Random walk model of subdiffusion in a system with a thin membrane

Kosztołowicz, Tadeusz

doi:10.1103/physreve.91.022102

Cited by 20 publications

(37 citation statements)

References 41 publications

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“…(28) and (34), we suppose that the following relation is satisfied for all functions occurring in Eqs. (23) and (24)…”

Section: From a Discrete To Continuous Space Variablementioning

confidence: 99%

“…In this situation, the membrane loses its selective property. To avoid this, we suppose that the probabilities q 1 and q 2 are functions of ǫ, in such a way that q 1 (0) = q 2 (0) = 1 [24,25]. The calculation presented in Appendix II shows that…”

Section: From a Discrete To Continuous Space Variablementioning

confidence: 99%

“…However, it provides somewhat nontrivial results which have a simple physical interpretation. We add that such models were used to model subdiffusion processes in membrane systems [24,25], as was mentionad earlier in this paper, as well as in the system with reactions [48,50].…”

Section: Final Remarksmentioning

confidence: 99%

“…One of the criteria that provides the utility of the used approximation is that the Green's functions obtained for a membrane system should depend on the parameters of membrane permeability including the case ofγ 1 =γ 2 (see the discussion presented in [24]). However, when α 1 = α 2 , several leading terms in the series which approximate the functions (39) and (40) with respect to s, are dependent on the ratioγ 1 /γ 2 alone; the number of leading terms increases when |α 1 − α 2 | becomes smaller.…”

Section: Final Remarksmentioning

confidence: 99%

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Subdiffusion in a system consisting of two different media separated by a thin membrane

Kosztołowicz

2017

International Journal of Heat and Mass Transfer

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We consider subdiffusion in a system which consists of two media separated by a thin membrane. The subdiffusion parameters may be different in each of the medium. Using the new method presented in this paper we derive the probabilities (the Green's functions) describing a particle's random walk in the system. Within this method we firstly consider the particle's random walk in a system with both discrete time and space variables in which a particle can vanish due to reactions with constant probabilities R1 and R2 defined separately for each medium. Then, we move from discrete to continuous variables. The reactions included in the model play a supporting role. We link the reaction probabilities with the other subdiffusion parameters which characterize the media by means of the formulae presented in this paper. Calculating the generating functions for the difference equations describing the random walk in the composite membrane system with reactions, which depend explicitly on R1 and R2, we are able to correctly incorporate the subdiffusion parameters of both the media into the Green's functions. Finally, placing R1 = R2 = 0 into the obtained functions we get the Green's functions for the composite membrane system without any reactions. From the obtained Green's functions, we derive the boundary conditions at the thin membrane. One of the boundary conditions contains the Riemann-Liouville fractional time derivatives, which show that the additional 'memory effect' is created by a discontinuoity of the system. The second boundary condition demands continuity of a flux at the membrane. keywords: subdiffusion, diffusion in composite systems with a thin membrane, random walk model

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“…(28) and (34), we suppose that the following relation is satisfied for all functions occurring in Eqs. (23) and (24)…”

Section: From a Discrete To Continuous Space Variablementioning

confidence: 99%

Section: From a Discrete To Continuous Space Variablementioning

confidence: 99%

Section: Final Remarksmentioning

confidence: 99%

Section: Final Remarksmentioning

confidence: 99%

See 2 more Smart Citations

Subdiffusion in a system consisting of two different media separated by a thin membrane

Kosztołowicz

2017

International Journal of Heat and Mass Transfer

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“…The choice of appropriate boundary conditions is crucial to achieve a suitable description of these complex phenomena in connection with the specific processes undergone by particles in the vicinity of the membrane, which involve also translocation through it [10][11][12]. Boundary conditions obtained from the experimental scenario also determine the dynamics of particles in the bulk and may lead to an anomalous diffusion which, frequently, manifest different diffusive regimes [13].…”

Section: Introductionmentioning

confidence: 99%

Anomalous diffusion and transport in heterogeneous systems separated by a membrane

et al. 2016

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Diffusion of particles in a heterogeneous system separated by a semipermeable membrane is investigated. The particle dynamics is governed by fractional diffusion equations in the bulk and by kinetic equations on the membrane, which characterizes an interface between two different media. The kinetic equations are solved by incorporating memory effects to account for anomalous diffusion and, consequently, non-Debye relaxations. A rich variety of behaviours for the particle distribution at the interface and in the bulk may be found, depending on the choice of characteristic times in the boundary conditions and on the fractional index of the modelling equations.

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Boundary conditions at a thin membrane for normal diffusion, classical subdiffusion, and slow subdiffusion processes

Kosztołowicz

Dutkiewicz

2020

Math Methods in App Sciences

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We consider three different diffusion processes in a system with a thin membrane: normal diffusion, classical subdiffusion, and slow subdiffusion. We conduct the considerations following the rule: If a diffusion equation is derived from a certain theoretical model, boundary conditions at a thin membrane should also be derived from this model with additional assumptions taking into account selective properties of the membrane. To derive diffusion equations and boundary conditions at a thin membrane, we use a particle random walk model in one-dimensional membrane system in which space and time variables are discrete. Then we move from discrete to continuous variables. We show that the boundary conditions depend on both selective properties of the membrane and a type of diffusion in the system.

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Random walk model of subdiffusion in a system with a thin membrane

Cited by 20 publications

References 41 publications

Subdiffusion in a system consisting of two different media separated by a thin membrane

Subdiffusion in a system consisting of two different media separated by a thin membrane

Anomalous diffusion and transport in heterogeneous systems separated by a membrane

Boundary conditions at a thin membrane for normal diffusion, classical subdiffusion, and slow subdiffusion processes

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