Strong Anomalous Diffusion Research Articles

Emergent anomalous transport and non-Gaussianity in a simple mobile–immobile model: the role of advection

We analyse the transport of diffusive particles that switch between mobile and immobile states with finite rates. We focus on the effect of advection on the density functions and mean squared displacements (MSDs). At relevant intermediate time scales we find strong anomalous diffusion with cubic scaling in time of the MSD for high Péclet numbers. The cubic scaling exists for short and long mean residence times in the immobile state . For long the plateau in the MSD at intermediate times, previously found in the absence of advection, also exists for high Péclet numbers. Initially immobile tracers are subject to the newly observed regime of advection induced subdiffusion for short immobilisations and high Péclet numbers. In the long-time limit the effective advection velocity is reduced compared to advection in the mobile phase. In contrast, the MSD is enhanced by advection. We explore physical mechanisms behind the emerging non-Gaussian density functions and the features of the MSD.

Diffusion and escape from polygonal channels: extreme values and geometric effects

Polygonal billiards are an example of pseudo-chaotic dynamics, a combination of integrable evolution and sudden jumps due to conical singular points that arise from the corners of the polygons. Such pseudo-chaotic behaviour, often characterised by an algebraic separation of nearby trajectories, is believed to be linked to the wild dependence that particle transport has on the fine details of the billiard table. Here, we address this relation through a detailed numerical study of the statistics of displacement in a family of polygonal channel billiards with parallel walls. We show that transport is characterised by strong anomalous diffusion, with a mean square displacement that scales in time faster than linear, and with a probability density of the displacement exhibiting exponential tails and ballistic fronts. In channels of finite length, the distribution of first-passage times is characterised by fat tails, with a mean first-passage time that diverges when the aperture angle is rational. These findings have non-trivial consequences for a variety of experiments.

Displacement autocorrelation functions for strong anomalous diffusion: A scaling form, universal behavior, and corrections to scaling

Strong anomalous diffusion is characterized by asymptotic power-law growth of the moments of displacement, with exponents that do not depend linearly on the order of the moment. The exponents concerning small-order moments are dominated by random motion, while higher-order exponents grow by faster trajectories, such as ballistic excursions or "light fronts". Often such a situation is characterized by two linear dependencies of the exponents on their order. Here, we introduce a simple exactly solvable model, the Fly-and-Die (FnD) model, that sheds light on this behavior and on the consequences of light fronts on displacement autocorrelation functions in transport processes. We present analytical expressions for the moments and derive a scaling form that expresses the long-time asymptotics of the autocorrelation function $\langle x(t_1)\,x(t_2)\rangle$ in terms of the dimensionless time difference $(t_2-t_1)/t_1$. The scaling form provides a faithful collapse of numerical data for vastly different systems. This is demonstrated here for the Lorentz gas with infinite horizon, polygonal billiards with finite and infinite horizon, the L\'evy-Lorentz gas, the Slicer Map, and L\'evy walks. Our analysis also captures the system-specific corrections to scaling.

Phoretic dynamics of colloids in a phase separating critical liquid mixture

This article presents the emergent dynamics of St\"ober and Janus colloids upon the spinodal phase separation of the suspending medium investigated by dynamic X-ray scattering. Both colloids manifest an enhanced dynamics dominated by strong velocity fluctuations and anomalous diffusion akin to superdiffusion.

Strong anomalous diffusion in two-state process with Lévy walk and Brownian motion

Strong anomalous diffusion phenomena are often observed in complex physical and biological systems, which are characterized by the nonlinear spectrum of exponents $q\nu(q)$ by measuring the absolute $q$-th moment $\langle |x|^q\rangle$. This paper investigates the strong anomalous diffusion behavior of a two-state process with L\'{e}vy walk and Brownian motion, which usually serves as an intermittent search process. The sojourn times in L\'{e}vy walk and Brownian phases are taken as power law distributions with exponents $\alpha_+$ and $\alpha_-$, respectively. Detailed scaling analyses are performed for the coexistence of three kinds of scalings in this system. Different from the pure L\'{e}vy walk, the phenomenon of strong anomalous diffusion can be observed for this two-state process even when the distribution exponent of L\'{e}vy walk phase satisfies $\alpha_+<1$, provided that $\alpha_-<\alpha_+$. When $\alpha_+<2$, the probability density function (PDF) in the central part becomes a combination of stretched L\'{e}vy distribution and Gaussian distribution due to the long sojourn time in Brownian phase, while the PDF in the tail part (in the ballistic scaling) is still dominated by the infinite density of L\'{e}vy walk.

