Nonergodicity of Blinking Nanocrystals and Other Lévy-Walk Processes

Margolin, Gennady; Barkai, Eli

doi:10.1103/physrevlett.94.080601

Cited by 185 publications

(191 citation statements)

References 25 publications

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“…We are not the first who point to the possibility of non-ergodicity of Levy flight, since it has been considered as a problem by Zumofen et. al [16] and others [17,18] in recent years. …”

Section: Resultsmentioning

confidence: 99%

Computer simulation study of the Levy flight process

Ghaemi¹,

Zabihinpour²,

Asgari

2009

Physica A: Statistical Mechanics and its Applications

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Abstract. Random walk simulation of the Levy flight shows a linear relation between the mean square displacement and time. We have analyzed different aspects of this linearity. It is shown that the restriction of jump length to a maximum value (l m ) affects the diffusion coefficient, even though it remains constant for l m greater than 1464. So, this factor has no effect on the linearity. In addition, it is shown that the number of samples does not affect the results. We have demonstrated that the relation between the mean square displacement and time remains linear in a continuous space, while continuous variables just reduce the diffusion coefficient. The results are also implied that the movement of a levy flight particle is similar to the case the particle moves in each time step with an average length of jumping . Finally, it is shown that the non-linear relation of the Levy flight will be satisfied if we use time average instead of ensemble average. The difference between time average and ensemble average results points that the Levy distribution may be a non-ergodic distribution.

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“…We are not the first who point to the possibility of non-ergodicity of Levy flight, since it has been considered as a problem by Zumofen et. al [16] and others [17,18] in recent years. …”

Section: Resultsmentioning

confidence: 99%

Computer simulation study of the Levy flight process

Ghaemi¹,

Zabihinpour²,

Asgari

2009

Physica A: Statistical Mechanics and its Applications

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“…Lévy walk [1][2][3] is an important concept which describes a wide spectrum of physical and biological processes involving stochastic transport [4][5][6][7][8][9][10]. Cold atoms moving in dissipative optical lattices [11], endosomal active transport in living cells [12], and T-cells migrating in the brain tissue [13] are just several examples where Lévy walks were reported.…”

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confidence: 99%

Emergence of Lévy walks in systems of interacting individuals

Fedotov¹,

Korabel²

2017

Phys. Rev. E

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We propose a model of superdiffusive Lévy walk as an emergent nonlinear phenomenon in systems of interacting individuals. The aim is to provide a qualitative explanation of recent experiments [G. Ariel et al., Nat. Commun. 6, 8396 (2015)] revealing an intriguing behavior: swarming bacteria fundamentally change their collective motion from simple diffusion into a superdiffusive Lévy walk dynamics. We introduce microscopic mean-field kinetic equations in which we combine two key ingredients: (1) alignment interactions between individuals and (2) non-Markovian effects. Our interacting run-and-tumble model leads to the superdiffusive growth of the mean-squared displacement and the power-law distribution of run length with infinite variance. The main result is that the superdiffusive behavior emerges as a cooperative effect without using the standard assumption of the power-law distribution of run distances from the inception. At the same time, we find that the collision and repulsion interactions lead to the density-dependent exponential tempering of power-law distributions. This qualitatively explains the experimentally observed transition from superdiffusion to the diffusion of mussels as their density increases [M. [15][16][17][18]. Recently an intriguing behavior of swarming bacteria was found: they fundamentally change their collective motion from simple diffusion into a superdiffusive Lévy walk dynamics [19]. The extraordinary feature of this movement is that the emergence of a superdiffusive motility is a result of the interactions between bacteria rather than the standard mechanism of controlling the individual frequency of tumbling. However, it is still an open question how Lévy walks emerge in systems of interacting self-propelled particles. The current theory of Lévy walk [8] assumes noninteracting particles and power-law distribution of traveled distances from the inception.The collective behavior of large groups of interacting individuals such as bird flocks, fish schools, and the collective migration of cells or bacteria is another rapidly growing area of active matter research [20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38]. There exist two main types of models used for a collective behavior: (1) Lagrangian models describing the movements of self-propelled particles in terms of nonlinear equations for the positions and velocities of all particles [20][21][22][23], and (2) kinetic models involving partial differential equations for the population densities [24][25][26][27][28][29][30]. In this Rapid Communication we are only concerned with the nonlinear kinetic and macroscopic equations. They have been used to describe interactions between individuals and investigate the formation of a large variety of spatiotemporal self-organized aggregations. Most of these models converge to the density-dependent diffusive transport and do not lead to non-Markovian Lévy walks.In this Rapid Communication we propose a kinetic nonlinear Lévy walk model for interacting individuals in which...

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“…More flattened plateaus are found when at least one of the exponents is ∼1.5 which is close to the transition point from the "U" to the "W" shape. 35 Meanwhile, the inter-peak region shows various degrees of tilting both sloping towards the on and off states. The tilt is more sensitive to the power law exponent than the curvature.…”

Section: B the Inter-peak Feature At Different Power Law Exponentsmentioning

confidence: 99%

“…34 The former corresponds to the case of strong ergodicity breaking since the histogram does not show a peak at the ensemble average, while the latter corresponds to weak ergodicity breaking due to the arising of a central peak around the ensemble average. 35 In order to investigate upon such effects for finite bin times typical for blinking experiments, the present analysis has been extended to power law exponents in the range (1.1-1.9) keeping all other parameters fixed. Figure 3 summarises the finding and reveals a behaviour somewhat reminiscent of the Lamperti function.…”

Section: B the Inter-peak Feature At Different Power Law Exponentsmentioning

confidence: 99%

Two-state theory of binned photon statistics for a large class of waiting time distributions and its application to quantum dot blinking

Volkan-Kacso

2014

The Journal of Chemical Physics

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A theoretical method is proposed for the calculation of the photon counting probability distribution during a bin time. Two-state fluorescence and steady excitation are assumed. A key feature is a kinetic scheme that allows for an extensive class of stochastic waiting time distribution functions, including power laws, expanded as a sum of weighted decaying exponentials. The solution is analytic in certain conditions, and an exact and simple expression is found for the integral contribution of "bright" and "dark" states. As an application for power law kinetics, theoretical results are compared with experimental intensity histograms from a number of blinking CdSe/ZnS quantum dots. The histograms are consistent with distributions of intensity states around a "bright" and a "dark" maximum. A gap of states is also revealed in the more-or-less flat inter-peak region. The slope and to some extent the flatness of the inter-peak feature are found to be sensitive to the power-law exponents. Possible models consistent with these findings are discussed, such as the combination of multiple charging and fluctuating non-radiative channels or the multiple recombination center model. A fitting of the latter to experiment provides constraints on the interaction parameter between the recombination centers. Further extensions and applications of the photon counting theory are also discussed. © 2014 AIP Publishing LLC. [http://dx

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Nonergodicity of Blinking Nanocrystals and Other Lévy-Walk Processes

Cited by 185 publications

References 25 publications

Computer simulation study of the Levy flight process

Computer simulation study of the Levy flight process

Emergence of Lévy walks in systems of interacting individuals

Two-state theory of binned photon statistics for a large class of waiting time distributions and its application to quantum dot blinking

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