The University of Arizona

The Principle of G-Equivariant Universality

The Principle of G-Equivariant Universality

Series: Analysis, Dynamics, and Applications Seminar
Location: Math 102
Presenter: Stephen J. Watson, Department of Mathematics, University of Glasgow

The statistical physics governing phase-ordering dynamics following a symmetry breaking first-order phase transition is an area of active research. The Coarsening/Ageing of the ensemble of phase domains, wherein  irreversible annihilation or joining of domains yields a growing characteristic domain length, is a omniprescent feature whose universal characteristics one would wish to understand. Driven kinetic Ising models and growing nano-faceted crystals are theoretically important examples of such Coarsening (Ageing) Dynamical Systems (CDS), since they additionally break thermodynamic fluctuation-dissipation relations.  Power-laws for the growth in time of the characteristic size of domains (e.g., lengths) of  CDS, and a concomitant scale-invariance of the associated length distributions, has so frequently been empirically observed that their presence has acquired the status of a principle; the so-called Dynamic-Scaling Hypothesis.  But the dynamical symmetries of a given CDS- its Coarsening Group $G$ - may include more than the global spatio-temporal scalings underlying the Dynamic Scaling Hypothesis.  In this talk, I will present a recently developed theoretical framework (Ref.[1]) that shows how the symmetry group G of a Coarsening (ageing) Dynamical System (CDS) necessarily yields G-equivariance (covariance) of the CDS's  universal statistical observables. We exhibit this theory for a variety of model systems, of both thermodynamic and driven type, with symmetries that may also be emergent (Ref. [2,3]) and/or hidden. We will close with a remarkable theoretical coarsening law that combines Lorentzian and Parabolic symmetries.

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