The hard sphere model is a simple to define and long studied mathematical model of gas, in which the only interactions are the hard-core constraint that two spheres cannot overlap in space. In three dimensions it is expected to exhibit a gas-to-crystal phase transition. Despite its simplicty, rigorous results on the model are rather sparse. I will introduce the model, discuss some of the main open questions, and present some results new and old.
We rigorously prove the filamentation phenomenon for a class of weak solutions to the Euler equations known as vortex caps. Vortex caps are characteristic functions representing time-evolving sets of Lagrangian type, with energy preserved at all times. The filamentation of vortex caps is characterized by L^1 -stability alongside unbounded growth of the perimeter of their interfaces. We recall the existence and stability results for vortex caps on the sphere, based on Yudovich theory.
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.
Random matrix products perhaps among some of the most extensively studied examples of random dynamical systems, and moreover are central to the study of one-dimensional disordered systems. We discuss recent results by the author (joint with S.
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 so
The work deals with the existence of solutions of an integro-differential equation in the case of the normal diffusion and the influx/efflux term proportional to the Dirac delta function in the presence of the drift term. The proof of the existence of solutions relies on a fixed point technique. We use the solvability conditions for the non-Fredholm elliptic operators in unbounded domains and discuss how the introduction of the transport term influences the regularity of the solutions.
The Ginzburg-Landau model is a phenomenological description of superconductivity. A key feature of type-II superconductors is the emergence of singularities, known as vortices, which occur when the external magnetic field exceeds the first critical field. Determining the location and number of these vortices is crucial. Furthermore, the presence of impurities in the material can influence the configuration of these singularities.
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.
As is well known, many materials freeze at low temperatures. Microscopically,
this means that their molecules form a phase where there is long range order
in their positions. Despite their ubiquity, proving that these freezing
transitions occur in realistic microscopic models has been a significant
challenge, and it remains an open problem in continuum models at positive
temperatures. In this talk, I will focus on lattice particle models, in which
