Non-Markovian Lévy diffusion in nonhomogeneous media

Srokowski, T.

doi:10.1103/physreve.75.051105

Cited by 22 publications

(20 citation statements)

References 48 publications

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“…In the following we interpret the Langevin equation in the Stratonovich sense [36], to ensure the correct limiting transition for the noise with infinitely short correlation times [37] 5 . Specifically, for D(x) we employ the power-law form [38,39]…”

Section: Heterogeneous Diffusion Modelmentioning

confidence: 99%

Anomalous diffusion and ergodicity breaking in heterogeneous diffusion processes

2013

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We demonstrate the non-ergodicity of a simple Markovian stochastic process with space-dependent diffusion coefficient D(x). For power-law forms D(x) |x| α , this process yields anomalous diffusion of the form x 2 (t) t 2/(2−α) . Interestingly, in both the sub-and superdiffusive regimes we observe weak ergodicity breaking: the scaling of the time-averaged mean-squared displacement δ 2 ( ) remains linear in the lag time and thus differs from the corresponding ensemble average x 2 (t) . We analyse the non-ergodic behaviour of this process in terms of the time-averaged mean-squared displacement δ 2 and its random features, i.e. the statistical distribution of δ 2 and the ergodicity breaking parameters. The heterogeneous diffusion model represents an alternative approach to non-ergodic, anomalous diffusion that might be particularly relevant for diffusion in heterogeneous media.

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Section: Heterogeneous Diffusion Modelmentioning

confidence: 99%

Anomalous diffusion and ergodicity breaking in heterogeneous diffusion processes

2013

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“…Examples include the motion of particles in specific Lorentz gases, e.g., a triangular lattice of immobile reflecting disks [1,2], protein folding [3][4][5] and polymer looping [6,7] in was investigated. Moreover, diffusion coefficients with a power-law form [44,67], or modelled by a Feller process [68] were studied. Cherstvy used a diffusion coefficient depending on the radial distance to a cell centre, corresponding to a radially dependent mobility [36] mimicking the local diffusivity variation in biological cells [11].…”

Section: Introductionmentioning

confidence: 99%

Particle dynamics and transport enhancement in a confined channel with position-dependent diffusivity

Mei

et al. 2020

New J. Phys.

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This work focuses on the dynamics of particles in a confined geometry with position-dependent diffusivity, where the confinement is modelled by a periodic channel consisting of unit cells connected by narrow passage ways. We consider three functional forms for the diffusivity, corresponding to the scenarios of a constant (D 0 ), as well as a low (D m ) and a high (D d ) mobility diffusion in cell centre of the longitudinally symmetric cells. Due to the interaction among the diffusivity, channel shape and external force, the system exhibits complex and interesting phenomena. By calculating the probability density function, mean velocity and mean first exit time with the Itô calculus form, we find that in the absence of external forces the diffusivity D d will redistribute particles near the channel wall, while the diffusivity D m will trap them near the cell centre. The superposition of external forces will break their static distributions. Besides, our results demonstrate that for the diffusivity D d , a high dependence on the x coordinate (parallel with the central channel line) will improve the mean velocity of the particles. In contrast, for the diffusivity D m , a weak dependence on the x coordinate will dramatically accelerate the moving speed. In addition, it shows that a large external force can weaken the influences of different diffusivities; inversely, for a small external force, the types of diffusivity affect significantly the particle dynamics. In practice, one can apply these results to achieve a prominent enhancement of the particle transport in two-or three-dimensional channels by modulating the local tracer diffusivity via an engineered gel of varying porosity or by adding a cold tube to cool down the diffusivity along the central line, which may be a relevant effect in engineering applications. Effects of different stochastic calculi in the evaluation of the underlying multiplicative stochastic equation for different physical scenarios are discussed.New J. Phys. 22 (2020) 053016 Y Li et al confined spaces, transport of particles in nanopores, zeolites and gel networks [8-10], or protein diffusion in the mammalian cell cytoplasm [11]. In many cases, such structured environments can be viewed as confined channels with different boundaries and properties [12][13][14]. The behaviours of systems in these confined channels have been studied by virtue of the analytical Fick-Jacobs (FJ) equation and numerical simulations of the dynamic equations [15][16][17][18][19][20]. Results for symmetric periodic channels indicate that the mean transport velocity may be proportional to large external forces in the confined environment [19][20][21][22]. For asymmetric channels, it was found that the shape of the confinement contributes to the moving direction and mean displacement [20,23], which can be applied to fine separation of mixtures and also the design of many devices [24,25]. Time-dependent and deformable boundaries were discussed to explore the activity in living cells [26,27]. In addition, the interaction...

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“…In this direction, generalizations of Equation (5) that preserve linearity have been made by the introduction of a diffusion coefficient which depends on time and space [24], spatial fractional derivatives (Lévy diffusion) [25][26][27], and a combination of them [28,29]. These anomalous cases and many others may be viewed as a continuous limit of…”

Section: Usual Random Walkmentioning

confidence: 99%

Random Walks Associated with Nonlinear Fokker–Planck Equations

Mendes

Lenzi

Malacarne

et al. 2017

Entropy

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Abstract:A nonlinear random walk related to the porous medium equation (nonlinear Fokker-Planck equation) is investigated. This random walk is such that when the number of steps is sufficiently large, the probability of finding the walker in a certain position after taking a determined number of steps approximates to a q-Gaussian distribution (G q,β (x) ∝ [1 − (1 − q)βx 2 ] 1/(1−q) ), which is a solution of the porous medium equation. This can be seen as a verification of a generalized central limit theorem where the attractor is a q-Gaussian distribution, reducing to the Gaussian one when the linearity is recovered (q → 1). In addition, motivated by this random walk, a nonlinear Markov chain is suggested.

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Non-Markovian Lévy diffusion in nonhomogeneous media

Cited by 22 publications

References 48 publications

Anomalous diffusion and ergodicity breaking in heterogeneous diffusion processes

Anomalous diffusion and ergodicity breaking in heterogeneous diffusion processes

Particle dynamics and transport enhancement in a confined channel with position-dependent diffusivity

Random Walks Associated with Nonlinear Fokker–Planck Equations

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