Pattern formation in particle systems: from spherical shells to regular simplices
speaker: Robert McCann (University of Toronto)abstract: Flocking and swarming models which seek to explain pattern formation in mathematical biology
often assume that organisms interact through a force which is attractive over large distances yet
repulsive at short distances. Suppose this force is given as a difference of power laws
and normalized so that its unique minimum occurs at unit separation.
We detail a phase transition in the mildly repulsive range of exponents, which separates a region
where the minimum energy configuration is uniquely attained by a uniform distribution of organisms over a spherical shell,
from a region in which it is uniquely attained --- apart from translations and rotations --- by equidistributing
the organisms over the vertices of a regular top-dimensional simplex (i.e. an equilateral triangle in two dimensions and regular tetrahedron in three).
We explore the sense in which such configurations are stable fixed points for the dynamics
of the aggregation equation, also known as the $2$-Wasserstein gradient flow of the associated interaction energy.
Based on work with Cameron Davies (University of Toronto) and Tongseok Lim (of Purdue's Krannert School of Management) 7879 at http:/www.math.toronto.edumccannpublications
timetable:
Mon 24 Oct, 9:00 - 9:45, Aula Magna Bruno Pontecorvo
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