 Okay, so first I want to thank the organizer for giving me a chance to represent my work. I come from Germany, so my work, this talk about effective long distance from a short distance, maybe these are two key words, two key words here, and this paper is published So I work together with the students, more than you, and Dr. Mihar Matala and also Professor Gershaw is my boss. Okay, so at first I give a short review of a 1D gaze and introduce our model, and this part is the key part of my talk. I introduce the interaction in the face space and I'll show some many body dynamics. Okay, so first is the short review of 1D gaze. So the 1D gaze has been invested as early as 1936 by Punks. In his paper he introduced a free particle in 1D system, so and they have a contact interaction. This means that these particles can only have interaction with the contact each other and have a contact strength, a G here, and in this paper he invited the G go to infinity limit. This is called a hard sphere, and later in 1960 a G-audio, so he solved the problem of quantum gas, the boss gas, and found some interesting like fermionization. And later, so in 1963, so LiP and the leading guy, they extended this work into an arbitrary quantum-like interaction. The G goes to any real numbers, so and they got exact solutions of this. And later a young training, he investigated the thermodynamics of this model. And this is the pure theory, and if you want to find the more interesting thing, you can go to the review papers on this. And in the recent years, because of the development of code atoms, people can realize this kind of a 2G gaze and found some and found some things like the fermionization or something like the interesting things. Okay, here in our model, we introduce this model here. In our model, there are some key ingredients different from the free particles. We trap these particles in harmonic trap, so we need a very strong harmonic trap. And we also add a purely driving field here, and we also consider the interactions. So I want to stress on that in this talk, we only consider the classical dynamics because in this time, in the classical regime, we already found some interesting things. The quantum work is being prepared. So let's write the Hamiltonian here. Let's explain the Hamiltonian more. So this is just the harmonic oscillators, many independent harmonic oscillators. Then I add a driving, so gamma is a scaled dimension is a driving strength. And omega here is just the ratio of driving frequency over your harmonic frequency. This can be near to one, two, three, some integrals here. And this is the interaction. Here we assume any type of interaction. So here, we have the driving field actually is very small. So any particle, the motion of any particle is still dominated by the global oscillation. And just their amplitude and face just changed a bit. So we can separate the total motion of one particle into a fast oscillation, like a fast oscillation mode and a slow mode like this here. This is x and p. So if there's no driving and no interaction, this capital X and capital P just a constant is a harmonic oscillator. But if you add some driving and interaction, this will change a bit. So we are interested in the slow dynamics of capital X and capital P. So you plug this back into this Hamiltonian, then you throw away this fast oscillation term. This is called rotating wave approximation. Then you get the time independent Hamiltonian like this. So actually this part is comes from the driving that we discussed in this paper. But in this talk, we focus on this interaction. So we ask what will be the interaction in the rotating frame from this real interaction. Okay. So first I want to show some equations of motion, which I do. I use the numerical code to simulate the dynamics. So in the rest of frame, so the equation of motion is given by these equations. Then you from this equation of motion, there's no approximation here. Then you get x, you can get some motion of this. Then you take the value of x and p every time period, delta t. This is called, sometimes it's called a Pungkai making. Then you got stroboscopic dynamics. So it's just like this. So this is the propeller of a plane. So of course it can also, so it shoots very, very slow motion, but in fact it's slow, it rotates very fastly. This is because of your camera. So your camera is taken 24 pictures every second. So you will just see this slow motion. So this is called stroboscopic dynamics. And you can also go to the rotating frame. This means you rotate with this, then see the dynamics. Then this dynamic is directly given by G, this Hamiltonian I introduced before. And this is the rotating wave approximation. We already use the rotating wave approximation. So there's no approximation here. There's the rotating wave approximation here. I will use both of these to check my results. So and the thing that we ask, what is this? This is a key problem. Then I want to solve this. And this problem solved in this paper. So here I only tell you the results. I don't give you some details. So first we introduce a phase space distance. So you have a particle here and here. Particle i, particle j introduce a phase space distance. Then you give me a real interaction. We ask calling interaction, contact interaction or some other kind of interaction. Then you do this Fourier transformation. You get the Fourier coefficient vq. Then you use this vq. And this is the 0th order, the Bitter function. You do the interval. Then you get this. This is basically the result. And maybe here it's a little math. It's a math. So what is the physical meaning of this? So you use the formula of Bitter function. Then you plug it back. You will get an equivalent form here. Now you see the meaning is very interesting. It's very clear now. So the interaction here is just the average interaction every time period. Because your two particles do this oscillation, then you average the interaction. Every time period you get the phase space interaction. I should have here. Okay. So then I will use these to calculate some examples here. So first we assume the interaction of two particles is a rectangular potential. This means if the two particles are far away, they have no interaction. Only in very short distance, they have a constant interaction. This is just a model interaction. So this is a short distance interaction. We assume that the distance is very short. Then we use our results to calculate. Now you see there's a long tail. Now it's a long tail. Then we see that and the long tail, the symbolic behavior is like 1 over r is a cooling type. Now we have a long distance interaction from a short distance interaction. This is what I mean from the beginning. Then we go to another example, where the cooling interaction. So we assume the real particles are cooling. Here you should know the cooling interaction we assume is very small. So they can also form some crystal structure in the control. The beta is very small. Then you use our results to calculate. You will find this is divergent. So to get out this divergent, we can introduce a cutoff here, toasty. Then you get a finite result. This is the renormalization. But you will ask how do we determine the toasty, the cutoff? So actually the physics meaning is that because the cooling interaction when they approach each other, they will go to infinity. So in