Diffusion through permeable interfaces: Fundamental equations and their application to first-passage and local time statistics

Kay, Toby; Giuggioli, Luca

doi:10.1103/physrevresearch.4.l032039

Cited by 20 publications

(24 citation statements)

References 40 publications

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“…It is instructive to relate the above probabilistic model of single-particle diffusion through a semi-permeable barrier to a recent study based on a Fokker–Planck description [20]. The latter was derived by taking a continuum limit of a continuous-time random walk model with a defect.…”

Section: The Snapping Out Brownian Motionmentioning

confidence: 99%

A probabilistic model of diffusion through a semi-permeable barrier

Bressloff

2022

Proc. R. Soc. A.

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Diffusion through semi-permeable structures arises in a wide range of processes in the physical and life sciences. Examples at the microscopic level range from artificial membranes for reverse osmosis to lipid bilayers regulating molecular transport in biological cells to chemical and electrical gap junctions. There are also macroscopic analogues such as animal migration in heterogeneous landscapes. It has recently been shown that one-dimensional diffusion through a barrier with constant permeability κ 0 is equivalent to snapping out Brownian motion (BM). The latter sews together successive rounds of partially reflecting BMs that are restricted to either the left or the right of the barrier. Each round is killed when its Brownian local time exceeds an exponential random variable parameterized by κ 0 . A new round is then immediately started in either direction with equal probability. In this article, we use a combination of renewal theory, Laplace transforms and Green’s function methods to show how an extended version of snapping out BM provides a general probabilistic framework for modelling diffusion through a semi-permeable barrier. This includes modifications of the diffusion process away from the barrier (e.g. stochastic resetting) and non-Markovian models of membrane absorption that kill each round of partially reflected BM. The latter leads to time-dependent permeabilities.

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Section: The Snapping Out Brownian Motionmentioning

confidence: 99%

A probabilistic model of diffusion through a semi-permeable barrier

Bressloff

2022

Proc. R. Soc. A.

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“…Another possible extension could be to discrete heterogeneous diffusive processes with resetting, for which the continuous limits with position-dependent diffusion coefficient have been recently analysed [85]. One may also study the discrete resetting dynamics in presence of permeable interface, for which a fundamental diffusion equation in the continuous limits has been derived from microscopic description very recently [86]. using a sum of terms as follows The first term on the right-hand side of Eq.…”

Section: Discussionmentioning

confidence: 99%

Discrete space-time resetting model: Application to first-passage and transmission statistics

Das¹,

Giuggioli²

2022

Preprint

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We consider the dynamics of lattice random walks with resetting. The walker moving randomly on a lattice of arbitrary dimensions resets at every time step to a given site with a constant probability r. We construct a discrete renewal equation and present closed-form expressions for different quantities of the resetting dynamics in terms of the underlying reset-free propagator or Green's function. We apply our formalism to the biased random walk dynamics in one-dimensional unbounded space and show how one recovers in the continuous limits results for diffusion with resetting. The resetting dynamics of biased random walker in one-dimensional domain bounded with periodic and reflecting boundaries is also analyzed. Depending on the bias the first-passage probability in periodic domain shows multi-fold non-monotonicity as r is varied. Finally, we apply our formalism to study the transmission dynamics of two lattice walkers with resetting in one-dimensional domain bounded by periodic and reflecting boundaries. The probability of a definite transmission between the walkers shows non-monotonic behavior as the resetting probabilities are varied.

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“…It is instructive to relate the above probabilistic model of single-particle diffusion through a semi-permeable barrier to a recent study based on a Fokker-Planck description [18]. The latter was derived by taking a continuum limit of a continuous-time random walk model with a defect.…”

Section: Thin-layer Approximationmentioning

confidence: 99%

“…Finally, taking the limit a → 0 + recovers the permeable barrier boundary condition. Moreover, equation (2.28) is equivalent to the FPE description derived in [18].…”

Section: Thin-layer Approximationmentioning

confidence: 99%

“…This has motivated a number of studies of random walks on lattices in which semipermeable barriers are represented by local defects [16,17]. Moreover, a Fokker-Planck description of single-particle diffusion through a semiperrmeable membrane has recently been derived by taking an appropriate continuum limit of a random walk model [18]. An alternative approach to modeling single-particle diffusion is to use stochastic differential equations (SDEs).…”

Section: Introductionmentioning

confidence: 99%

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A probabilistic model of diffusion through a semi-permeable barrier

Bressloff¹

2022

Preprint

View full text Add to dashboard Cite

Diffusion through semipermeable structures arises in a wide range of processes in the physical and life sciences. Examples at the microscopic level range from artificial membranes for reverse osmosis to lipid bilayers regulating molecular transport in biological cells to chemical and electrical gap junctions. There are also macroscopic analogs such as animal migration in heterogeneous landscapes. It has recently been shown that one-dimensional diffusion through a barrier with constant permeability κ0 is equivalent to snapping out Brownian motion (BM). The latter sews together successive rounds of partially reflecting BMs that are restricted to either the left or right of the barrier. Each round is killed when its Brownian local time exceeds an exponential random variable parameterized by κ0. A new round is then immediately started in either direction with equal probability. In this paper we use a combination of renewal theory, Laplace transforms and Green's function methods to show how an extended version of snapping out BM provides a general probabilistic framework for modeling diffusion through a semipermeable barrier. This includes modifications of the diffusion process away from the barrier (eg. stochastic resetting) and non-Markovian models of membrane absorption that kill each round of partially reflected BM. The latter leads to timedependent permeabilities.

show abstract

Diffusion through permeable interfaces: Fundamental equations and their application to first-passage and local time statistics

Cited by 20 publications

References 40 publications

A probabilistic model of diffusion through a semi-permeable barrier

A probabilistic model of diffusion through a semi-permeable barrier

Discrete space-time resetting model: Application to first-passage and transmission statistics

A probabilistic model of diffusion through a semi-permeable barrier

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