Time-averaged Einstein relation and fluctuating diffusivities for the Lévy walk

Froemberg, D.; Barkai, Eli

doi:10.1103/physreve.87.030104

Cited by 85 publications

(112 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%

“…Although the ensemble-averaged MSDs show subdiffusion as well as superdiffusion, the TAMSDs always increase linearly with time in SEDLF [6]. This behavior is completely different from that in Lévy walk [2,17]. Moreover, the distribution of the TAMSD with a fixed converge to a time-independent distribution which is not the same as a universal distribution in CTRW (Mittag-Leffler distribution):…”

Section: Introductionmentioning

confidence: 87%

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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%

Section: Introductionmentioning

confidence: 87%

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: 99%

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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“…As examples we mention continuous time random walk processes with scale free distributions of waiting times [7,[25][26][27][28][29]31], correlated continuous time random walks [44], as well as diffusion processes with space [32][33][34][35][36] and time [32,38,45,46] dependent diffusion coefficients and their combinations [47]. We also mention ultraslow diffusion processes with a logarithmic form for x 2 (t) and linear lag time dependence (8) of the time averaged MSD [48] as well as the ultraweak ergodicity breaking of superdiffusive Lévy walks [49]. For finite measurement time even ergodic processes exhibit a statistical scatter of the amplitude of time averaged observables.…”

Section: Observablesmentioning

confidence: 99%

Ergodicity breaking and particle spreading in noisy heterogeneous diffusion processes

Cherstvy

Metzler

2015

The Journal of Chemical Physics

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We study noisy heterogeneous diffusion processes with a position dependent diffusivity of the form D(x) ∼ D0|x| α in the presence of annealed and quenched disorder of the environment, corresponding to an effective variation of the exponent α in time and space. In the case of annealed disorder, for which effectively α = α(t) we show how the long time scaling of the ensemble mean squared displacement (MSD) and the amplitude variation of individual realizations of the time averaged MSD are affected by the disorder strength. For the case of quenched disorder, the long time behavior becomes effectively Brownian after a number of jumps between the domains of a stratified medium. In the latter situation the averages are taken over both an ensemble of particles and different realizations of the disorder. As physical observables we analyze in detail the ensemble and time averaged MSDs, the ergodicity breaking parameter, and higher order moments of the time averages.

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Time-averaged Einstein relation and fluctuating diffusivities for the Lévy walk

Cited by 85 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

Ergodicity breaking and particle spreading in noisy heterogeneous diffusion processes

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