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
He 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.
In nature, the consumers (predator) use many strategies for hunting the resources (prey). The hunting cooperation is an effective strategy for the consumers for giving more accuracy to the hunting. ...This strategy works on perturbing the resources for facilitating the hunting. In this paper, we are concerned to analyze a superdiffusive resource-consumer system with hunting cooperation functional response. The presence of supperdiffusion represents the effect of the fear of the resources and the organized hunting strategy of the consumer. In fact, the effect of the superdiffusion is successfully established, where it is shown that the superdiffusion undergoes a complex patterns. Also, it noticed that the investigated system has a complex patters as Hopf bifurcation (HB), Turing bifurcation (TB), and Turing-Hopf bifurcation (THB). Furthermore, the super-diffusion can influence the stability of some equilibria. The obtained results are tested numerically.
•We propose a new approach for the analysis of experimental trajectories based on the fractional Lévy stable motion.•It takes into account non-Gaussian, stable distributions and the memory parameter, ...which governs the diffusion type.•We apply the procedure to the analysis of the Golding-Cox mRNA dataset (I. Golding et al., Cell 123, 1025, 2005).•The methods are universal and can be applied to any experimental data which exhibit a non-Gaussian behavior.
In this paper we propose a new approach for the analysis of experimental data based on the fractional Lévy stable motion (FLSM) and apply it to the Golding–Cox mRNA dataset. The FLSM takes into account non-Gaussian α-stable distributions and is characterized by the memory parameter d=H−1/α, where H is the Hurst exponent. The sign of d indicates the type of diffusion: d=0 for Lévy diffusion, d<0 for subdiffusion and d>0 for superdiffusion. By estimating the memory parameter for trajectories, we obtain their classification along the x and y coordinates independently. It appears that most of the trajectories are subdiffusive, other follow the Lévy-diffusion, but none of them is superdiffusive. We also justify presence of the non-Gaussian α-stable distribution by five different goodness-of-fit tests. We note that the classification procedure presented here can be applied to other experimental data which exhibit a non-Gaussian behavior.
Many predator species attempt to locate prey by following seemingly random paths. Although the underlying physical mechanism of the search remains largely unknown, such search paths are usually ...modeled by some type of random walk. Here, we introduce the stochastic pursuit-evasion equations that consider the bidirectional interaction between predators and prey. This assumption results in a modulated persistent random walk that is characterized by three interesting properties: power-law or tempered power-law distributed running times, superdiffusive or transient superdiffusive dynamics, and strong directional persistence. Furthermore, the proposed model exhibits a transition from Brownian to Lévy-like motion with intensifying predator–prey interaction. Interestingly, persistent random walks with pure-power law distributed running times emerge at the limit of highest predator–prey interaction. We hypothesize that the system ultimately self-organizes into a critical interaction to avoid extinction.
•Introduction of the stochastic pursuit-evasion model.•Emergence of a modulated persistent random walk with tempered power-law distributed running times.•Emergence of a modulated persistent random walk with superdiffusive or transient superdiffusive dynamics.•Emergence of a modulated persistent random walk with strong directional persistence.•Qualitative comparison with foraging dynamics.
In this paper, we study the time–space fractional order (fractional for simplicity) nonlinear subdiffusion and superdiffusion equations, which can relate the matter flux vector to concentration ...gradient in the general sense, describing, for example, the phenomena of anomalous diffusion, fractional Brownian motion, and so on. The semi-discrete and fully discrete numerical approximations are both analyzed, where the Galerkin finite element method for the space Riemann–Liouville fractional derivative with order
1
+
β
∈
1
,
2
and the finite difference scheme for the time Caputo derivative with order
α
∈
(
0
,
1
)
(for subdiffusion) and
(
1
,
2
)
(for superdiffusion) are analyzed, respectively. Results on the existence and uniqueness of the weak solutions, the numerical stability, and the error estimates are presented. Numerical examples are included to confirm the theoretical analysis. During our simulations, an interesting diffusion phenomenon of particles is observed, that is, on average, the diffusion velocity for
0
<
α
<
1
is slower than that for
α
=
1
, but the diffusion velocity for
1
<
α
<
2
is faster than that for
α
=
1
. For the spatial diffusion, we have a similar observation.
