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When a flux of Brownian particles is injected in a narrow window located onthe surface of a bounded domain, these particles diffuse and can eventuallyescape through a cluster of narrow windows. At steady-state, we computeasymptotically the distribution of concentration between the different windows.The solution is obtained by solving Laplace's equation using Green's functiontechniques and second order asymptotic analysis, and depends on the influxamplitude, the diffusion properties as well as the geometrical organization ofall the windows, such as their distances and the mean curvature. We explore therange of validity of the present asymptotic expansions using numericalsimulations of the mixed boundary value problem. Finally, we introduce a lengthscale to estimate how deep inside a domain a local diffusion current canspread. We discuss some applications in biophysics.

Modeling and asymptotic analysis of the concentration difference in a nanoregion between an influx and outflux diffusion across narrow windows