The Mean Escape Time for a Narrow Escape Problem with Multiple Switching Gates

Ammari, Habib; Garnier, Josselin; Kang, Hyeonbae; Lee, Hyundae; Sølna, Knut

doi:10.1137/100817103

Cited by 15 publications

(16 citation statements)

References 22 publications

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“…As we explained above, the average NET is smallest when arc length s between two gates is around π. Meanwhile, Fig 2 and Fig 3 both suggest that the average NET decrease as the switching rate k between the two gates increases, this is consistent with known theoretical result [11]. Each curve in Fig 4 predicts an decrease in the average NET with angular velocity w. We plot here for switching rate k = 0.1 between the two gates, for 0 < k < 1, it has similar trends.…”

Section: Resultssupporting

confidence: 87%

Mean escape time for randomly switching narrow gates in a steady flow

Wang

Duan

Geng

et al. 2020

Computers & Mathematics with Applications

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The escape of particles through a narrow absorbing gate in confined domains is a abundant phenomenon in various systems in physics, chemistry and molecular biophysics. We consider the narrow escape problem in a cellular flow when the two gates randomly switch between different states with a switching rate k between the two gates. After briefly deriving the coupled partial differential equations for the escape time through two gates, we compute the mean escape time for particles escaping from the gates with different initial states. By numerical simulation under nonuniform boundary conditions, we quantify how narrow escape time is affected by the switching rate k between the two gates, arc length s between two gates, angular velocity w of the cellular flow and diffusion coefficient D. We reveal that the mean escape time decreases with the switching rate k between the two gates, angular velocity w and diffusion coefficient D for fixed arc length, but takes the minimum when the two gates are evenly separated on the boundary for any given switching rate k between the two gates. In particular, we find that when the arc length size ε for the gates is sufficiently small, the average narrow escape time is approximately independent of the gate arc length size. We further indicate combinations of system parameters (regions located in the parameter space) such that the mean escape time is the longest or shortest. Our findings provide mathematical understanding for phenomena such as how ions select ion channels and how chemicals leak in annulus ring containers, when drift vector fields are present.

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Section: Resultssupporting

confidence: 87%

Mean escape time for randomly switching narrow gates in a steady flow

Wang

Duan

Geng

et al. 2020

Computers & Mathematics with Applications

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“…For example, several works-often in the context of biochemical reactions-study the escape of diffusing molecules in the presence of a switching boundary in which the molecules can exit only when the boundary is in a particular state [5], [6], [16], [23], [24]. Similarly, other works study the escape of diffusing molecules when the molecules themselves switch states and can exit only in a particular state [1], [4], [19], [20], [22]. In addition, the membrane voltage fluctuations of a single neuron due to diffusion of ions through stochastically gated channels continues to be an important problem [13], [17].…”

mentioning

confidence: 99%

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

Lawley¹,

Keener²

2015

SIAM J. Appl. Dyn. Syst.

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We give a new derivation of Robin boundary conditions and interface jump conditions for the diffusion equation in one dimension. To derive a Robin boundary condition, we consider the diffusion equation with a boundary condition that randomly switches between a Dirichlet and a Neumann condition. We prove that, in the limit of infinitely fast switching rate with the proportion of time spent in the Dirichlet state, denoted by ρ, approaching zero, the mean of the solution satisfies a Robin condition, with conductivity parameter determined by the rate at which ρ approaches zero. We carry out a similar procedure to derive an interface jump condition by considering the diffusion equation with a no flux condition in the interior of the domain that is randomly imposed/removed. Our results also provide the effective deterministic boundary condition for a randomly switching boundary.

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“…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%

“…We generalize this result by using different techniques. Exit statistics for a single diffusing particle in a domain with switching boundary conditions are computed in [1,6,7,35,36]. Hitting probabilities for multiple particles are derived in [8] for a specific example.…”

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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The Mean Escape Time for a Narrow Escape Problem with Multiple Switching Gates

Cited by 15 publications

References 22 publications

Mean escape time for randomly switching narrow gates in a steady flow

Mean escape time for randomly switching narrow gates in a steady flow

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

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

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