Continuous time random walk model with asymptotical probability density of waiting times via inverse Mittag-Leffler function

Liang, Yingjie; Chen, Wen

doi:10.1016/j.cnsns.2017.10.014

Cited by 20 publications

(10 citation statements)

References 34 publications

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“…This general formulation has led to many variations over the years (24) , we chose to implement the uncoupled CTRW simulation presented by Fulger et al (25), due to its relative simplicity and speed. In short, spatial increments are drawn from a symmetric -stable levy distribution, temporal increments are drawn from a Mittag-Leffler distribution.…”

Section: Resultsmentioning

confidence: 99%

Single particle diffusion characterization by deep learning

Granik

Nehme

Weiss

et al. 2019

Preprint

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Diffusion plays a crucial role in many biological processes including signaling, cellular organization, transport mechanisms, and more. Direct observation of molecular movement by single-particle tracking experiments has contributed to a growing body of evidence that many cellular systems do not exhibit classical Brownian motion, but rather anomalous diffusion. Despite this evidence, characterization of the physical process underlying anomalous diffusion remains a challenging problem for several reasons. First, different physical processes can exist simultaneously in a system. Second, commonly used tools to distinguish between these processes are based on asymptotic behavior, which is inaccessible experimentally in most cases. Finally, an accurate analysis of the diffusion model requires the calculation of many observables, since different transport modes can result in the same diffusion power-law α, that is obtained from the commonly used mean-squared-displacement (MSD) in its various forms. The outstanding challenge in the field is to develop a method to extract an accurate assessment of the diffusion process using many short trajectories with a simple scheme that is applicable at the non-expert level.Here, we use deep learning to infer the underlying process resulting in anomalous diffusion. We implement a neural network to classify single particle trajectories according to diffusion type -Brownian motion, fractional Brownian motion (FBM) and Continuous Time Random Walk (CTRW). We further use the net to estimate the Hurst exponent for FBM, and the diffusion coefficient for Brownian motion, demonstrating its applicability on simulated and experimental data. The networks outperform time averaged MSD analysis on simulated trajectories while requiring as few as 25 time-steps. Furthermore, when tested on experimental data, both network and ensemble MSD analysis converge to similar values, with the network requiring half the trajectories required for ensemble MSD. Finally, we use the nets to extract diffusion parameters from multiple extremely short trajectories (10 steps).

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Section: Resultsmentioning

confidence: 99%

Single particle diffusion characterization by deep learning

Granik

Nehme

Weiss

et al. 2019

Preprint

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“…Recently, fractional derivatives have been widely used in rock creep, − non-Darcian flow, − and anomalous diffusion − due to their nonlocal properties. As a direct expansion of the fractional derivative, the variable-order fractional derivative is recognized as a powerful tool for modeling complex physical phenomena .…”

Section: Introductionmentioning

confidence: 99%

Modeling Approaches to Permeability of Coal Based on a Variable-Order Fractional Derivative

et al. 2023

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In the past decades, many permeability models have been proposed to characterize the coal permeability evolution in the elastic state. Considering that the coal near the mining working face is often in the plastic or post-peak failure state, it is significant to characterize the coal permeability evolution in the plastic and post-peak failure stages for safe mining. In this study, two permeability models are developed using a variable-order fractional derivative to characterize the coal permeability evolution during the whole process of elastic, plastic, and post-peak failure stages. The results indicate that both models have the ability to better describe the coal permeability evolution during the whole process in areas near the mining working face. In addition, the physical and mechanical interpretation of the variable-order function is given as an indicator of fracture development, indicating the sensitivity of stress variation to coal fracture development. Moreover, the relationships among the effective stress, damage variable, and post-peak permeability of coal are discussed.

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“…In this study, we would like to investigate the feasibility of the M-L distribution in fitting the fatigue data based on the relative entropy method [ 17 ]. Nowadays, the M-L distribution has been applied as a novelty statistical tool to describe non-exponential statistical phenomena in diverse fields [ 18 , 19 ], such as bridge fatigue life assessment [ 12 ] and modeling of an anomalous diffusion with hereditary effects for the importance of the M-L function in the fractional calculus [ 20 , 21 ]. We choose the M-L distribution as a tool to describe the distribution of fatigue data, since it has an apparent hereditary effect and power decay or heavy-tailed traits [ 22 , 23 , 24 , 25 , 26 , 27 ].…”

Section: Introductionmentioning

confidence: 99%

Relative Entropy Method Applied for the Fatigue Life Distribution of Carbon Fiber/Epoxy Composites

Yuan

Liang

2021

Entropy

Self Cite

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This paper verifies the feasibility of the relative entropy method in selecting the most suitable statistical distribution for the experimental data, which do not follow an exponential distribution. The efficiency of the relative entropy method is tested through the fractional order moment and the logarithmic moment in terms of the experimental data of carbon fiber/epoxy composites with different stress amplitudes. For better usage of the relative entropy method, the efficient range of its application is also studied. The application results show that the relative entropy method is not very fit for choosing the proper distribution for non-exponential random data when the heavy tail trait of the experimental data is emphasized. It is not consistent with the Kolmogorov–Smirnov test but is consistent with the residual sum of squares in the least squares method whenever it is calculated by the fractional moment or the logarithmic moment. Under different stress amplitudes, the relative entropy method has different performances.

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Continuous time random walk model with asymptotical probability density of waiting times via inverse Mittag-Leffler function

Cited by 20 publications

References 34 publications

Single particle diffusion characterization by deep learning

Single particle diffusion characterization by deep learning

Modeling Approaches to Permeability of Coal Based on a Variable-Order Fractional Derivative

Relative Entropy Method Applied for the Fatigue Life Distribution of Carbon Fiber/Epoxy Composites

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