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Many body quantum dynamics and nonlinear dispersive PDE

Introductory Workshop: Randomness and long time dynamics in nonlinear evolution differential equations August 24, 2015 - August 28, 2015

August 26, 2015 (11:30 AM PDT - 12:30 PM PDT)
Speaker(s): Natasa Pavlovic (University of Texas, Austin)
Location: MSRI: Simons Auditorium
Primary Mathematics Subject Classification No Primary AMS MSC
Secondary Mathematics Subject Classification No Secondary AMS MSC
Video

14344

Abstract

The derivation of nonlinear dispersive PDE, such as the nonlinear Schr\"{o}dinger (NLS) from many body quantum dynamics is a central topic in mathematical physics, which has been approached by many authors in a variety of ways. In particular, one way to derive NLS is via the Gross-Pitaevskii (GP) hierarchy, which is an infinite system of coupled linear non-homogeneous PDE that describes the dynamics of a gas of infinitely many interacting bosons, while at the same time retains some of the features of a dispersive PDE.

 

In these lectures we will discuss the process of going from a quantum many body system of bosons to the NLS via the GP. The most involved part in such a derivation of NLS consists in establishing uniqueness of solutions to the GP, which was originally obtained by Erd\"os-Schlein-Yau. A key ingredient in their proof is a powerful combinatorial method that resolves the problem of the factorial growth of number of terms in iterated Duhamel expansions. In the lectures we will focus on approaches to the uniqueness step that are motivated by the perspective coming from nonlinear dispersive PDE, including the approach of Klainerman-Machedon and the approach that we developed with Chen-Hainzl-Seiringer based on the quantum de Finetti's theorem. Also we will look into what else the nonlinear PDE such as the NLS can tell us about the GP hierarchy and quantum many body systems, following results that we obtained with Chen, Chen-Tzirakis and Chen-Hainzl-Seiringer.

Supplements
24220?type=thumb Pavlovic Notes 678 KB application/pdf Download
Video/Audio Files

14344

H.264 Video 14344.mp4 371 MB video/mp4 rtsp://videos.msri.org/data/000/024/104/original/14344.mp4 Download
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