reality, the two particle never touches other. They have a very small distance, r, c. Then you can use the energy conservation law to estimate the r, c. Then you introduce the cutoff. So we also introduce a gamma called collision factor. This is because when they collide, the orbit will change a bit. So this is a factor. So anyway, this is important. You first calculate this, then you get the result. So now I use this procedure to get the smallest distance r, c, and the cutoff. Then you plug it into the here. Now you get this form. Now you see that the phase space interaction now is still in the form of cooling type. But with the renormalized coupling and the strength here, beta is given by this. It depends on your kinetic energy. This is the logarithm depends on. So this is nearly a constant. So I check this. So here, this is two particles. I put one particle here, one particle here. If there is no interaction, because we also have a driving. So these two particles will do this small circle here. Then if we add the interaction, the circle will become larger here. So you can see that here I use both methods. No approximation and rotating wave approximations. They are just the overlaid each other. So this means that this formula is good under our approximation. Okay. So now I go to a very general case, like inverse parallel potential here. So if A is 1 over 2, say it's just a cooling interaction. If A is 1, it goes a little deeper here, the steeper here. And A is 3 and even the steeper. And you will go to infinity. It's just a hard curve interaction. You can model the hard curve interaction. Okay. Then I use this, use our formula. I calculate the phase space interaction like this. So you can see the blue is the cooling interaction. It's still like a cooling interaction. But if A is 1, you will see that now this interaction becomes a constant. Then you will go to even deeper. You will see it's increased now. It's not decreased now. It's increased. And for the hard curve interaction, it will become a linear. And this is a summary of the results here. For the hard curve interaction, go to linear. And this is for a general case, 1, the integral and a half integral like this. And some interesting, I show you some interesting cases here. If A is 1, then it's a constant. So I checked numerically. You can see no matter I change the strength, so the non-rotating wave approximation and the rotating wave approximation, they are very good with each other. Then if there's a good infinity, it's linear. So it's like a quark interaction. So it's just, but why? Here, why it's become a linear? A simple explanation that if the two particles, the hard curve, they like to move together, synchronize together. If you push them a little bit, they will collide. So they don't like this kind of state. They like it just alternate together. This is why it's linear. Okay. So I just show you some interesting simulation here. So if you put two particles here, I assume their distances are, so if there are no interactions, no interactions. I also said the driving is very weak. So the total dynamics is dominant by their interaction. So if I put two particles here, if there are no interactions, so they're just a fixed point. Because the harmonic oscillation is just a periodic oscillation. You'll see so. But periodically, it's just a fixed point. Then if they have interactions, they will rotate. They will rotate with some frequency. It's just like this. If you put two particles here, they will just rotate. And this rotating frequency can be given by this formula. So the use here is to renormalize the phase-space interaction. So this is the analytical result. And we check it with numerical simulations here. These dots are from the real time, the numerical simulation without the rotating wave approximation. And the solid curve is just from an analytical result. So you see that actually, this picture also shows the validity of our approximation. So our approximation only holds for large r. So if the two particles are large enough, our approximation is very good. But if they are close here, the short distance, if they are too close, they are a large difference. So the large distance means the kinetic energy is very large. So this is the kinetic energy. So our approximation is valid in the case of kinetic energy. It's a very large interaction energy. Okay. Then I will show this three body dynamics. Two body dynamics can be desirable. But three body dynamics in general cannot solve it analytically. I just show some results. So you can see here, if I put three particles, this is the particle, this is the particle 2, this is the particle 3, if they are, this is very close to each other, and this is a little far from each other. So for cooling interaction, because the phase-space interaction is still a cooling interaction, this means their interaction is much stronger than this one. So actually these two particles, they rotate very fast, like this show this one. And the center of this system and it rotates with the further particles together with a large circle, and this and low frequency. This is what you see here. This is the center of mass motion. But for the hardcore, for the hardcore interaction, then the interaction is linear. This means the interaction between particle 1 and particle 3 is larger than particle 3 and particle 2. So you can see you will not see this kind of behavior. So it's different from this. You don't see this kind of faster frequency because of this sub-system. You cannot see this kind of fast oscillation. Okay. And you can see that, yeah, I compared with the rotating wave approximation and non-rotating wave approximation. Okay, the last slide I will show you some, okay, this is also, so here, this is a real-time evolution. I put the same initial condition. This is three particles. And for the cooler interaction, you will see that these two particles just rotate with each other. Then the whole system rotates with this. And the rotating frequency is very, very slow. But for the hardcore interaction, so because the interaction is linear, you will see a very different picture. Here, now, these two particles, they don't have a center. They just rotate with this, actually. And also, you will also see the direction. The direction is opposite. This is along this direction and this direction is different. And also, this is very fast. Yeah. This is because the integration is linear, like this. Okay. So, we will summary on my work. So, we found that long distance integration can be produced by a short distance integration. And we found this is maybe useful because in the code atom system, you only have contact interaction. But normally, so in the code atom center, you don't have a cooling like interaction. So, here, we found that you can even use the contact interaction to produce a long distance cooling like interaction. Maybe it's useful for the code atoms. And we explained this by this. So, we separated the motion of your particle into a fast oscillation mode and a slow, also, and a slow evolution. The Facebook interaction describes the interaction between the slow modes and the fast oscillation modes that take a role like a force carrier, something like this. And technically, we remote the divergence by introducing some renormalizing methods. Okay. So, the next step is we will, if I study through the quantum regime. Okay. Thank you.