First, I would like to thank Enric for this invitation. I was here two years ago. I think

it's the exact same room. It's great pressure to be back here. Looks like today we have

SJTU invasion of Eilongan. Today I will share with you some work I did in the last few years

in the direction of interacting particle system for both classical and the quantum systems.

So this is what I try to cover. First, talk about the random batch methods for classical

interacting particle systems. So the joint work with Lady, also my young colleague, and Jiangguo Liu

at Duke University. Then one particular application which I think is important is

in molecular dynamics. Then I will move on to talk about quantum and body problems and quantum

molecular. So basically the problem I try to solve is very fundamental equations that everybody

knows. F equals MA. So it's Newton's second law. So it takes the form of a system of ODEs. So here

we have a bunch of M particles with positions Xi. For the I's particle, position Xi and the VI's

is velocity. So the dots here are just time derivative. So the first line is the definition,

right? DX, DT equals velocity. And second line is F equals MA. So here assume all the particles are

exactly the same. So MA says is one. So the left-hand side is clearly acceleration. The

derivative of time derivative of velocity is A, acceleration. The right-hand side is just F. So here

F comes from interacting of particles. So particles, they have force act upon each other. What's

important is the summation term. So each particle interacts with all the other particles. So here

essentially we have M minus 1 summations. So this is usually called second order interacting

particle systems, where phi is the potential. So minus negative gradient. A negative gradient

of potential is just force. So it's force between the particles. Sometimes you take so-called

over-damped limit, you get first order system. You limit the velocity, for example, in certain

asymptotic regimes. Then you have this sort of long-distance equation, interaction particle,

long-distance equations. You still have these interacting particles, potential force, K. K is

interacting with force. We normalize it, divide it by M minus 1. And here V is the external

potential. You have some external force. You get an external potential. You may also assume this

noisy background. So these are the systems we try to solve. Talking about applications,

it's everywhere. If you have a long enough career, there's a big probability you'll solve this

equation. So here I just listed a few. Well, in physics and chemistry, molecular dynamics,

for example. In astrophysics, where you have gravitational force between stars, galaxies.

In biology, in the last two decades, people use so-called agent-based models. So you model

interaction between different biological agents. And social science, economics, people they trade,

for example. That's kind of interaction between different economic agents. Also, the particle

method is also a numerical method to solve usually high-dimensional PDEs. It's called

Lagrangian methods versus Eulerian, which solve PDEs. So here you solve PDEs by a particle method.

Usually, for example, in kinetic equations and mean field equations, very high-dimensional PDE.

So difficult here, actually, is the size of the problem. So here, notice that we have a

big summation. And the number of particles usually is astronomical. For example, in cosmology,

astrophysics, n can be as high as 10 to the 20. In plasmas, also 10 to the 20. The numerical

particle methods, the largest computer you can do is 10 to the 9 to 10 to 12,

than many other applications. So the difficulty here is that we have to add a lot of terms. So

it's really the size of the problem. It's not that we don't know how to solve all these. Everybody

here knows how to solve all these. But the problem is that this summation, you see is all the n

summation for each i. But we have n of them, so it's all the n square, complexity of all the n

square. That's the difficulty here. And n square usually is forbidden, sort of forbidden for class

computation. And there are very famous algorithms developed to address this issue, for example,

Fourier transform. And also, particularly for the problem I'm going to talk about, fast multiple

methods introduced by Rocklin and Gringard. Those are two of the so-called top 10 algorithms of

last century. So it's about how to reduce all the n square to all the n, n log n. So today I will

show you a method we introduced. It's called random batch methods. So the goal is to reduce

from n square, from n square to all the m. And actually it's a very simple method and also very