•This paper presents the space-time fractional diffusion equation.•The diffusion is related with the electromagnetic transient transmission lines.•The generalization of the equations in space-time ...exhibit anomalous behavior.•An analysis of the fractional time constant is presented.•The solutions are given in terms of the Mittag-Leffler function.•To keep the dimensionality an auxiliary parameter σ is introduced.
In this paper, the space-time fractional diffusion equation related to the electromagnetic transient phenomena in transmission lines is studied, three cases are presented; the diffusion equation with fractional spatial derivative, with fractional temporal derivative and the case with fractional space-time derivatives. For the study cases, the order of the spatial and temporal fractional derivatives are 0 < β, γ ≤ 2, respectively. In this alternative representation we introduce the appropriate fractional dimensional parameters which characterize consistently the existence of the fractional space-time derivatives into the fractional diffusion equation. The general solutions of the proposed equations are expressed in terms of the multivariate Mittag-Leffler functions; these functions depend only on the parameters β and γ and preserve the appropriated physical units for any value of the fractional derivative exponent. Furthermore, an analysis of the fractional time constant was made in order to indicate the change of the medium properties and the presence of dissipation mechanisms. The proposed mathematical representation can be useful to understand electrochemical phenomena, propagation of energy in dissipative systems, irreversible thermodynamics, quantum optics or turbulent diffusion, thermal stresses, models of porous electrodes, the description of gel solvents and anomalous complex processes.
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.
In this mini-review, we addressed the transient-anomalous diffusion by MRI, starting from the assumption that transient-anomalous diffusion is ubiquitously observed in biological tissues, as ...demonstrated by different single-particle-tracking optical experiments. The purpose of this review is to identify the main pitfalls that can be encountered when venturing into the field of anomalous diffusion quantified by diffusion-MRI methods. Therefore, the theory of anomalous diffusion deriving from its mathematical definition was reported and connected with the consolidated description and the established procedures of conventional diffusion-MRI of tissues. We highlighted the two different modalities for quantifying subdiffusion and superdiffusion parameters of anomalous diffusion. Then we showed that most of the papers concerning anomalous diffusion, actually deal with pseudo-superdiffusion due to the use of a superdiffusion signal representation. Pseudo-superdiffusion depends on water diffusion multi-compartmentalization and local magnetic in-homogeneities that mimic the superdiffusion of spins. In addition to the relatively large production of pseudosuperdiffusion images, anomalous diffusion research is still in its early stages due to the limited flexibility of conventional clinical MRI scanners that currently prevent the acquisition of diffusion-weighted images by varying the diffusion time (the necessary acquisition modality to quantify transient-subdiffusion in human tissues). Moreover, the wide diffusion gradient pulses complicates the definition of a reliable function representative of anomalous diffusion signal behavior to fit data. Nevertheless, it is important and possible to address these limitations, as one of the potentialities of anomalous diffusion imaging is to increase the resolution, sensitivity, and specificity of MRI.
In this article, we obtain the rates of convergence for superdiffusion in the Boltzmann–Grad limit of the periodic Lorentz gas, which is one of the fundamental models for studying diffusions in ...deterministic systems. In their seminal work, Marklof and Strömbergsson (2011) proved the Boltzmann–Grad limit of the periodic Lorentz gas, following which Marklof and Tóth (2016) established a superdiffusive central limit theorem in large time for the Boltzmann–Grad limit. Based on their work, we apply Stein’s method to derive the convergence rates for the superdiffusion in the Boltzmann–Grad limit of the periodic Lorentz gas. The convergence rate in Wasserstein distance is obtained for the discrete-time displacement, while the result for the Berry–Essen type bound is presented for the continuous-time displacement.
Abstract
We introduce non-local dynamics on directed networks through the construction of a fractional version of a non-symmetric Laplacian for weighted directed graphs. Furthermore, we provide an ...analytic treatment of fractional dynamics for both directed and undirected graphs, showing the possibility of exploring the network employing random walks with jumps of arbitrary length. We also provide some examples of the applicability of the proposed dynamics, including consensus over multi-agent systems described by directed networks.
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