# Workshop on Quantum and Kinetic Transport

## Introduction

This workshop aims at bringing in internationally leading and young experts on mathematical analysis, modeling and computation of quantum and classical kinetic transport problems, in order to present the state-of-the-art results of the field, and to foster future collaborations between researchers overseas and in China.

## Acknowledgement

This conference is supported by Institute of Natural Sciences, MOE Key Lab on Scientific and Engineering Computing, National Natural Science Foundation of China, and Network of International Centers of Education in China.

April 8-9, 2019

## Venue

Conference Room 1, Floor 3, No. 5 Science Building, Minhang Campus, Shanghai Jiao Tong University

## Organizer

• Shi Jin, Shanghai Jiao Tong University, Shanghai, China

## Schedule

Time April 8 (Mon.) April 9 (Tue.)
08:45-09:00 Registraton
09:00-09:45 Francois Golse Mohammed Lemou
09:50-10:35 Jian-Guo Liu Nicolas Crouseilles
10:40-11:10 Coffee break Coffee break
11:15-12:00 Lei Li Yuhua Zhu
12:00-14:00 Lunch Lunch
14:00-14:45 Min Tang
14:50-15:35 Lihui Chai
15:40-16:10 Coffee break
16:20-17:05 Zhennan Zhou
18:00 Banquet

## Program

### Kinetic models for spin-magnetization coupling in ferromagnetic materials

Lihui Chai, Sun Yat-Sen University

Abstract:

The Schr"odinger-Poisson-Landau-Lifshitz-Gilbert (SPLLG) system is an effective microscopic model that describes the coupling between conduction electron spins and the magnetization in ferromagnetic materials, based on which, we rigorously prove the existence of weak solutions to SPLLG and derive the Vlasov-Poisson-Landau-Lifshitz-Gilbert systm as the semiclassical limit connected to the mean-field model. We further discuss the diffusion llimit of this semilassical limit system.

### Splitting methods for rotations: application to Vlasov equations

Nicolas Crouseilles, INRIA

Abstract:

In this talk, a splitting strategy is introduced to approximate two-dimensional rotation motions. Unlike standard approaches based on directional splitting which usually lead to a wrong angular velocity and then to large error, the splitting studied here turns out to be exact in time. Combined with spectral methods, the so-obtained numerical method is able to capture the solution to the associated partial differential equation with a very high accuracy.

Then, the method is used to design highly accurate time integrators for Vlasov type equations.

Finally, several numerical illustrations and comparisons with methods from the literature are discussed. This is a joint work with Joackim Bernier (University Rennes 1) and Fernando Casas (University Jaume I).

### Merging-splitting dynamics for animal group size without detailed balance

Jian-Guo Liu, Duke University

Abstract:

We study coagulation-fragmentation equations inspired by a simple model derived in fisheries science to explain data on the size distribution of schools of pelagic fish. The equations lack detailed balance and admit no H-theorem, but we are anyway able to develop a rather complete description of equilibrium profiles and large-time behavior, based on complex function theory for Bernstein and Pick (Herglotz) functions. The generating function for discrete equilibrium profiles also generates the Fuss-Catalan numbers that count all ternary trees with $n$ nodes. The structure of equilibrium profiles and other related sequences is explainedthrough a new and elegant characterization of the generating functions of completely monotone sequences, as those Pick functions analytic and nonnegative on $(-\infty,1)$. Self-similar solutions with infinite first moment exist that attract data with the same initial tail behavior.

### Semiclassical Schrödinger equation with random inputs: analysis and the Gaussian wave packet transform based numerical scheme

Zhennan Zhou, Peking University

Abstract:

In this work, we study the semiclassical limit of the Schrödinger equation with random inputs, and show that the semiclassical Schrödinger equation produces O(epsilon) oscillations in the z variable in general. However, with the Gaussian wave packet transform, the original Schrödinger equation is mapped to an ODE system for the wave packet parameters coupled with a PDE for the quantity w in rescaled variables. Further, we show that the w equation does not produce epsilon dependent oscillations in the rescaled spatial variable, and thus it is more amenable for numerical simulations. We propose multi-level sampling strategy in implementing the Gaussian wave packet transform, where in the most costly part, simulating the w equation, it is sufficient to use ε independent samples. We also provide extensive numerical tests as well as meaningful numerical experiments to justify the properties of the numerical algorithm, and hopefully shed light on possible future directions.