 Yeah, sorry. Yeah. Yeah, you could share your risk right now. No, it doesn't work. Doesn't work? Do you have? Yeah, it's okay. So please start your talk. Oh, you should unmute your microphone, Maria, and Google Meet. Yeah. Okay, yeah, sorry. Yeah, sorry for the inconvenience. We tried before and it worked perfectly and I don't know why it didn't work for the second time. Don't worry, it's okay. Please. Yeah. Yeah, hi, everybody. Yeah, patterns of polarity fields. And I was thinking to start with all the tanks because I already have flowers on my first slide. Yeah, I would like to thank everyone for being here on a Friday morning, afternoon, evening, or night, and speakers for the interesting talks. I really enjoyed them. I would also like to thank the organizers for the opportunity and also for doing all the work, and especially for letting me know about the whole event. Now, let me come back to the flowers. I want to make a point. Flowers like many other multicellular organisms represent interesting patterns and shapes. Flowers or animals or other multicellular organisms have different organs that are developed during development and they are like well-defined shapes and structures that are formed by collective interaction and organizations of cells. It's a good example of out-of-equilibrium system and a self-organized system. Yeah, these structures are usually characterized by well-established sizes and shapes. And even if we zoom in in many systems, we see that there are a lot of precise structures and patterns, even a smaller scale. For example, in tissues which are like 2D systems, 2D sheets of cells, we can see a lot of pattern formation going on. And I want to highlight the formation of large-scale patterns of polarity fields in the tissues. Here, for example, you can see the shoot apical meristem of plants where new organs are developed. It's a very important organ in the plant and you can see that all the cytoskeleton here acting, filaments are aligned very well and they form a very reproducible pattern, which is important for the development. And of course it's interesting to know how cells collectively organize to form such pattern. And let me first simplify the system and say what exactly polarity means in such a system. It can be anisotropic distribution of molecules, for example, at one part of the cell or the orientation of cytoskeleton filaments, or it can also be cell shape or other polarity. It can be expressed by different cell. And this talk would be very theoretical. We were inspired by the pattern seen in clam tissues and we were thinking of this theoretical model and we found interesting results, but we didn't apply the model to the system because of other complexities. So the model is similar to classical spin models on square lattices, like XY model, and we have at each square we have a polarity which is defined by the orientation, like the XY model, and the magnitude here, which can vary from one side to the other one. And for the rest of the talk I also show the magnitude with a color code, just to clarify it now. And then we use a modified XY energy for this system. We have interactions between neighbors, the first term, and we have also, we can also have coupling with an external field, which in the case of tissues can be like a signal from outside or some stress, or I mean there are some candidates for that. This is very similar to XY model, but we consider the polarity magnitude values. And then we consider dynamics of the system and we have two different, we consider two different dynamics for the system. For polarity orientation we assume a non-conservative dissipative dynamic, like the first equation, and we also have the noise term, which is like a white noise. And for the polarity magnitude we assume the dynamics is much slower than the orientation, and therefore we consider a sort of transport between cells, which is like the P model for the conserved dissipative dynamics. Just very classical models, but different dynamics. And we also have diffusion, which is like the noise term here and connected to the temperature. So if we consider just the external field at the beginning, there is no coupling between neighbors and we just rewrite the equations. I don't go into details, but you know, just quite a trivial calculation. We see an interesting pattern. Here in the first figure, this one, there is no transport possible between cells, and also there is no diffusion. And you see when we start from a polarity distribution, here we start from non-uniform distribution. Actually the polarity magnitude would stay constant at each side, and the orientation at finite temperature would be random, as we expect from XY model, you know, for temperatures higher than zero. But as soon as we start having this transform between neighbors, there are some sides with high polar magnitude. They are formed, and then they, in these sides, the polarity is aligned with the external field. However, everywhere else is very random, where the polarity magnitude is small. Of course, when we have diffusion, this can also stay. However, if we just have diffusion, we would have a uniform polarity magnitude distribution all over the tissue, but everything would be random at finite temperature. And if we look at the order parameters, two-order parameters, like the average... Can I ask you a question, please? Yes, sure. Could you go back to the previous slide? Yes. So, did you look at the spatial orientational correlations as well, as a function of distance