When molecules are contained in interstices or pores or blocked by tissue walls, their
motion is hindered by the restricting geometry. Restricted motion of molecules in complex
structures is important in many areas of science, engineering and medicine. Examples
can be taken from biology (nutrient transport, perfusion, membrane function, blood flow),
catalysis, foodstuff, materials (concrete, cement, polymer networks, self-organizing
materials), and geology (fluid movement in hydrocarbon reservoirs, ground water
migration, contamination). Transport and relaxation of signal in restricted geometry are
controlled by eigenvalues and eigenfunctions of the Laplacain in the restricted Geometry
with partially absorbing boundaries. The local geometry leaves a finger-print, so to say, on
the diffusion coefficient, disperson coefficient and signal relaxation rate. For example the
diffusion coefficient changes from the putative bulk time independent value D0 to a time
dependent D(t). In this talk we focus on two problems i) Certain short-time asymptotics
are independent of details of the eigenvalues and eigenfunctions, but depends on the
geometrical features such as surface to volume ratio. ii) We describe how to solve the
problem of Taylor dispersion in the presence of partially absorbing boundaries using an
exact stochastic formulation in terms of the transverse diffusion eigenmodes. The
cumulants grow asymptotically linearly with time ensuring a Gaussian distribution in the
long-time limit. The effective velocity and skewness are enhanced while Taylor dispersion
is suppressed due to absorption at the boundary.
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