Memory Driven Pattern Formation

Details Memory Driven Pattern Formation Memory Driven Pattern Formation

2004-06-11

arxiv.org/abs/cond-mat/0406269v1

Physics

Condensed Matter

Statistical Mechanics

14 pages, 12 figures

Scientific paper

10.1016/j.physleta.2004.09.018

The diffusion equation is extended by including spatial-temporal memory in such a manner that the conservation of the concentration is maintained. The additional memory term gives rise to the formation of non-trivial stationary solutions. The steady state pattern in an infinite domain is driven by a competition between conventional particle current and a feedback current. We give a general criteria for the existence of a non-trivial stationary state. The applicability of the model is tested in case of a strongly localized, time independent memory kernel. The resulting evolution equation is exactly solvable in arbitrary dimensions and the analytical solutions are compared with numerical simulations. When the memory term offers an spatially decaying behavior, we find also the exact stationary solution in form of a screened potential.

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