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Homogenization of lateral diffusion on a random surface

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posted on 2023-06-09, 06:42 authored by Andrew B Duncan
We study the problem of lateral diffusion on a static, quasi-planar surface generated by a stationary, ergodic random field possessing rapid small-scale spatial fluctuations. The aim is to study the effective behavior of a particle undergoing Brownian motion on the surface, viewed as a projection on the underlying plane. By formulating the problem as a diffusion in a random medium, we are able to use known results from the theory of stochastic homogenization of SDEs to show that, in the limit of small scale fluctuations, the diffusion process behaves quantitatively like a Brownian motion with constant diffusion tensor D. In one dimension, the effective diffusion coefficient is given by 1/Z2, where Z is the average line element of the surface. In two-dimensions, D will not have a closed-form expression in general. However, we are able to derive variational bounds for the effective diffusion tensor. Moreover, in the special case when D is isotropic, we show that D = 1/Z I, where Z is the average area element of the random surface. We also describe a numerical scheme for approximating the effective diffusion tensor and illustrate this scheme with three examples.

History

Publication status

  • Published

File Version

  • Published version

Journal

Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal

ISSN

1540-3459

Publisher

SIAM

Issue

4

Volume

13

Page range

1478-1506

Department affiliated with

  • Mathematics Publications

Research groups affiliated with

  • Probability and Statistics Research Group Publications

Full text available

  • Yes

Peer reviewed?

  • Yes

Legacy Posted Date

2017-06-15

First Open Access (FOA) Date

2017-06-15

First Compliant Deposit (FCD) Date

2017-06-15

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