Time-fractional diffusion equation with time dependent diffusion coefficient

Fa, Kwok Sau; Lenzi, E. K.

doi:10.1103/physreve.72.011107

Cited by 37 publications

(28 citation statements)

References 23 publications

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“…In the simplest case when the time-dependent diffusivity D t is deterministically prescribed, T t is not random, so that Q(t; T ) = δ(T −T t ) and thus Υ(t; λ) = exp −λT t ). When the particle undergoes a continuous-time random walk with long stalling periods characterized by an anomalous waiting exponent 0 < α < 1 [12], one gets Υ(t; λ) = E α (−D α t α λ), where E α (z) is the Mittag-Leffler function, and D α is the (constant) generalized diffusion coefficient [63,64]. One can also consider Lévy-noise-driven processes to model diffusivity with heavy tails [65], geometric Brownian motion to get a nonstationary evolution, or a customized stochastic process to produce the desired distribution of the stationary diffusivity [66].…”

Section: Discussionmentioning

confidence: 99%

Diffusion-limited reactions in dynamic heterogeneous media

Lanoiselée¹,

Moutal²,

Grebenkov³

2018

Nat Commun

148

129

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Most biochemical reactions in living cells rely on diffusive search for target molecules or regions in a heterogeneous overcrowded cytoplasmic medium. Rapid rearrangements of the medium constantly change the effective diffusivity felt locally by a diffusing particle and thus impact the distribution of the first-passage time to a reaction event. Here, we investigate the effect of these dynamic spatiotemporal heterogeneities onto diffusion-limited reactions. We describe a general mathematical framework to translate many results for ordinary homogeneous Brownian motion to heterogeneous diffusion. In particular, we derive the probability density of the first-passage time to a reaction event and show how the dynamic disorder broadens the distribution and increases the likelihood of both short and long trajectories to reactive targets. While the disorder slows down reaction kinetics on average, its dynamic character is beneficial for a faster search and realization of an individual reaction event triggered by a single molecule.

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Section: Discussionmentioning

confidence: 99%

Diffusion-limited reactions in dynamic heterogeneous media

Lanoiselée¹,

Moutal²,

Grebenkov³

2018

Nat Commun

148

129

View full text Add to dashboard Cite

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“…The same conclusion that the exponent of the anomalous diffusion does not depend on the prescription has been obtained in Refs. [58,73,74] for equations describing diffusion without the presence of an external force. …”

Section: Discussionmentioning

confidence: 99%

Influence of external potentials on heterogeneous diffusion processes

Kazakevičius

Ruseckas

2017

2017 International Conference on Noise and Fluctuations (ICNF)

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In this paper we consider heterogeneous diffusion processes with the power-law dependence of the diffusion coefficient on the position and investigate the influence of external forces on the resulting anomalous diffusion. The heterogeneous diffusion processes can yield subdiffusion as well as superdiffusion, depending on the behavior of the diffusion coefficient. We assume that not only the diffusion coefficient but also the external force has a power-law dependence on the position. We obtain analytic expressions for the transition probability in two cases: when the power-law exponent in the external force is equal to 2η − 1, where 2η is the power-law exponent in the dependence of the diffusion coefficient on the position, and when the external force has a linear dependence on the position. We found that the power-law exponent in the dependence of the mean square displacement on time does not depend on the external force, this force changes only the anomalous diffusion coefficient. In addition, the external force having the power-law exponent different from 2η − 1 limits the time interval where the anomalous diffusion occurs. We expect that the results obtained in this paper may be relevant for a more complete understanding of anomalous diffusion processes.

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“…It can also reproduce the asymptotic behavior of the random-walk model and time fractional dynamic equation for t = t β(2+θ)/2 [9], where 0 < β < 1. Now we want to show two interesting processes which can be obtained from Eqs.…”

Section: Langevin Equation and Its Corresponding Fokker-planck Equationmentioning

confidence: 99%

Time-fractional diffusion equation with time dependent diffusion coefficient

Cited by 37 publications

References 23 publications

Diffusion-limited reactions in dynamic heterogeneous media

Diffusion-limited reactions in dynamic heterogeneous media

Influence of external potentials on heterogeneous diffusion processes

Solution of Fokker-Planck equation for a broad class of drift and diffusion coefficients

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