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Calogero Particles and Fluids: A Review (Lecture 3) by Alexios Polychronakos

112 views · Published 15 December 2018 · 1:37:35 · Indexed 6 October 2026

Channel: International Centre for Theoretical Sciences · 2018 · Science & Technology

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PROGRAM: 
INTEGRABLE SYSTEMS IN MATHEMATICS, CONDENSED MATTER AND STATISTICAL PHYSICS

ORGANIZERS: 
Alexander Abanov, Rukmini Dey, Fabian Essler, Manas Kulkarni, Joel Moore, Vishal Vasan and Paul Wiegmann

DATE : 16 July 2018 to 10 August 2018

VENUE: Ramanujan Lecture Hall, ICTS Bangalore


This program aims to address various aspects of integrability and its role in the advancement of Mathematics, Mathematical Physics, Condensed Matter Physics and Statistical Mechanics. The scope of the proposed program is therefore highly interdisciplinary, being aimed at mathematicians and physicists who are working on different aspects of integrability. The purpose is to bring together experienced as well as young scientists, graduate students and postdoctoral fellows who are working on various aspect of quantum as well as classical systems in which integrability plays a paramount role.


Dates:
July 16 - July 27 (First Division: Integrable systems in Mathematics and Mathematical Physics)
July 30 - Aug 10 (Second Division: Integrable systems in Condensed Matter / Statistical Physics)


The above two divisions are only made just as a broad reference for gathering like-minded mathematicians and physicists. But, we aim to strongly encourage cross-disciplinary discussions throughout the 4 week program under the common theme of integrability.

During the first two weeks of the program, there is also a parallel meeting Quantum Fields, Geometry and Representation Theory and we expect fruitful interactions.

Topics to be discussed under the division of Mathematics and Mathematical Physics include (but not restricted to):

Constant Mean Curvature Surfaces (classical and discrete) and their Relation to Integrable Systems
Integrable systems and geometric asymptotics
Quantum Integrable Systems
Analytical methods for partial differential equations (PDEs) inspired by integrable systems
Applications of integrable PDEs in mathematical physics (for e.g, Benjamin-Ono, Nonlinear Schrodinger, Korteweg–de Vries equations)
Topics to be discussed under the division of Condensed Matter and Statistical Physics include (but not restricted to):

Nonequilibrium dynamics and transport: Integrability to many-body localization
Perturbed conformal and integrable field theories with applications to low dimensional strongly correlated systems
Bethe ansatz and applications to spin chains
Hydrodynamics and collective behavior of many body systems
Calogero, Lieb -Liniger, Yang-Gaudin models and their applications
There will also be some pedagogic lectures on the below topics:

Mathematics and Mathematical Physics:

A. Bobenko (TU Berlin) - “CMC Surfaces (classical and discrete) and their Relation to Integrable Systems”
David Smith (NUS, Yale) - “The Unified Transform Method for linear evolution equations”
Tom Trogdon (University of California-Irvine, USA) - “The computational theory of Riemann–Hilbert problems”
Paul Wiegmann (Chicago) - "Hofstadter problem: Integrability and Complexity"
Ritwik Mukherjee (NISER, Bubaneswar) - " Quantum Cohomology and WDVV equation"
Condensed Matter and Statistical Physics:

Fabian Essler (Oxford) - “Integrability out of equilibrium”
Joel Moore (Berkeley) - “Nonequilibrium dynamics and transport: Integrability to many-body localization”
Alexander Abanov (Simons Center, Stony Brook) - "Hydrodynamics, variational principles and integrability"
Alexios Polychornakos (CCNY-CUNY) - "Physics and Mathematics of Quantum and Classical Calogero models"
Fabio Franchini (University of Zagreb, Croatia) - "Basic Lectures on Bethe Ansatz"
 

CONTACT US:  [email protected]

PROGRAM LINK: https://www.icts.res.in/program/integrability2018

Table of Contents (powered by https://videoken.com)
0:00:00 Start 
0:01:20 Review
0:02:37 Spectrum of spin-particle model in brief
0:03:29 Ferro: f 0, bosons. Ground state:
0:04:37 Asymptotic bethe Ansatz approach
0:10:06 Sidebar: Yang-Baxter equation and integrability 
0:18:12 Scattering result plus periodicity of wavefunction gives the spectrum on a space of period 2x 2nk;
0:24:52 Particles with spin:
0:27:45 The freezing trick
0:32:27 From harmonic spin-Calogero model we obtain a spin chain model with spins sitting on an inhomogeneous lattice
0:35:23 Parting words
0:40:24 Folding the Calogero model: new systems Sorry, no time!
0:43:48 Euler description (space-fixed)
0:47:14 Continuum formula valid if near-neighbor particles do not contribute a macroscopic fraction of the potential
0:53:20 QM collective field method
0:56:52 Free Calogero:
1:02:58 Consider the Calogero potential contributed by a particle x(0) = X
1:08:22 Use
1:12:58 Sound waves
1:18:40 Solitons
1:20:25 and Waves
1:22:04 Dual formulation: the Generating Function approach
1:24:22 After some algebra and symmetrization's of indices we obtain
1:26:06 The dual equations 
1:27:00 Deriving hydrodynamics from dual system
1:28:05 Therefore
1:29:45 Two soliton solution for quartic potential
1:33:33 Q&A

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