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 International Conference on Complex Systems (ICCS2007)

Multi-scale diffusions on biological interfaces

Fabrice Debbasch
ERGA/LERMA Universite Paris 6

Claire Chevalier
ERGA/LERMA Universite Paris 6

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     Last modified: September 14, 2007

Abstract
Lateral diffusions on 2D interfaces play an important role in many
biological phenomena. In practice, the geometry of an interface $I$
can only be determined with a certain finite spatial and temporal
resolution, which separates the so-called large scales from the small
ones. The interface is then usually modelled by a surface $S$ whose
geometry varies only on large scales, and the large
scale aspects of lateral diffusions on the interface $I$ are expected
to be captured by the large scale aspects of stochastic processes on
the surface $S$. In other words, one assumes that small scale
variations of the geometry do not significantly influence the large
scale aspects of diffusions.

This assumption is however invalid because the coupling between
geometry and diffusions is non-linear. We first present the general
tools necessary to a multi-scale comparison of Brownian motions in
different geometries. These tools are then applied to study diffusions
on nearly flat surfaces whose geometries fluctuate at small, non
observable scales only. We prove by an explicit perturbative
calculation that, generically, the relative density differences
between Brownian motions on these surfaces and the corresponding
Brownian motions on a plane increase exponentially with time at both
small and large scales.
This is a memory effect and geometry fluctuations thus have
generically a cumulative influence on diffusions at all scales. The
dependence of the associated blow-up time on the scale separation and
on the amplitude of the geometry fluctuations is also discussed.







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