Abstract : The kinetic theory describes the non-equilibrium dynamics of a many-body system from the statistical viewpoint, which is acknowledged to be a significant model to bridge the microscopic and macroscopic regimes in classical mechanics. On the other hand, many novel quantum phenomena emerge in the physics and material fields, where the microscopic description is the quantum many-body system.
Hence, applying the kinetic philosophy to study the many-body systems in the quantum field becomes pretty natural, and this Minisymposium aims at fostering the development of multiscale modeling, mathematical analysis, and numerical simulation about the interplay between kinetic theory and quantum dynamics.
[04305] Emergent phenomena in an interacting Bose gas
Format : Online Talk on Zoom
Author(s) :
Michael Hott (University of Minnesota)
Thomas Chen (The University of Texas at Austin)
Abstract : The study of kinetic equations describing collisions between a BEC and the surrounding normal fluid go back to Kirkpatrick and Dorfmann '83, '85 and Eckern '84. Ever since, this subject has attracted a lot of attention as it relates to condensation. In this context, mathematicians have studied the quartic quantum Boltzmann equation in the presence of a BEC. In this talk, we will discuss some of the progress made on the PDE level of the quantum Boltzmann equation. Then, we will focus on the validity of the kinetic equations. We will describe the crucial scale separations needed to extract a Boltzmann equation from the quantum dynamics. Moreover, we will see how the interference of sound waves can produce some surprising effects if a Bose gas is trapped in a volume of unit size. This is based on joint work with Thomas Chen.
[05382] Fluid limits from Quantum Boltzmann equation
Format : Talk at Waseda University
Author(s) :
Ning Jiang (Wuhan University)
Abstract : In his 2015 Ecole Polytechnique thesis, T.Zakrevskiy formally derived some fluid dynamics from quantum Boltzmann equation (Fermi-Dirac statistics). We rigorously justify two types of limits: incompressible Navier-Stokes-Fourier and compressible Euler (then acoustic) systems, by establishing some new nonlinear estimates on triple terms, and uniform estimates with respect to Kundsen number. A particular novelty is that the compressible Euler system derived from the quantum Boltzmann equation has a pressure law which is different and more general with that from the classical Boltzmann equation.
[04436] An explicit coercivity estimate of the linearized quantum Boltzmann operator
Format : Talk at Waseda University
Author(s) :
YULONG ZHOU (Sun Yat-Sen University)
Abstract : The Boltzmann-Bose-Einstein equation describes a large system of Bose-Einstein particles in the weak-coupling regime. If the particle interaction is governed by the inverse power law, the corresponding collision kernel has angular singularity. We present a coercivity estimate of the linearized Boltzmann-Bose-Einstein operator for such kernel. The estimate may not be sharp but explicitly reveals the dependence on the fugacity parameter. Joint work with Prof. Tong Yang.
[04331] Frozen Gaussian Approximation for open quantum system
Format : Talk at Waseda University
Author(s) :
Geshuo Wang (National University of Singapore)
Zhenning Cai (National University of Singapore)
Siyao Yang (National University of Singapore)
Abstract : We study the system-bath dynamics for open quantum systems applying frozen Gaussian approximation, which proposes an approximated ansatz for the wave function, converting the direct calculation of the Schrödinger equation into some ODEs of the parameters in the ansatz. We then derive the Dyson series under such approximation. To further improve the computational efficiency, we develop a fast algorithm known as the inchworm algorithm for the current framework.
Abstract : Motivated by the need to develop accurate numerical methods for computing the electronic properties of twisted bilayer graphene, we consider the problem of numerically computing the dynamics of a general aperiodic discrete (tight-binding) Schrödinger equation in an infinite domain. We prove that, under appropriate conditions, these dynamics can be rigorously approximated by those of a finite-dimensional truncated model. The key role in the proof is played by speed of propagation estimates derived from Combes-Thomas estimates. Besides the general aperiodic medium, we further improve our truncation analysis and Combes-Thomas Estimate for aperiodic medium with low dimensional structure, general van der Waal heterostructures. We then present a range of numerical experiments showing the effectiveness of our analysis.
[04339] On the kinetic description of the objective molecular dynamics
Format : Talk at Waseda University
Author(s) :
Kunlun Qi (University of Minnesota)
Li Wang (University of Minnesota)
Abstract : In this talk, we will introduce a multiscale hierarchy framework for objective molecular dynamics (OMD), a reduced molecular dynamics with certain symmetry, that connects it to the statistical kinetic equation, and the macroscopic hydrodynamic model. In the mesoscopic regime, we exploit two interaction scalings that lead to either a mean-field type or a Boltzmann-type equation. At the macroscopic level, we also derive the corresponding reduced Euler and Navier-Stokes systems by conducting a detailed asymptotic analysis.
