Random time averaged diffusivities for Lévy walks

Froemberg, D.; Barkai, Eli

doi:10.1140/epjb/e2013-40436-1

Cited by 54 publications

(50 citation statements)

References 47 publications

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“…On the other hand, in Lévy walk, the proportional constant of the EAMSD in a non-equilibrium ensemble such as an ordinary renewal process differs from that in an equilibrium one [16][17][18]45]. We note that the TAMSD coincides with the EAMSD in an equilibrium ensemble as the measurement time goes to infinity.…”

Section: Mean Square Displacementmentioning

confidence: 72%

Phase Diagram in Stored-Energy-Driven Lévy Flight

Akimoto

Miyaguchi

2014

J Stat Phys

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Phase diagram based on the mean square displacement (MSD) and the distribution of diffusion coefficients of the time-averaged MSD for the stored-energy-driven Lévy flight (SEDLF) is presented. In the SEDLF, a random walker cannot move while storing energy, and it jumps by the stored energy. The SEDLF shows a whole spectrum of anomalous diffusions including subdiffusion and superdiffusion, depending on the coupling parameter between storing time (trapping time) and stored energy. This stochastic process can be investigated analytically with the aid of renewal theory. Here, we consider two different renewal processes, i.e., ordinary renewal process and equilibrium renewal process, when the mean trapping time does not diverge. We analytically show the phase diagram according to the coupling parameter and the power exponent in the trapping-time distribution. In particular, we find that distributional behavior of time-averaged MSD intrinsically appears in superdiffusive as well as normal diffusive regime even when the mean trapping time does not diverge.

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Section: Mean Square Displacementmentioning

confidence: 72%

Phase Diagram in Stored-Energy-Driven Lévy Flight

Akimoto

Miyaguchi

2014

J Stat Phys

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“…Another interesting question is how the sensitivity to initial conditions translates to the time-averaged diffusivity, where, instead of averaging over an ensemble of trajectories at a given time, the mean-square displacement is computed from a time average over a single trajectory. The dependence of the time-averaged diffusivity on the initial conditions has so far only been discussed for special cases [71][72][73][74], and whether it will correspond to the stationary or nonstationary expression obtained here or to neither is an open question.…”

Section: Discussionmentioning

confidence: 93%

Scaling Green-Kubo Relation and Application to Three Aging Systems

et al. 2014

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The Green-Kubo formula relates the spatial diffusion coefficient to the stationary velocity autocorrelation function. We derive a generalization of the Green-Kubo formula that is valid for systems with longrange or nonstationary correlations for which the standard approach is no longer valid. For the systems under consideration, the velocity autocorrelation function hvðt þ τÞvðtÞi asymptotically exhibits a certain scaling behavior and the diffusion is anomalous, hx 2 ðtÞi ≃ 2D ν t ν . We show how both the anomalous diffusion coefficient D ν and the exponent ν can be extracted from this scaling form. Our scaling GreenKubo relation thus extends an important relation between transport properties and correlation functions to generic systems with scale-invariant dynamics. This includes stationary systems with slowly decaying power-law correlations, as well as aging systems, systems whose properties depend on the age of the system. Even for systems that are stationary in the long-time limit, we find that the long-time diffusive behavior can strongly depend on the initial preparation of the system. In these cases, the diffusivity D ν is not unique, and we determine its values, respectively, for a stationary or nonstationary initial state. We discuss three applications of the scaling Green-Kubo relation: free diffusion with nonlinear friction corresponding to cold atoms diffusing in optical lattices, the fractional Langevin equation with external noise recently suggested to model active transport in cells, and the Lévy walk with numerous applications, in particular, blinking quantum dots. These examples underline the wide applicability of our approach, which is able to treat very different mechanisms of anomalous diffusion.

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“…Apart from the processes discussed herein, non-ergodic behaviour also occurs in other stochastic processes, including the ultraweakly non-ergodic Lévy walks [125][126][127][128] where the disparity between ensemble and time averaged MSDs only amounts to a constant factor. Diffusion on random, fractal percolation clusters was shown to be ergodic.…”

Section: Discussionmentioning

confidence: 96%

“…Additional recent studies of Lévy walks analyse their response to an external bias and the power spectral properties. [125][126][127]129 Lévy flights and walks are used as statistical models in many fields, for example, to quantify blind search processes of animals for sparse food sources. [130][131][132] In the science of movement ecology, the so-called Lévy foraging hypothesis has become widely accepted.…”

Section: Superdiffusive Continuous Time Random Walks and Ultraweak Ermentioning

confidence: 99%

Non-Ergodicity and Ageing in Anomalous Diffusion

Metzler

2015

Order, Disorder and Criticality

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Anomalous diffusion, the deviation from classical Brownian motion with its linear-in-time mean squared displacement, is a widespread phenomenon in a large variety of complex systems, ranging from amorphous semiconductors to biological cells. Anomalous diffusion processes are no longer confined by the central limit theorem, that enforces the convergence of Brownian motion to the Gaussian shape of the probability density function. Instead, anomalous processes have a rich variety of physical origins and non-traditional mathematical properties. We here provide a summary of various anomalous diffusion models, paying special attention to their non-ergodic and ageing behaviour. Whether a process is ergodic or not is of vital importance when measurements are evaluated in terms of time averaged observables such as the mean squared displacement. For non-ergodic processes these can no longer be interpreted by comparison to the typically calculated ensemble averaged observables. Breaking of ergodicity occurs in many anomalous diffusion processes. We also consider their ageing properties, the explicit dependence of observables on the time span between the original system initiation and the start of the measurement.

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Random time averaged diffusivities for Lévy walks

Cited by 54 publications

References 47 publications

Phase Diagram in Stored-Energy-Driven Lévy Flight

Phase Diagram in Stored-Energy-Driven Lévy Flight

Scaling Green-Kubo Relation and Application to Three Aging Systems

Non-Ergodicity and Ageing in Anomalous Diffusion

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