 Počke. OK. Ja. Hvala, vse. Čakaj do organizacijs, za organizacijs vse večne konference in za njihoj posledanje, da se tukaj počkati na spontenstvenosti možnosti v Milingu, in v biti začal. Čakaj? OK. Video. OK. Miling je več vsočen pattern of collective motion. It's a collective circular motion around the common centre, which has been often observed in fist schools, but also in other animal species, like, for example, ants. So there are many theoretical models of collective motion which show milling and most of these include avoidance, alignment and attraction between individuals, among individuals. Though the mechanisms which are underlying milling formation are still unclear, Arevoidance, alignment, and attraction, all necessary? Can simpler models generate patterns like milling? Well, the answer to this last question is yes. There is, for example, a model by Strombom of 2011, which has only attraction, which does show milling, for example. Though, how important is the alignment interaction? Is milling with only alignment possible? There is the Vitzert model, which has been mentioned already a few times today also, which is a model with only alignment, and does not show any milling. Or actually, it does show some milling if you confine particles in circular geometry, and if you add also some excluded volume interactions between the particles. So, I'm here interested in the minimal modifications, which we have to do on the Vitzert model in order to obtain milling without any walls, so in bulk. So, as most of you indeed already know, the Vitzert model is a two-dimensional model of point particles, which are characterized by position and orientation, which have all the same constant speed. There is only alignment, no avoidance and no attraction, and the particles align to the average orientation of neighbors, including themselves, in a certain interaction range, which can be chosen as unit of length. So, for example, that yellow particle there will interact with the neighboring particles, and the next time step go directly to the average orientation of the neighbors. On top of that, there is a random noise, and indeed the phase transition from disorder to order, but as said, no milling. So, how can we get milling from the Vitzert models? What can we modify? Let's consider an idealized case, indeed of four particles on a circle of radius r without any noise, which move at constant speed b and with a certain angular velocity. So, the first thing that we may not can say is that we actually need a limitation on the field of view. Why that? Because if the field of view is complete, as in the original Vitzert model, then the particle there will interact with the particle in front and with the particle in the back, and this orientation cancel each other out, and therefore this particle will then simply continue to move straight. But now if we introduce a limitation on the field of view and find angle in the back, then this particle here will interact only with this particle here, for example, and not anymore with the particle in the back, and therefore will tend to rotate counterclockwise in this case. On top of that, we note that that condition has to be satisfied, which is actually just the definition of any circular motion, which means so the speed has to be not too high, not too large compared to the angular velocity, a bit like a satellite around the Earth. If it's too fast, then it simply flies away, and if it's too slow, it crashes into the Earth. On top of that, we notice that the interaction range is here in this case the radius times square root of 2. The radius is therefore smaller than the interaction range divided by square root of 2, and that means that the radius is of the order of magnitude of the interaction range, which is what we will observe in the milling, which I will show you. So therefore we modify the Bitzek model, introducing indeed limitation on the field of view with a blind angle in the back, and a limited maximal angular velocity, which means that a particle cannot anymore instantaneously let's say go to the average orientation of the neighbors, but can only rotate towards it by indeed a maximal angular velocity. Implementing this, and indeed starting from a random starting configuration in a quadratic box with periodic boundary conditions, we see that indeed milling spontaneously emerged, and actually here we have also multiple mills and counter rotating mills. So actually varying with the few parameters of the model we not only obtain mills, but also other patterns of collective motion like flocks, lines, bands, fronts. Therefore we need a way to identify and measure, quantify and measure indeed milling. We can use the average velocity, so the polar order parameter, which is 1 when particles are ordered, and 0 when particles are disordered, and we can use the average absolute value of the normalized angular momentum, which is 1 when particles are milling, though where rcm is the distance of particle i from the center of mass of the group, it is belonging to, and where group is a set of particles, where the distance between particles is smaller than half of the interaction range. And this works well for identifying also multiple and counter rotating mills indeed. But indeed the angular momentum alone is not sufficient to identify milling, because there are cases like for example the case of the band pattern, where this quantity is actually high. And therefore it turns out that a good condition for identifying milling is that the average velocity has to be lower than a certain threshold, as well as the average momentum has to be higher than a certain threshold. So performing 100 simulations of 2,000 times steps in the quadratic box of size 20 in units of interaction range with periodic boundary conditions, as we have already seen, we can measure time averages of the average velocity and average momentum in the last 500 steps, where the system is already in the steady state. For each of the runs, we get either milling state or not, and then we can compute a milling proportion, so just the number of runs where we have milling divided by the total number of runs. So the three parameters of the model are indeed the ratio of speed to maximum angular velocity, the field of view, the noise and the particle density. We see where indeed we can get some milling. As we have seen milling in these mergers, we have seen it in the video, and here we see a bit more in detail where. So we need some intermediate field of views, so about 200 degrees, and the ratio of speed to angular velocity has to be around one, which was indeed also what we expected. A bit more in detail here, where we can see the dependence on the ratio of speed to angular velocity in empty circles and the dependence on the field of view in full circles. Varing also the particle density and the noise, we get a bigger picture, and from which we can conclude that indeed we need not too high noise, obviously we can say, and also the density has to be high enough in order to get milling. Again a bit more in detail, we see here the milling proportion as a function of noise and as a function of particle density, and indeed we see the milling to no milling transition by either increasing noise or decreasing density, which is a bit similar to the, which reminds at least to the order, disorder transition in the Vitzek model by increasing noise or decreasing density. We observe only milling, but also flux, lines, bands and fronts. We studied a bit where they happen, for which values of the different parameters and for example we see that these lines happen at low noise and low field of view. Increasing the noise, we have first fronts here emerging and then bands here emerging, milling is already almost disappeared and increasing the noise further of course at certain points the disorder state will invade the entire phase space. So, let's consider now two types of particles. So, one type is in this case too slow in order to get milling, so it is indeed flocking and one other type which has the right set of parameters values in order to get milling. What happens if we put them together in the same box? Will they mill all together? Will nobody mill? Will they segregate? So, running simulation in this particular case we see that actually the milling particles are still milling here and you see that they have kind of trapped the non-milling particles inside their mills and they have kind of induced this rotational motion also to the non-milling particles. So, actually varying the speeds of the non-milling particles we can get not only the case as we have already just seen of milling induction with the non-milling particles inside the mill but we can also get milling induction with the non-milling particles outside the mill if the speed of the non-milling particles is higher than the speed of the milling particles and actually not only milling induction but we can get also milling destruction if the speed of the non-milling particles is too slow or too high in that case they simply get out of the milling particles because they are either too too slow and they are kind of obstacles to this, they cannot really follow the rotational motion or they are too fast and they wash away the rotational motion. Yeah, actually that's what is it was already this. So, we have seen that this modified Vitzek model milling for intermediate field of use and when the ratio between speed and angular velocity is about one for low noise and for high enough density and this is indeed a case of milling using only alignment which is the first model studying this. We also have seen the existence of milling induction and milling destruction effect and all these are important insights for the understanding of milling formation maybe also in real animal schools. Thank you.