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Series: Math Physics Seminar

TBA

Series: Math Physics Seminar

We introduce a quantum version of the Kac Master equation,and we explain issues like equilibria, propagation of chaos and the corresponding quantum Boltzmann equation. This is joint work with Eric Carlen and Maria Carvalho.

Series: Math Physics Seminar

We investigate a dynamical system consisting of $N$ particles moving on a $d$-dimensional torus under the action of an electric field $E$ with a Gaussian thermostat to keep the total energy constant. The particles are also subject to stochastic collisions which randomize direction but do not change the speed. We prove that in the van Hove scaling limit, $E\to 0$ and $t\to t/E^2$, the trajectory of the speeds $v_i$ is described by a stochastic differential equation corresponding to diffusion on a constant energy sphere.Our results are based on splitting the system's evolution into a ``slow'' process and an independent ``noise''. We show that the noise, suitably rescaled, converges to a Brownian motion. Then we employ the Ito-Lyons continuity theorem to identify the limit of the slow process.

Series: Math Physics Seminar

We will talk about discrete versions of the Bethe-Sommerfeld conjecture. Namely, we study the spectra of multi-dimensional periodic Schrödinger operators on various discrete lattices with sufficiently small potentials. In particular, we provide sharp bounds on the number of gaps that may perturbatively open, we characterize those energies at which gaps may open, and we give sharp arithmetic criteria on the periods that ensure no gaps open. We will also provide examples that open the maximal number of gaps and estimate the scaling behavior of the gap lengths as the coupling constant goes to zero. This talk is based on a joint work with Svetlana Jitomirskaya and another work with Jake Fillman.

Series: Math Physics Seminar

Electrons possess both spin and charge. In one dimension, quantum theory predicts that systems of interacting electrons may behave as though their charge and spin are transported at different speeds.We discuss examples of how such many-particle effects may be simulated using neutral atoms and radiation fields. Joint work with Xiao-Feng Shi

Series: Math Physics Seminar

Localization properties of quantum many-body systems have been a very active subject in theoretical physics in the most recent decade. At the same time, finding rigorous approaches to understanding many-body localization remains a wide open challenge. We will report on some recent progress obtained for the case of quantum spin chains, where joint work with A. Elgart and A. Klein has provided a proof of several manifestations of MBL for the droplet spectrum of the disordered XXZ chain.

Series: Math Physics Seminar

TBA

Series: Math Physics Seminar

This talk will be focused on the large deviation theory (LDT) for Schr\"odinger cocycles over a quasi-periodic or skew-shift base. We will also talk about its connection to positivity and regularity of the Lyapunov exponent, as well as localization. We will also discuss some open problems of the skew-shift model.

Series: Math Physics Seminar

Consider a relativistic electron interacting with a nucleus of nuclear charge Z and coupled to its self-generated electromagnetic field. The resulting system of equations describing the time evolution of this electron and its corresponding vector potential are known as the Maxwell-Dirac-Coulomb (MDC) equations. We study the time local well-posedness of the MDC equations, and, under reasonable restrictions on the nuclear charge Z, we prove the existence of a unique local in time solution to these equations.

Series: Math Physics Seminar

Recent advances in fluid dynamics reveal that the recurrent flows observed in moderate Reynolds number turbulence result from close passes to unstable invariant solutions of Navier-Stokes equations. By now hundreds of such solutions been computed for a variety of flow geometries, but always confined to small computational domains (minimal cells).Pipe, channel and plane flows, however, are flows on infinite spatial domains. We propose to recast the Navier-Stokes equations as a space-time theory, with the unstable invariant solutions now being the space-time tori (and not the 1-dimensional periodic orbits of the classical periodic orbit theory). The symbolic dynamics is likewise higher-dimensional (rather than a single temporal string of symbols). In this theory there is no time, there is only a repertoire of admissible spatiotemporal patterns.We illustrate the strategy by solving a very simple classical field theory on a lattice modelling many-particle quantum chaos, adiscretized screened Poisson equation, or the ``spatiotemporal cat.'' No actual cats, graduate or undergraduate, have showninterest in, or were harmed during this research.