Spatial distributions at equilibrium under heterogeneous transient subdiffusion

Berry, Hugues; Soula, HÃ©di A.

doi:10.3389/fphys.2014.00437

Cited by 12 publications

(23 citation statements)

References 40 publications

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“…We note that heterogeneous diffusivities can, for instance, play a role in the formation of gradients of morphogen molecules in a developing cell tissue [51], a process known to involve features of anomalous diffusion. It also features a division of fluxes of the molecules into fluxes through cells, across the outer cell membranes, and transport in extracellular spaces [52]. Heterogeneous diffusion of water molecules in brain tissues [53] and strongly heterogeneous structures of cardiac muscle tissue with nontrivial cell-cell coupling [54] could be another example.…”

Section: Discussionmentioning

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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Section: Discussionmentioning

confidence: 99%

Ergodicity breaking and particle spreading in noisy heterogeneous diffusion processes

Cherstvy

Metzler

2015

The Journal of Chemical Physics

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“…To show the dominance over the standard drift, we take the Laplace transform of Eq. (17). The equation will be the same as (12) but with a modified flux move ultra-slowly towards x = 0 since the mean position of particles decreases to zero logarithmically.…”

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confidence: 99%

“…In the continuous limit, this master equation for symmetric random walks, r i = 0.5, reduces to the fractional diffusion equation (1). For an asymmetric random walk, this master equation reduces to the fractional Fokker-Planck equation (17). Figure 1 shows the normalised histograms for N = 10 4 particles performing the symmetric random walk with an uniform initial distribution; r i = 0.5, L = 1, k = 50, τ 0 = 10 −3 and µ i = 0.4 + 0.5(i − 1)/(k − 1).…”

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confidence: 99%

Asymptotic Behavior of the Solution of the Space Dependent Variable Order Fractional Diffusion Equation: Ultraslow Anomalous Aggregation

Fedotov

Han

2019

Phys. Rev. Lett.

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We find the asymptotic representation of the solution of the variable-order fractional diffusion equation, which remains unsolved since it was proposed in [Checkin et. al., J. Phys. A, 2005]. We identify a new advection term that causes ultra-slow spatial aggregation of subdiffusive particles due to dominance over the standard advection and diffusion terms, in the long-time limit. This uncovers the anomalous mechanism by which non-uniform distributions can occur. We perform Monte Carlo simulations of the underlying anomalous random walk and find good agreement with the asymptotic solution.Anomalous diffusion has attracted immense interest in the past due to many physical, chemical and biological processes characterized by the mean square displacement (MSD) involving the fractional exponent µ:Anomalous diffusion is observed also in many other areas, for instance, in finance and economics [7]. An influential paper by Metzler and Klafter [2] reviews anomalous diffusion in the scope of a constant exponent µ. However, anomalous transport in realistic inhomogeneous and complex environments [8], such as lipid granules [9], porous media [10] and entangled polymer liquids [11], requires a multi-fractional approach involving the space-dependent variable-order fractional exponent [12][13][14][15][16][17][18]. Important examples of anomalous transport involving multi-fractional exponents are lateral diffusion of proteins on crowded lipid membranes [19], intracellular subdiffusion of proteins [20], mRNA [21] and organelles [22] due in part to inhomogeneous crowding [23] and weak interactions between components in the cell [24]. Recent observations show that lysosomes, which are key organelles for cellular metabolism, predominantly move subdiffusively and maintain a non-uniform spatial distribution in the cell [24]. The majority of these organelles are concentrated in the perinuclear area. A fundamental unresolved question is how lysosomes are self-organized spatially to coordinate their roles [24]. In this Letter, we propose a new anomalous mechanism by which nonuniform distribution of subdiffusing organelles can occur.A generic model for anomalous diffusion in inhomogeneous media is the space-dependent variable-order fractional diffusion equation [12][13][14][15][16] ∂p(x, t) ∂twhere p(x, t) is the probability density function (PDF) of a particle at position x and time t. This function can be also interpreted as the mean number density of subdiffusive particles. In Eq. (1), D µ(x) = a 2 /2τis the fractional diffusion coefficient with the microscopic time scale τ 0 , length scale a, and space-dependent fractional exponent µ(x) ∈ (0, 1). The Riemann-Liouville deriva- (2) also involves spatial dependence. Equation (1) was first derived by Chechkin, Gorenflo and Sokolov [12], and since then, many attempts have been made to find a solution through composite regions with constant anomalous exponents and numerically [12,14,25]. However, Eq. (1) remains unsolved for the general case of a spacedependent anomalous exponent µ(x).In thi...

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“…MSD x 2 (t) and time averaged MSD δ 2 (∆) (thick blue curves) as well as individual time traces δ 2 (∆) (red curves) for overdamped RDPs. The asymptotes(20) and(22)for the MSD and the time averaged MSD are shown by the dashed curves. The asymptotes often superimpose with the results of simulations.…”

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confidence: 99%

Anomalous diffusion in time-fluctuating non-stationary diffusivity landscapes

Cherstvy

Metzler

2016

Phys. Chem. Chem. Phys.

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We investigate the ensemble and time averaged mean squared displacements for particle diffusion in a simple model for disordered media by assuming that the local diffusivity is both fluctuating in time and has a deterministic average growth or decay in time. In this study we compare computer simulations of the stochastic Langevin equation for this random diffusion process with analytical results. We explore the regimes of normal Brownian motion as well as anomalous diffusion in the sub- and superdiffusive regimes. We also consider effects of the inertial term on the particle motion. The investigation of the resulting diffusion is performed for unconfined and confined motion.

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Spatial distributions at equilibrium under heterogeneous transient subdiffusion

Cited by 12 publications

References 40 publications

Ergodicity breaking and particle spreading in noisy heterogeneous diffusion processes

Ergodicity breaking and particle spreading in noisy heterogeneous diffusion processes

Asymptotic Behavior of the Solution of the Space Dependent Variable Order Fractional Diffusion Equation: Ultraslow Anomalous Aggregation

Anomalous diffusion in time-fluctuating non-stationary diffusivity landscapes

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