# The universal behavior of modulated stripe patterns

When |
Sep 13, 2022
from 11:00 to 12:00 |
---|---|

Where | Salle des thèses |

Attendees |
Alan C. Newell |

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Natural patterns arise when microscopic systems are driven far from equilibrium by some external stress. At a critical threshold, some, but not all, of the symmetries of the underlying system are broken. For example, natural striped patterns occur in stressed systems with continuous translational and rotational symmetries. At the phase transition, the former but not the latter symmetry is broken. In large boxes, the resulting pattern is a mosaic of patches of stripes of almost constant wavelength with different orientations meeting and melding along line and point defects. For a broad class of microscopic gradient systems, the evolution of the macroscopic order parameter, the gradient of the pattern phase, including the far-field behaviors of all the defects, is captured by a universal and canonical equation, the phase diffusion equation, derived from an averaged energy functional consisting of coordinate invariant combinations of the metric and curvature two-forms of the phase surface. Moreover, because of a nontrivial property of the map from physical to order-parameter space, the evolution equation can be (almost) linearized. The defects are also canonical. In two dimensions, they are concave and convex disclinations with topological indices of integer multiples of 1/2. In three dimensions, they are loops with concave and convex cross sections with integer multiples of 1/2,1/3 and 1, pattern quarks, and leptons. It is remarkable that a wide class of systems enjoying only the simplest of symmetries can, when stressed, evolve objects with fractional charges.