism of the motion of nanovehicles on hexagonal boron-nitride: A molecular dynamics study

Nanocars have been proposed to transport nanomaterials on the surface. Study of the mechanism of the motion of nanocars has attracted a lot of interests due to the potential ability of these nano-vehicles in the construction of nanostructures with bottom-up approach. Using molecular dynamics simulations, we study the motion of two nano-vehicles named “Nanocar” and “Nanotruck” on hexagonal boron-nitride monolayer. The obtained results reveals that, boron-nitride is an appropriate option to obtain higher mobility of nanocars compared with metal substrates. Considering different temperatures reveals that nanocars start to move on the BN at 200 K, while long range motions are observed at 400 K and higher temperatures. The flexibility of Nanocar chassis wastes a portion of its energy and reduces its displacement range. The anomaly parameters show that C60, Nanocar and Nanotruck experience super-diffusive regime at 100 K and higher temperatures. Using diffusion coefficient and activation energy, rotational motion of the nanocars is evaluated. To accurately investigate the mechanism of nanocars motion, we find rotation of the wheels around their axles and the speed of Nanotruck in directions parallel and perpendicular to the chassis. The mentioned parameters indicate that wheel rolling mechanism is the minority mode of the nanocars motion and the translations commonly occur through sliding. At low temperatures, two stable configurations are found for the Nanocar, and reconfiguration is energetically possible at 200 K and higher temperatures. The results presented in this work are expected to facilitate the fabrication of nanostructures using molecular machines.

Moments of the superdiffusive elephant random walk with general step distribution

We consider the elephant random walk with general step distribution. We calculate the first four moments of the limiting distribution of the position rescaled by nα in the superdiffusive regime where α is the memory parameter. This extends the results obtained by Bercu in [Ber17].

Introducing smooth amnesia to the memory of the Elephant Random Walk

This paper is devoted to the asymptotic analysis of the amnesic elephant random walk (AERW) using a martingale approach. More precisely, our analysis relies on asymptotic results for multidimensional martingales with matrix normalization. In the diffusive and critical regimes, we establish the almost sure convergence and the quadratic strong law for the position of the AERW. The law of iterated logarithm is given in the critical regime. The distributional convergences of the AERW to Gaussian processes are also provided. In the superdiffusive regime, we prove the distributional convergence as well as the mean square convergence of the AERW.

Dynamics of active particles with space-dependent swim velocity.

We study the dynamical properties of an active particle subject to a swimming speed explicitly depending on the particle position. The oscillating spatial profile of the swim velocity considered in this paper takes inspiration from experimental studies based on Janus particles whose speed can be modulated by an external source of light. We suggest and apply an appropriate model of an active Ornstein Uhlenbeck particle (AOUP) to the present case. This allows us to predict the stationary properties, by finding the exact solution of the steady-state probability distribution of particle position and velocity. From this, we obtain the spatial density profile and show that its form is consistent with the one found in the framework of other popular models. The reduced velocity distribution highlights the emergence of non-Gaussianity in our generalized AOUP model which becomes more evident as the spatial dependence of the velocity profile becomes more pronounced. Then, we focus on the time-dependent properties of the system. Velocity autocorrelation functions are studied in the steady-state combining numerical and analytical methods derived under suitable approximations. We observe a non-monotonic decay in the temporal shape of the velocity autocorrelation function which depends on the ratio between the persistence length and the spatial period of the swim velocity. In addition, we numerically and analytically study the mean square displacement and the long-time diffusion coefficient. The ballistic regime, observed in the small-time region, is deeply affected by the properties of the swim velocity landscape which induces also a crossover to a sub-ballistic but superdiffusive regime for intermediate times. Finally, the long-time diffusion coefficient decreases as the amplitude of the swim velocity oscillations increases because the diffusion is mainly determined by those regions where the particles are slow.

New Insights on the Reinforced Elephant Random Walk Using a Martingale Approach

This paper is devoted to the asymptotic analysis of the reinforced elephant random walk (RERW) using a martingale approach. In the diffusive and critical regimes, we establish the almost sure convergence, the law of iterated logarithm and the quadratic strong law for the RERW. The distributional convergences of the RERW to some Gaussian processes are also provided. In the superdiffusive regime, we prove the distributional convergence as well as the mean square convergence of the RERW. All our analysis relies on asymptotic results for multi-dimensional martingales with matrix normalization.

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.

Great Barrier Reef degradation, sea surface temperatures, and atmospheric CO2 levels collectively exhibit a stochastic process with memory

Quantifying ecological memory could be done at several levels from the rate of physiological changes in an ecosystem all the way down to responses at the genetic level. One way of unlocking the information encoded in a collective environmental memory is to examine the recorded time-series data generated by different components of an ecosystem. In this paper, we probe into the case of the Great Barrier Reef (GBR) which is threatened by elevated sea surface temperatures (SST) and ocean acidification attributed to rising atmospheric CO2 levels. Specifically, we investigate the interrelated dynamics between the degradation of the GBR, SST, and rising atmospheric CO2 levels, by considering three datasets: (a) the mean percentage hard coral cover of the GBR from the archives of the Australian Institute of Marine Science; (b) SST close to the GBR from the National Oceanic and Atmospheric Administration; and (c) the Keeling curve for atmospheric CO2 levels measured by the Mauna Loa Observatory. We show that fluctuating observables in these datasets have the same memory behavior described by a non-Markovian stochastic process. All three datasets show a good match between empirical and analytical mean square deviation. An explicit analytical form for the corresponding probability density function is obtained which obeys a modified diffusion equation with a time dependent diffusion coefficient. This study provides a new perspective on the similarities of and interaction between the GBR’s declining hard coral cover, SST, and rising atmospheric CO2 levels by putting all three systems into one unified framework indexed by a memory parameter \(\mu \) and a characteristic frequency \(\nu \). The short-time dynamics of CO2 levels and SST fall in the superdiffusive regime, while the GBR exhibits hyperballistic fluctuation in percent coral cover with the highest values for \(\mu \) and \(\nu \).