Georgia Institute of Technology College of Sciences

We study differential inclusions of the type $A v=0$ and $v \in K$, where $v$ is a vector field satisfying a linear PDE system $A$ and $K$ is a compact set. We are particularly interested in the case when $K$ consists of two vectors (\textit{two-state problem}). We consider Dirichlet boundary conditions for $v$, in which case the differential inclusion typically has no solutions. We study a suitable relaxation of the system, in which we penalize the surface energy required to switch between the two states. We study the asymptotics of the regularized energy integral. We show that the asymptotics depend polynomially on the regularization parameter with a quantification which — somewhat surprisingly — depends on the order of the linear PDE system $A$. Joint work with A. R\”{u}land, C. Tissot, A. Tribuzio.

Programmable matter explores how collections of computationally limited agents acting locally and asynchronously can achieve some useful coordinated behavior.  We take a stochastic approach using techniques from randomized algorithms and statistical physics to develop distributed algorithms for emergent collective behaviors that give guarantees and are robust to failures.  By analyzing the Gibbs distribution of various fixed-magnetization models from equilibrium statistical mechanics, we show that particles moving stochastically according to local affinities can solve various useful collective tasks. Finally, we will briefly introduce new tools that may prove fruitful in nonequilibrium settings as well.

I will present joint work in progress with Gero Friesecke.  We introduce a two parameter family of variational problems; varying the first parameter interpolates between a regularized version of Monge's optimal transport (OT) problem and Kantorovich's relaxed version.  The first limit problem has the advantage over Monge's original problem of always admitting a solution.  In cases where a (sufficiently regular) Monge map exists, the solution will be of such a form; if not, the limit problem essentially minimizes the transportation cost among all best approximations of the target measure by  pushforwards of the source.  When the source measure is discrete, we show that this is equivalent to the optimal quantization of the target measure, with the additional constraint that the weights of the approximating discrete masses are prescribed.  The second parameter controls the regularity of the pseudo-Monge map. In both the high and low regularity limits, the problem converges to the classical Kantorovich problem, under mild assumptions.

Part of the motivation for this problem is to understand whether the strictly correlated electron ansatz is valid in the semi-classical limit of density functional theory (DFT).  We will briefly discuss the corresponding application of OT to DFT, and outline what is known about the existence of Monge solutions (or, equivalently, the validity of the strictly correlated electron ansatz).

We consider the mean field Bose gas on the unit torus at temperatures proportional to the critical temperature of the Bose—Einstein condensation phase transition. We discuss trace norm convergence of the Gibbs state to a state given by a convex combination of quasi-free states. Two consequences of this relation are precise asymptotic formulas for the two-point function and the distribution of the number of particles in the condensate. A crucial ingredient of the proof is a novel abstract correlation inequality. This is joint work with Nam Panh Tanh and Marcin Napiorkowski.