# 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

Room 703, No.6 Science Building, North Nanyang Road, 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 Group Photo & 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.

PPT_Kinetic models for spin-magnetization coupling in ferromagnetic materials

### 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).

PPT_Splitting methods for rotations: application to Vlasov equations

### A notion of empirical measure in quantum mechanics. Applications to the mean-field limit (in collaboration with Thierry Paul)

Francois Golse, Ecole Polytechnique Paris

Abstract:

We propose a quantum analogue of the notion of empirical measure in the context of the quantum mechanics of systems of identical particles. We write a dynamical equation for this new object, and prove that the Hartree equation coincides with this dynamics for a special class of quantum empirical measures corresponding to the chaotic limit. This is on a par with the theory of Klimontovich solutions of the Vlasov equation. As an application, we give an O(1/\sqrt{N}) convergence rate for the mean-field limit in quantum mechanics uniformly in the Planck constant, for smooth interaction potentials.

PPT_ A notion of empirical measure in quantum mechanics. Applications to the mean-field limit (in collaboration with Thierry Paul)

### Highly-oscillatory evolution equations with varying frequency: asymptotics and numerics

Mohammed Lemou, Universite de Rennes 1

Abstract:

I will start by a brief presentation of recent numerical methods that we recently developed to solve a class of highly oscillatory problems including some kinetic and Schrödinger equations. In these studies, and more generally in the analysis of highly-oscillatory evolution problems, it is commonly assumed that a single frequency is present and that it is either constant or, at least, bounded from below by a strictly positive constant uniformly in time. Allowing for the possibility that the frequency actually depends on time and vanishes at some instants introduces additional difficulties from both the asymptotic analysis and numerical simulation points of view. I will explain a first step towards the resolution of these difficulties. In particular, we show that it is still possible in this situation to infer the asymptotic behavior of the solution at the price of more intricate computations and we derive a second order uniformly accurate numerical method.

PPT_Highly-oscillatory evolution equations with varying frequency: asymptotics and numerics

### Random Batch Method and its application to sampling

Lei Li, Shanghai Jiao Tong University

Abstract:

The first order interacting particle systems are ubiquitous. For example, they can be viewed as the overdamped Langevin equations. We first introduce a random algorithm, called Random Batch Method (RBM), for simulating first order systems. The algorithms are motivated by the mini-batch idea in machine learning and statistics. Under some special conditions, we show the convergence of RBMs for the first marginal distribution under Wasserstein distance. Compared with traditional tree code and fast multipole expansion algorithms, RBM works for kernels that do not necessarily decay. We then apply RBM to Stein Variational Gradient Descent, a recent algorithm in statistics and machine learning, to obtain an efficient sampling method. This talk is based on joint works with Shi Jin (Shanghai Jiao Tong University), Jian-Guo Liu (Duke University), Jianfeng Lu (Duke University） and Zibu Liu (Duke University).

PPT_ Random Batch Method and its application to sampling

### 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.

PPT_Merging-splitting dynamics for animal group size without detailed balance

### The role of intracellular signaling in the stripe formation in engineered E. coli populations

Min Tang, Shanghai Jiao Tong University

Abstract:

Recent experiments showed that engineered Escherichia coli colonies grow and self-organize into periodic stripes with high and low cell densities in semi-solid agar. The stripes develop sequentially behind a radially propagating colony front, similar to the formation of many other periodic patterns in nature. These bacteria were created by genetically coupling the intracellular chemotaxis pathway of wild-type cells with a quorum sensing module through the protein CheZ. In this paper, we develop multiscale models to investigate how this intracellular pathway affects stripe formation. We first develop a detailed hybrid model that treats each cell as an individual particle and incorporates intracellular signaling via an internal ODE system. To overcome the computational cost of the hybrid model caused by the large number of cells involved, we next derive a mean-field PDE model from the hybrid model using asymptotic analysis. We show that this analysis is justified by the tight agreement between the PDE model and the hybrid model in 1D simulations. Numerical simulations of the PDE model in 2D with radial symmetry agree with experimental data semi-quantitatively. Finally, we use the PDE model to make a number of testable predictions on how the stripe patterns depend on cell-level parameters, including cell speed, cell doubling time and the turnover rate of intracellular CheZ.

PPT_The role of intracellular signaling in the stripe formation in engineered E. coli populations

### 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.

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