Reaction Spreading in Systems With Anomalous Diffusion

We briefly review some aspects of the anomalous diffusion, and its relevance in reactive systems. In particular we consider strong anomalous diffusion characterized by the moment behaviour ⟨x (t )q ⟩ ∼ t qν (q ) , where ν (q ) is a non constant function, and we discuss its consequences. Even in the apparently simple case ν (2) = 1 / 2, strong anomalous diffusion may correspond to non trivial features, such as non Gaussian probability distribution and peculiar scaling of large order moments. When a reactive term is added to a normal diffusion process, one has a propagating front with a constant velocity. The presence of anomalous diffusion by itself does not guarantee a changing in the front propagation scenario; a key factor to select linear in time or faster front propagation has been identified in the shape of the probability distribution tail in absence of reaction. In addition, we discuss the reaction spreading on graphs, underlying the major role of the connectivity properties of these structures, characterized by the connectivity dimension .

Complementary Densities of Lévy Walks: Typical and Rare Fluctuations

Strong anomalous diffusion is a recurring phenomenon in many fields, ranging from the spreading of cold atoms in optical lattices to transport processes in living cells. For such processes the scaling of the moments follows ⟨|x (t )|q ⟩ ∼ t qν (q ) and is characterized by a bi-linear spectrum of the scaling exponents, qν (q ). Here we analyze Levy walks, with power law distributed times of flight ψ (τ ) ∼ τ −(1+α ) , demonstrating sharp bi-linear scaling. Previously we showed that for α > 1 the asymptotic behavior is characterized by two complementary densities corresponding to the bi-scaling of the moments of x (t ). The first density is the expected generalized central limit theorem which is responsible for the low-order moments 0 . The second one, a non-normalizable density (also called infinite density) is formed by rare fluctuations and determines the time evolution of higher-order moments. Here we use the Faa di Bruno formalism to derive the moments of sub-ballistic super-diffusive Levy walks and then apply the Mellin transform technique to derive exact expressions for their infinite densities. We find a uniform approximation for the density of particles using Levy distribution for typical fluctuations and the infinite density for the rare ones. For ballistic Levy walks 0 < 1 we obtain mono-scaling behavior which is quantified.

Strange kinetics of bulk-mediated diffusion on lipid bilayers.

Diffusion at solid-liquid interfaces is crucial in many technological and biophysical processes. Although its behavior seems to be deceivingly simple, recent studies showing passive superdiffusive transport suggest that diffusion on surfaces may hide rich complexities. In particular, bulk-mediated diffusion occurs when molecules are transiently released from the surface to perform three-dimensional excursions into the liquid bulk. This phenomenon bears the dichotomy where a molecule always return to the surface but the mean jump length is infinite. Such behavior is associated with a breakdown of the central limit theorem and weak ergodicity breaking. Here, we use single-particle tracking to study the statistics of bulk-mediated diffusion on a supported lipid bilayer. We find that the time-averaged mean square displacement (MSD) of individual trajectories, the archetypal measure in diffusion processes, does not converge to the ensemble MSD but it remains a random variable, even in the long observation-time limit. The distribution of time averages is shown to agree with a Lévy flight model. Our results also unravel intriguing anomalies in the statistics of displacements. The time-averaged MSD is shown to depend on experimental time and investigations of fractional moments show a scaling 〈|r(t)|(q)〉∼t(qν(q)) with non-linear exponents, i.e. ν(q) ≠ const. This type of behavior is termed strong anomalous diffusion and is rare among experimental observations.

Scaling forms of particle densities for Lévy walks and strong anomalous diffusion.

We study the scaling behavior of particle densities for Lévy walks whose transition length r is coupled with the transition time t as |r|∝t^{α} with an exponent α>0. The transition-time distribution behaves as ψ(t)∝t^{-1-β} with β>0. For 1<β<2α and α≥1, particle displacements are characterized by a finite transition time and confinement to |r|<t^{α} while the marginal distribution of transition lengths is heavy tailed. These characteristics give rise to the existence of two scaling forms for the particle density, one that is valid at particle displacements |r|≪t^{α} and one at |r|≲t^{α}. As a consequence, the Lévy walk displays strong anomalous diffusion in the sense that the average absolute moments 〈|r|^{q}〉 scale as t^{qν(q)} with ν(q) piecewise linear above and below a critical value q_{c}. We derive explicit expressions for the scaling forms of the particle densities and determine the scaling of the average absolute moments. We find that 〈|r|^{q}〉∝t^{qα/β} for q<q_{c}=β/α and 〈|r|^{q}〉∝t^{1+αq-β} for q>q_{c}. These results give insight into the possible origins of strong anomalous diffusion and anomalous behaviors in disordered systems in general.

