Efficient Monte Carlo Methods for Simulating Diffusion-Reaction Processes in Complex Systems

Grebenkov, Denis

doi:10.1142/9789814590297_0023

Cited by 9 publications

(13 citation statements)

References 79 publications

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“…Notice that for R c = 0, we retrieve the exact Eqs. (28) and (29). Let us now compare the approximate solution to previously known results in two limits R c → 1 and R c 1.…”

Section: Approximate Expression For the Mfptmentioning

confidence: 84%

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Exit Time Distribution in Spherically Symmetric Two-Dimensional Domains

et al. 2014

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The distribution of exit times is computed for a Brownian particle in spherically symmetric twodimensional domains (disks, angular sectors, annuli) and in rectangles that contain an exit on their boundary. The governing partial differential equation of Helmholtz type with mixed Dirichlet-Neumann boundary conditions is solved analytically. We propose both an exact solution relying on a matrix inversion, and an approximate explicit solution. The approximate solution is shown to be exact for an exit of vanishing size and to be accurate even for large exits. For angular sectors, we also derive exact explicit formulas for the moments of the exit time. For annuli and rectangles, the approximate expression of the mean exit time is shown to be very accurate even for large exits. The analysis is also extended to biased diffusion. Since the Helmholtz equation with mixed boundary conditions is encountered in microfluidics, heat propagation, quantum billiards, and acoustics, the developed method can find numerous applications beyond exit processes.

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“…Notice that for R c = 0, we retrieve the exact Eqs. (28) and (29). Let us now compare the approximate solution to previously known results in two limits R c → 1 and R c 1.…”

Section: Approximate Expression For the Mfptmentioning

confidence: 84%

“…To our knowledge, the results in Eqs. (27a), (27b), (28) and (29) are new. Last, as described in Fig.…”

Section: Exact Explicit Expression For the Mfpt In Angular Sectormentioning

confidence: 99%

Exit Time Distribution in Spherically Symmetric Two-Dimensional Domains

et al. 2014

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“…In turn, the intricate exploration of the partially reactive surface via diffusionmediated jumps in complicated structures such as multiscale porous media or domains with irregular or fractal boundaries, remains poorly understood. In this light, efficient numerical techniques such as fast Monte Carlo methods [82,67,68,83] or semi-analytical solutions [84,85] become particularly important.…”

Section: Conclusion: Is the Finite Reactivity Important?mentioning

confidence: 99%

Imperfect Diffusion-Controlled Reactions

Grebenkov

2019

Chemical Kinetics

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This chapter aims at emphasizing the crucial role of partial reactivity of a catalytic surface or a target molecule in diffusion-controlled reactions. We discuss various microscopic mechanisms that lead to imperfect reactions, the Robin boundary condition accounting for eventual failed reaction events, and the construction of the underlying stochastic process, the so-called partially reflected Brownian motion. We show that the random path to the reaction event can naturally be separated into the transport step toward the target, and the exploration step near the target surface until reaction. While most studies are focused exclusively on the transport step (describing perfect reactions), the exploration step, consisting is an intricate combination of diffusion-mediated jumps between boundary points, and its consequences for chemical reactions remain poorly understood. We discuss the related mathematical difficulties and recent achievements. In particular, we derive a general representation of the propagator, show its relation to the Dirichlet-to-Neumann operator, and illustrate its properties in the case of a flat surface.

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“…The presence of a boundary drastically changes the properties of Brownian motion (e.g., the reflecting boundary forces the process to remain inside the domain) so that earlier wavelet representations cannot directly incorporate the effect of the boundary. Since restricted diffusion is relevant for most physical, chemical and biological applications, various Monte Carlo methods have been developed for simulating this stochastic process, computing the related statistics (e.g., the first passage times [80]), and solving the underlying boundary value problems [81][82][83][84]. The slow convergence of Monte Carlo techniques (typically of the order of 1/ √ M in the number of trials) requires fast generation of Brownian paths.…”

Section: Restricted Diffusionmentioning

confidence: 99%

“…Due to its efficiency, fast random walk algorithms have been used to simulate diffusion-limited aggregates (DLA) [86,87], to generate the harmonic measure on fractals [78,88,89], to model diffusion-reaction phenomena in spherical packs [90,91], to compute the signal attenuation in pulsed-gradient spin-echo experiments [92,93], etc. In this section, we focus on multiscale tools to estimate the distance, while other aspects of fast random walk algorithms can be found elsewhere [84].…”

Section: Restricted Diffusionmentioning

confidence: 99%

A multiscale guide to Brownian motion

Belyaev

Jones

2015

J. Phys. A: Math. Theor.

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We revise the Lévy's construction of Brownian motion as a simple though still rigorous approach to operate with various Gaussian processes. A Brownian path is explicitly constructed as a linear combination of wavelet-based "geometrical features" at multiple length scales with random weights. Such a wavelet representation gives a closed formula mapping of the unit interval onto the functional space of Brownian paths. This formula elucidates many classical results about Brownian motion (e.g., non-differentiability of its path), providing intuitive feeling for non-mathematicians. The illustrative character of the wavelet representation, along with the simple structure of the underlying probability space, is different from the usual presentation of most classical textbooks. Similar concepts are discussed for fractional Brownian motion, Ornstein-Uhlenbeck process, Gaussian free field, and fractional Gaussian fields. Wavelet representations and dyadic decompositions form the basis of many highly efficient numerical methods to simulate Gaussian processes and fields, including Brownian motion and other diffusive processes in confining domains.

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Efficient Monte Carlo Methods for Simulating Diffusion-Reaction Processes in Complex Systems

Cited by 9 publications

References 79 publications

Exit Time Distribution in Spherically Symmetric Two-Dimensional Domains

Exit Time Distribution in Spherically Symmetric Two-Dimensional Domains

Imperfect Diffusion-Controlled Reactions

A multiscale guide to Brownian motion

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