between sites? Yeah, we looked at it, and actually, there is almost no correlation. There's no correlation, even at low temperatures? No, it's not low. I mean, here, it's the finite temperature. At low temperature, yeah, at zero temperature, everything is aligned with each other. It's just the finite temperature. It's above the transition of the XY model. We look at the order parameters. One is the classical one, which is defined also for the XY model, the average alignment, P1, and the other one, we call the whole magnetism, and it's like multiplied by rho. And we see that the average alignment, when there is no transport, it just both actually both order parameters follow each other, and it looks like the XY model. However, when we allow for the transport between neighbors, the second order parameter persists for a while, and then for, I mean, for a finite temperature here, they are plotted, sorry, as a function of temperature, and then it drops to zero. So, it's sort of like... Sorry, only five minutes left. Oh, okay, yes. And then we can also look at the interactions with the neighbors. Again, similar equation, and here, there is no extent of it. And it's also interesting that for such a system, we can have, you know, these polarity islands, we can also... I mean, they can also be established, and these domains where the polarity magnitude is high and versus or aligned with each other. However, outside these domains, the polarity orientation is random, as you can see in this system for a finite temperature. So, within the high polarity domain, they are aligned. The coupling between neighbors actually make them aligned. And when we increase temperature, as you can see in this picture, the polar domains actually become bigger, and the number gets smaller, and after a temperature, a certain temperature, of course, we expect that it would have a very random distribution of polarities. And here, you can also look at the average domain size as a function of noise or temperature, and you see that for different... the constants of this transport, it actually, it's almost constant at the beginning, but then near the transition point, it increases, and then, of course, it vanishes. Yeah, now I just briefly summarized that this model is like a very... I mean, a very simple model based on modified XY model, but with the potential transport of polarity particles, and this simple model can express non-trivial dynamics in the system, non-trivial pattern formation, what actually I showed were like a equilibrium state. There is also publish in this paper, and of course, I would like to thank my collaborator, Areski, for all the support, and also Christof Goden. Yeah, and thank you too. Thank you very much for your talk. We have a time for one or two questions. Any questions? I have a question, maybe. Ali? Yes, please. Mariam, can you go to the slide where you have the patterns driven by interaction... By interaction. Yeah, the last slide, to summarize. Oh, the summary, okay. Yes. Is there any particular structure to these islands? So are they all of the same size? Always? Yeah, I mean, the sizes are very similar in the square lattice, and I mean, here also the diffusion is not that big. It also depends on the diffusion. So if there was more diffusion, then you would expect a broader distribution of shapes, I guess. Exactly, exactly. But the square shape is basically imposed by the square lattice? Yes, exactly, yeah. And as you see here, it's not a square anymore because at least at this temperature, diffusion is smoothing the border somehow. Okay. And I'm just wondering in this such kind of the models, polarity models, do you have something like a cluster-less-tallness that we can see that's in the two-dimensional model, the other vector, vectorial two-dimensional model? You can see the cluster-less-tallness transition? Yeah, I mean, that's like the first one, right? Mm-hmm. Isn't it like a zero temperature? Actually, I'm not sure, I mean, I'm not sure if I remember correctly the name of the transition, but it's like the zero-temperature transition, right? Not zero. The cluster-less-tallness is some kind of the topological transition that we have some kind of the Heisenberg two-dimensional spin models. Well, actually, there is no order below the temperature, I mean, below the cluster-less-tallness critical temperature, so I'm not sure if it's the same. Yeah, but I should look at it because yeah, I mean, I'm not from the field of condense matter. Yeah, so if I know, I mean, I look at a few different transitions, but now I don't have in mind actually which one. But did you say that the zero-temperature, they're all ordered or? Yes, yeah, I mean, at zero-temperature they're all ordered. Yeah, so we have like the classical, as you can see here in order of parameters, we have the classical transition. I mean, it's a change, of course, but we have a transition at zero temperature, but we have another one. But the pictures of patterns are basically in some steady state, I didn't get this, or? The ones I showed are steady states. And because there is no diffusion, they don't kind of approach anymore to each other or? Yes, yeah. There seems also the distance between each kind of island is more or less similar. So they are homogeneous. Yeah, exactly. Also, when we do like a sort of analytical calculation, we can see that I'm interested to study the case where diffusion is higher and also considering the full random diffusion model. I think, yeah, should be. Okay, thanks. Thank you.