Max Planck Institute for Dynamics and Self-Organization -- Department for Nonlinear Dynamics and Network Dynamics Group
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Tuesday, 04.12.2007 17 c.t.

Growth of impenetrable particle's aggregates: Continuum description of a contact infection spread and DLA

by Dr. Eugene B. Postnikov
from Humboldt Universität Berlin/Kursk University, Russia

Contact person: Vitaly Belik


Seminarraum Haus 2, 4. Stock (Bunsenstr.)


The general topic of the presented talk is a continuous PDE-based description of an aggregate's growth via contact processes. The following examples are considered: The process of an epidemic spread in a population of individuals with low mobility within the SIR scheme and the process of Diffusion Limited Aggregation (a cluster formation by a sequential adhesion of randomly walking particles). Mathematically, both processes could be represented as the system of non-linear partial differetial equations with the diffusion coefficient linear in the density of susceptible individuals (random walkers/aggregated particles in the case of DLA). The solution gives propagating infection waves of asymmetric shapes, resembling Kendall waves observed in real infections. Moreover, it is shown that that the appropiate SIR-PDE-system allows a complete separation of variables. This provides the approximate expressions for the complete leading front and the tail of the infection wave for a wide range of parameters. In the case of DLA-growth, the considered approach allows to evaluate the fractal dimension using the solution of these mean-field equations. The results are in a quite good agreement with values found by the direct numerical simulations. The generalization on the case of the cluster description with different immiscible particles is also considered.

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