Infinite densities for Lévy walks.

Motion of particles in many systems exhibits a mixture between periods of random diffusive-like events and ballistic-like motion. In many cases, such systems exhibit strong anomalous diffusion, where low-order moments 〈|x(t)|(q)〉 with q below a critical value q(c) exhibit diffusive scaling while for q>q(c) a ballistic scaling emerges. The mixed dynamics constitutes a theoretical challenge since it does not fall into a unique category of motion, e.g., the known diffusion equations and central limit theorems fail to describe both aspects. In this paper we resolve this problem by resorting to the concept of infinite density. Using the widely applicable Lévy walk model, we find a general expression for the corresponding non-normalized density which is fully determined by the particles velocity distribution, the anomalous diffusion exponent α, and the diffusion coefficient K(α). We explain how infinite densities play a central role in the description of dynamics of a large class of physical processes and discuss how they can be evaluated from experimental or numerical data.

Non-Normalizable Densities in Strong Anomalous Diffusion: Beyond the Central Limit Theorem

Strong anomalous diffusion, where ⟨|x(t)|(q)⟩ ∼ tqν(q) with a nonlinear spectrum ν(q) ≠ const, is wide spread and has been found in various nonlinear dynamical systems and experiments on active transport in living cells. Using a stochastic approach we show how this phenomenon is related to infinite covariant densities; i.e., the asymptotic states of these systems are described by non-normalizable distribution functions. Our work shows that the concept of infinite covariant densities plays an important role in the statistical description of open systems exhibiting multifractal anomalous diffusion, as it is complementary to the central limit theorem.

Experimental evidence of strong anomalous diffusion in living cells

We evaluated the transport of polymeric particles internalized into living cancer cells. The mean-square displacement demonstrates superdiffusion with a scaling exponent of 1.25. Scaling exponents of a range of displacement moments are bilinear with moment order, exhibiting slopes of 0.6 and 0.8. Thus, we present experimental evidence of strong anomalous diffusion. Bilinearity indicates that particle motion is composed of subdiffusive regimes separated by active yet nonballistic flights. We discuss the results in terms of particle interactions with their microenvironment.

Hypothesis of strong chaos and anomalous diffusion in coupled symplectic maps

We investigate the high-dimensional Hamiltonian chaotic dynamics in N coupled area-preserving maps. We show the existence of an enhanced trapping regime caused by trajectories performing a random walk inside the area corresponding to regular islands of the uncoupled maps. As a consequence, we observe long intermediate regimes of power law decay of the recurrence time statistics (with exponent γ = 0.5) and of ballistic motion. The asymptotic decay of correlations and anomalous diffusion depend on the stickiness of the N-dimensional invariant tori. Detailed numerical simulations show weaker stickiness for increasing N suggesting that such paradigmatic class of Hamiltonian systems asymptotically fulfill the demands of the usual hypotheses of strong chaos.

Algebraic correlation functions and anomalous diffusion in the Hamiltonian mean fieldmodel

We numerically compute correlation functions of momenta and diffusionof angles with homogeneous initial conditions in the quasi-stationarystates of the Hamiltonian mean field model. This is an example, in anN-body Hamiltonian system, of anomalous transport properties characterized bynon-exponential relaxations and long-range temporal correlations. Kinetic theory predicts astriking transition between weak anomalous diffusion and strong anomalous diffusion. Thenumerical results are in excellent agreement with the quantitative predictions of theanomalous transport exponents. It is noteworthy that, at statistical equilibrium also,the system exhibits long-range temporal correlations: the correlation function isinversely proportional to time with a logarithmic correction instead of the usuallyexpected exponential decay, leading to weak anomalous transport properties.

CLUSTER spacecraft observation of a thin current sheet at the Earth’s magnetopause

On March 30, 2002 within 17 min the four CLUSTER spacecraft crossed the Earth’s high-latitude magnetopause three times. We found that all three magnetopause current sheets are nearly co-planar on the scales of the CLUSTER spacecraft separation. The thickness of the current sheets changes from 1 to 40 magnetosheath-proton thermal gyro-radii. The thinnest (first crossed) magnetopause current sheet is locally open and might be electron driven. The frozen-in condition of the thermal ions is violated, the Hall term (Lorentz force term) is comparable to the electric field. Substantial electrostatic wave intensities are observed at lower-hybrid frequencies on the magnetospheric side of the thinnest magnetopause current sheet. We obtained that the quasi-linear diffusion rate due to lower-hybrid drift waves would not explain the formation of the adjacent boundary layer and a fast magnetopause thickening. Instead, the diffusion is enhanced due to strong, non-linear wave-particle interactions. We estimate the anomalous collision- and diffusion rates by calculating the correlation between the fluctuations of current density and magnetic field. The resulting strong anomalous diffusion can account for the formation of the adjacent magnetopause boundary layers and eventually the fast magnetopause thickening.

