Diffusion of FiniteSize Particles in Confined Geo

Authors: Maria Bruna S Jonathan Chapman

Publish Date: 2013/05/10

Volume: 76, Issue: 4, Pages: 947-982

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Abstract

The diffusion of finitesize hardcore interacting particles in two or threedimensional confined domains is considered in the limit that the confinement dimensions become comparable to the particle’s dimensions The result is a nonlinear diffusion equation for the oneparticle probability density function with an overall collective diffusion that depends on both the excludedvolume and the narrow confinement By including both these effects the equation is able to interpolate between severe confinement for example singlefile diffusion and unconfined diffusion Numerical solutions of both the effective nonlinear diffusion equation and the stochastic particle system are presented and compared As an application the case of diffusion under a ratchet potential is considered and the change in transport properties due to excludedvolume and confinement effects is examinedThis publication was based on work supported in part by Award No KUKC101304 made by King Abdullah University of Science and Technology KAUST MB acknowledges financial support from EPSRC We are grateful to the organizers of the workshop “Stochastic Modelling of ReactionDiffusion Processes in Biology” which has led to this Special IssueThis appendix is devoted to the derivation of 10 in the twodimensional channel NC2 case The derivation of the threedimensional cases NC3 and PP or other simple geometries follows similarly see Sect A3 for an outline of the conditions/calculations to be carried outSketch of the original channel geometry solid black lines and the effective configuration space for the center of a second circular particle given by the boundary function partial varOmegabf x 1 dash red lines which depends on the position of the first particle bf x 1 The collision boundary function mathcalC bf x 1 forms part of the effective boundary partialvarOmegabf x 1 Color figure onlineEquation 31 is halfway through transformation mathcalT 1 cf Fig 1 since the first half of the equation depends only on bf x 1 while the integral mathcalI still depends on the twoparticle density P near the collision surface mathcalC bf x 1At this stage it is common to use a closure approximation such as Pbf x 1bf x 2t = pbf x 1t pbf x 2t to evaluate mathcalI and obtained a closed equation for p Rubinstein and Keller 1989 However the pairwise particle interaction—and therefore the correlation between their positions—is exactly localized near the collision surface mathcalC bf x 1 Instead in the next section we will use an alternative method based on matched asymptotic expansions to evaluate mathcalI systematically Bruna and Chapman 2012bWe suppose that when two particles are far apart x 1−x 2≫1 they are independent at leading order whereas when they are close to each other x 1−x 2∼ϵ they are correlated We denote these two regions of the configuration space varOmega epsilon2 the outer region and the inner region respectively We use the xcoordinate to distinguish between the two regions because the inner region spans the channel’s cross section Importantly this implies that the outer region is disconnected

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