On the simulation of diffusion processes close to boundaries

Green, N. J. B.

doi:10.1080/00268978800101871

Cited by 25 publications

(29 citation statements)

References 13 publications

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“…Their model assumes constant diffusion coefficient and vanishing drift, for which they find the reactive constant in terms of the absorbtion probability and the diffusion coefficient. Previous simulation schemes that recover the Robin boundary condition [1], [24], [25], [26], [27] make use of the explicit solution to the half space FPE with linear drift term and constant diffusion coefficient with a Robin condition. In [28, and references therein] the specular reflection method near a reflecting boundary has been shown to be superior to other methods, such as rejection, multiple rejection and interruption.…”

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confidence: 99%

Partially Reflected Diffusion

Singer

Schuss

Osipov

et al. 2008

SIAM J. Appl. Math.

109

120

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Abstract. The radiation (reaction, Robin) boundary condition for the diffusion equation is widely used in chemical and biological applications to express reactive boundaries. The underlying trajectories of the diffusing particles are believed to be partially absorbed and partially reflected at the reactive boundary, however, the relation between the reaction constant in the Robin boundary condition and the reflection probability is not well defined. In this paper we define the partially reflected process as a limit of the Markovian jump process generated by the Euler scheme for the underlying Itô dynamics with partial boundary reflection. Trajectories that cross the boundary are terminated with probability P √ ∆t and otherwise are reflected in a normal or oblique direction. We use boundary layer analysis of the corresponding master equation to resolve the non-uniform convergence of the probability density function of the numerical scheme to the solution of the Fokker-Planck equation in a half space, with the Robin constant κ. The boundary layer equation is of the Wiener-Hopf type. We show that the Robin boundary condition is recovered if and only if trajectories are reflected in the co-normal direction σn, where σ is the (possibly anisotropic) constant diffusion matrix and n is the unit normal to the boundary. Otherwise, the density satisfies an oblique derivative boundary condition. The constant κ is related to P by κ = rP √ σn, where r = 1/ √ π and σn = n T σn. The reflection law and the relation are new for diffusion in higher-dimensions.

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confidence: 99%

Partially Reflected Diffusion

Singer

Schuss

Osipov

et al. 2008

SIAM J. Appl. Math.

109

120

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“…The full RF simulations generate the diffusive trajectories of all the reactive particles involved using discrete time-step jumps (or flights) of normally distributed random numbers, this is equivalent to exact sampling of the trajectory implied by the use of a diffusion equation. The RF model (with Bessel bridge interpolation to determine whether encounter has taken place) 30 has been described in detail elsewhere, 4,8,31−33 and only certain aspects of it will be described here. This simulation method is used here as a measure of "reality" because it corresponds to an exact sampling of the many-body diffusion equation which is the basic model of the diffusion kinetics.…”

Section: Full Rf Simulationsmentioning

confidence: 99%

“…In this limit, it is straightforward to show analytically that the integral I(N 0 ) is independent of α. 30 The full dependence is shown in Figure 12…”

Section: General Casementioning

confidence: 99%

Scavenging and recombination kinetics in radiation chemistry

Al-Samra

Green

2017

Phys. Chem. Chem. Phys.

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This work describes stochastic models developed to study the competition between radical scavenging and recombination for simple model systems typical of radiation chemistry, where the reactive particles are tightly clustered and reactions are assumed fully diffusion limited. Three models are developed: a Monte Carlo random flights model with a periodic boundary condition for scavengers, Monte Carlo simulations in which the scavenging rate is calculated from the Smoluchowski theory for diffusion-limited reactions and a modification of the independent reaction times method where the scavengers close to the spur are explicitly included and the scavengers further away are treated as a continuum. The results indicate that the Smoluchowski theory makes a systematic overestimate of the scavenging rate when such competition is present. A correction for the Smoluchowski rate constant is suggested, an analytical justification is presented and it is tested against the simulations, and shown to be a substantial improvement.

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“…Therefore, the particles that did encounter each other must not be propagated according to the free-space Green's function, but the propagation needs to take into account the reflective encounter, cp. [23,22,25,26,18,30,29]. In a naive BD simulation, reflecting bcs are frequently incorporated in an analogous manner to absorbing bcs: The terminal positions of only those particles that overlap after a time step ∆t are reset to the boundary r ∈ ∂S [29].…”

Section: Propagationmentioning

confidence: 99%

“…We emphasize that the general idea to use Green's functions of Fokker-Planck equations to enhance BD simulations is far from new, cp. [23,22,25,26,18,30,11,29]. Indeed, the method has been proposed for the case of purely absorbing boundary conditions and irreversible reactions already in [6] .…”

Section: Introductionmentioning

confidence: 99%

Coarse-Grained Stochastic Particle-based Reaction-Diffusion Simulation Algorithm

Prüstel,

Meier-Schellersheim

2011

Preprint

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On the simulation of diffusion processes close to boundaries

Cited by 25 publications

References 13 publications

Partially Reflected Diffusion

Partially Reflected Diffusion

Scavenging and recombination kinetics in radiation chemistry

Coarse-Grained Stochastic Particle-based Reaction-Diffusion Simulation Algorithm

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