I will show that each metamaterial has a groupoid behind it and that all physically sound Hamiltonians can be generated from the left-regular representation of the groupoid’s C*-algebra. The K-theories of this groupoid algebra classify the various dynamical features that can be observed in the metamaterial. Furthermore, in the presence of defects, the space of units of the groupoid displays closed invariant subspaces leading to a decomposition of this unit space that prompts and exact sequence of C*-algebras. The so called bulk-defect correspondence principle can be formulated in terms of this exact sequence and a specific Kasparov product. Furthermore, since exact sequences of C*-algebras are classified by certain Kasparov groups, we can use Kasparov’s theory to investigate and classify geometric defects of metamaterials. Examples will be provided.
Metamaterials by groupoid methods and Kasparov’s K-theory
Nov. 03, 2022 3:35pm (Virtual)
Probability
Jin Feng (University of Kansas)
X
The deterministic Carleman equation can be considered as an one dimensional two speed fictitious gas model. Its associated (2 scale) hydrodynamic limit gives a nonlinear heat equation. The first rigorous derivation of such limit was given by Kurtz in 1973. In this talk, starting from a more refined stochastic model giving the Carleman equation as mean field, we derive a macroscopic fluctuation structure associated with the hydrodynamic limit.
The purpose of such derivation is to illustrate a new, Hamiltonian-based, functional analytic and variational approach for the understanding of hydrodynamics limit program in general.
A large deviation result is established through an abstract Hamilton-Jacobi method applied to this specific setting. The principle idea is to identify a two scale averaging structure in the context of Hamiltonian convergence in the space of probability measures. This is achieved through 1). A change of coordinate to the density-flux description of the problem; 2). Extending the weak KAM theory (in Hamiltonian dynamical system) to infinite particle context for explicitly identifying effective Hamiltonian; 3). Developing results for a new class of PDEs: Hamilton-Jacobi equation in the space of measures.
This is a joint work with Toshio Mikami and Johannes Zimmer.