|InterJournal Complex Systems, 2076
|Manuscript Number: |
Submission Date: 20080226
|Multi-scale diffusions on biological interfaces|
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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