Multiscaling in earthquakes diffusion

[1] Earthquakes can be viewed as a relaxation process of the earth crust loaded with stresses mainly at the lithospheric plate boundaries. When an earthquake occurs the stress in the crust shall be redistributed generating new earthquakes where and when the stress assume again a value greater than the slipping threshold. This observation leads to study the earthquakes diffusion mechanism. Here we investigate the fluctuations of the average free walk between an earthquake and another, evaluating the scaling of the distance between earthquakes with the waiting time τ. We find scaling exponents varying with the order of the statistical moments and always less than 1. This implies strong anomalous diffusion that we interpret as due to different mechanisms of diffusion depending by the scale.

CTRW pathways to the fractional diffusion equation

The foundations of the fractional diffusion equation are investigated based on coupled and decoupled continuous time random walks (CTRW). For this aim we find an exact solution of the decoupled CTRW, in terms of an infinite sum of stable probability densities. This exact solution is then used to understand the meaning and domain of validity of the fractional diffusion equation. An interesting behavior is discussed for coupled memories (i.e., L\'evy walks). The moments of the random walk exhibit strong anomalous diffusion, indicating (in a naive way) the breakdown of simple scaling behavior and hence of the fractional approximation. Still the Green function $P(x,t)$ is described well by the fractional diffusion equation, in the long time limit.

Advection of passive particles in the Kolmogorov flow.

A statistical analysis of the advection of passive particles in a flow governed by driven two-dimensional Navier-Stokes equations (Kolmogorov flow) is presented. Different regimes are studied, all corresponding to a chaotic behavior of the flow. The diffusion is found to be strongly asymmetric with a very weak transport perpendicular to the forcing direction. The trajectories of the particles are characterized by the presence of traps and flights. The trapping time distributions show algebraic decrease, and strong anomalous diffusion is observed in transient phases. Different regimes lead to different types of diffusion, i.e., no universal behavior of diffusion is observed, and both time and space properties are needed to define anomalous transport. (c) 2001 American Institute of Physics.

Simple stochastic models showing strong anomalous diffusion

We show that {\it strong} anomalous diffusion, i.e. $\mean{|x(t)|^q} \sim t^{q \nu(q)}$ where $q \nu(q)$ is a nonlinear function of $q$, is a generic phenomenon within a class of generalized continuous-time random walks. For such class of systems it is possible to compute analytically nu(2n) where n is an integer number. The presence of strong anomalous diffusion implies that the data collapse of the probability density function P(x,t)=t^{-nu}F(x/t^nu) cannot hold, a part (sometimes) in the limit of very small x/t^\nu, now nu=lim_{q to 0} nu(q). Moreover the comparison with previous numerical results shows that the shape of F(x/t^nu) is not universal, i.e., one can have systems with the same nu but different F.

Numerical study of strong anomalous diffusion

The superdiffusion behavior, i.e., 〈x 2(t)〉 ∼ t 2ν , with ν>1/2, in general is not completely characterized by a unique exponent. We study some systems exhibiting strong anomalous diffusion, i.e., 〈| x( t)| q 〉∼ t qν( q) where ν(2)>1/2 and qν( q) is not a linear function of q. This feature is different from the weak superdiffusion regime, i.e., ν( q)= const>1/2, as in random shear flows. The strong anomalous diffusion can be generated by nontrivial chaotic dynamics, e.g. Lagrangian motion in 2d time-dependent incompressible velocity fields, 2d symplectic maps and 1d intermittent maps. Typically the function qν( q) is piecewise linear. This corresponds to two mechanisms: a weak anomalous diffusion for the typical events and a ballistic transport for the rare excursions. In order to have strong anomalous diffusion one needs a violation of the hypothesis of the central limit theorem, this happens only in a very narrow region of the control parameters space. In the presence of strong anomalous diffusion one does not have a unique exponent and therefore one has the failure of the usual scaling of the probability distribution, i.e., P( x, t)= t − ν F( x/ t ν ). This implies that the effective equation at large scale and long time for P( x, t), can obey neither the usual Fick equation nor other linear equations involving temporal and/or spatial fractional derivatives.