A New Derivation of Robin Boundary Conditions through Homogenization of a Stochastically Switching Boundary

Lawley, Sean D.; Keener, James P.

doi:10.1137/15m1015182

Cited by 49 publications

(43 citation statements)

References 23 publications

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“…The reactivity is thus related to the probability of reaction event at the encounter [12][13][14]. Robin boundary condition with homogeneous (constant) reactivity κ was often employed to describe many chemical and biochemical reactions and permeation processes [15][16][17][18][19][20][21][22][23][24][25][26], to model stochastic gating [27][28][29][30], or to approximate the effect of microscopic heterogeneities in a random distribution of reactive sites [31][32][33] (see a recent overview in [34]). In spite of its practical importance, diffusion-controlled reactions with heterogeneous surface reactivity κ(s) remain much less studied.…”

Section: Introductionmentioning

confidence: 99%

Spectral theory of imperfect diffusion-controlled reactions on heterogeneous catalytic surfaces

Grebenkov

2019

The Journal of Chemical Physics

108

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We propose a general theoretical description of chemical reactions occurring on a catalytic surface with heterogeneous reactivity. The propagator of a diffusion-reaction process with eventual absorption on the heterogeneous partially reactive surface is expressed in terms of a much simpler propagator toward a homogeneous perfectly reactive surface. In other words, the original problem with general Robin boundary condition that includes in particular mixed Robin-Neumann condition, is reduced to that with Dirichlet boundary condition. Chemical kinetics on the surface is incorporated as a matrix representation of the surface reactivity in the eigenbasis of the Dirichlet-to-Neumann operator. New spectral representations of important characteristics of diffusion-controlled reactions, such as the survival probability, the distribution of reaction times, and the reaction rate, are deduced. Theoretical and numerical advantages of this spectral approach are illustrated by solving interior and exterior problems for a spherical surface that may describe either an escape from a ball or hitting its surface from outside. The effect of continuously varying or piecewise constant surface reactivity (describing, e.g., many reactive patches) is analyzed.

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Section: Introductionmentioning

confidence: 99%

Spectral theory of imperfect diffusion-controlled reactions on heterogeneous catalytic surfaces

Grebenkov

2019

The Journal of Chemical Physics

108

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“…Example 4 (Deriving Robin boundary and interface jump conditions). It was recently shown that the classical Robin boundary condition and interface jump condition can be derived as averages of certain switching conditions [23]. By analyzing the BVP given by Theorem 1 for the mean of the switching PDE in Example 2, it was shown that the mean of the solution with the switching condition converges to a solution with a Robin condition in a certain fast switching limit.…”

Section: Pde Examplesmentioning

confidence: 99%

“…where f ± := lim x→L/2± f (x). Starting with this BVP, it was proven that in a certain fast switching limit, the mean, E[u(x, t)], converges to the solution of the heat equation on [0, L] with an interface jump condition at x = L/2 [23].…”

Section: Pde Examplesmentioning

confidence: 99%

“…Several recent models impose randomly switching boundary conditions on either a partial differential equation (PDE) [8,22,23,25] or stochastic differential equation (SDE) [1,4,6,7,35,36]. These models appear in a diverse set of fields, including neuroscience, insect physiology, medicine, biochemistry, intermittent search, and in the derivation of classical objects in dynamical systems.…”

Section: Introductionmentioning

confidence: 99%

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Boundary Value Problems for Statistics of Diffusion in a Randomly Switching Environment: PDE and SDE Perspectives

Lawley¹

2016

SIAM J. Appl. Dyn. Syst.

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Abstract. Driven by diverse applications, several recent models impose randomly switching boundary conditions on either a PDE or SDE. The purpose of this paper is to provide tools for calculating statistics of these models and to establish a connection between these two perspectives on diffusion in a random environment. Under general conditions, we prove that the moments of a solution to a randomly switching PDE satisfy a hierarchy of BVPs with lower order moments coupling to higher order moments at the boundaries. Further, we prove that joint exit statistics for a set of particles following a randomly switching SDE satisfy a corresponding hierarchy of BVPs. In particular, the M -th moment of a solution to a switching PDE corresponds to exit statistics for M particles following a switching SDE. We note that though the particles are non-interacting, they are nonetheless correlated because they all follow the same switching SDE. Finally, we give several examples of how our theorems reveal the sometimes surprising dynamics of these systems.keywords. random PDE, stochastic hybrid system, piecewise deterministic Markov process, randomly switching boundary, switched dynamical system.

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“…This makes sense because it is very hard for neurotransmitter to escape a small region near L because as soon as it is released, the boundary conditions switch and it is reabsorbed. More generally, fast switching between Dirichlet and Neumann always becomes pure Dirichlet if the proportion of time in each state is fixed [15]. This phenomenon can be understood in terms of the mean absorption time of a Brownian motion to a switching boundary.…”

Section: 1mentioning

confidence: 99%

Neurotransmitter concentrations in the presence of neural switching in one dimension

Lawley¹,

Best²,

Reed³

2016

DCDS-B

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In volume transmission, neurons in one brain nucleus send their axons to a second nucleus where neurotransmitter is released into the extracellular space. One would like methods to calculate the average amount of neurotransmitter at different parts of the extracellular space, depending on neural properties and the geometry of the projections and the extracellular space. This question is interesting mathematically because the neuron terminals are both the sources (when they are firing) and the sinks (when they are quiescent) of neurotransmitter. We show how to formulate the questions as boundary value problems for the heat equation with stochastically switching boundary conditions. In one space dimension, we derive explicit formulas for the average concentration in terms of the parameters of the problems in two simple prototype examples and then explain how the same methods can be used to solve the general problem. Applications of the mathematical results to the neuroscience context are discussed.2010 Mathematics Subject Classification. Primary: 35R60, 60H15; Secondary: 92C20.

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A New Derivation of Robin Boundary Conditions through Homogenization of a Stochastically Switching Boundary

Cited by 49 publications

References 23 publications

Spectral theory of imperfect diffusion-controlled reactions on heterogeneous catalytic surfaces

Spectral theory of imperfect diffusion-controlled reactions on heterogeneous catalytic surfaces

Boundary Value Problems for Statistics of Diffusion in a Randomly Switching Environment: PDE and SDE Perspectives

Neurotransmitter concentrations in the presence of neural switching in one dimension

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