 of the session today is Rosalba Garcia-Malian from University of Cambridge, and she's going to talk about inter-production in non-reciprocal, of non-reciprocal interaction, so, thank you. Let me check this. I think I need to go ahead. I don't know yet, so. On this side, you can plug this into the, and then maybe that. Oh, I think it's okay. Yeah, no, I don't mind. No, no, I think it was the same for me, and then work nevertheless. Okay, if not, you can. Yeah, I'm just going to start. Well, okay, so thank you so much to the organizers for this wonderful conference, and also thank you for giving me the chance to present my work. I'm going to talk about entropy production of non-reciprocal interactions, and this is joint work with Zillow, Jong, and Guna Prusna. So, first of all, well, there are many types of non-reciprocal interactions, and sometimes they can be used to model certain type of phenomena, so here, there's an example of a bird in a flock, and sometimes we can model this phenomena by letting the birds aligning their velocity with their neighbors. So, but these birds are not able to see all their neighbors. In fact, they have a vision cone, and so these interactions have a directionality. One bird will try to align with the bird in front of them, but not the other way around. And so another example is this one represented by three particles. Each of these three particles are subject to a harmonic potential, and on top of that, they may interact with one another. So, in this drawing, for example, the green particle is coupled to both purple and orange, but the orange particle is not coupled to any other particle, and in that sense, these type of interactions are non-reciprocal. So, this system was generalized in the paper by Lozan Klopp, and one of the conclusions is that non-reciprocal interactions are out of equilibrium in general, except for some particular choice of the parameters. So, in the system I'm going to show, there's a similar type of condition for detailed balance. So, this is the outline of my talk. I'm going to present the model of dock and ship. Then I'm going to talk about the two-point correlation function and the entropy production. So, the main motivation is to understand how non-reciprocal interactions are related to time irreversibility, and also another very strong motivation for us is to understand how to implement pairwise interactions in a microscopic theory. So, we derive a field theory that is able to retain these microscopic interactions. Okay, so this is the model of dock and ship. We consider two species of particles. One is dock, the other one is ship. Particles are subject to diffusion and self-propulsion. And to simplify things, we can think that they don't have any self-propulsion. So, essentially, they can be passive particles just diffusing around. Now, on top of that, they may interact by pairwise potentials. So, a particle that is in one species will interact with particles from another species, and they will do so by these pairwise interactions. The non-reciprocity comes about when these pair potentials are different. So, for example, the dock may be attracted to the ship, and the ship may be repelled by the dock, and so one is chasing the other one. Okay, so here's a simulation. So, here we've got these black dots represent ship, and the red cross represents the dock. And so this dock is running towards wherever this ship accumulates. And this ship run away from the dock, and as a result, there are these corridors or channels of empty space. And this is another example. So here, the particles have a very small diffusion constant, so they are quite localized. And anything that's happening in this system is because of these non-reciprocal interactions between this dock and the ship. Sorry? No, no, no, it looks like the other one. Ah. I recommend to watch some videos. Okay, so this is the mathematical description of the system. We have this system of coupled over-domed Langevan equations, and so X is the position of the dock and Y are the position of each ship. Each of them have a self-propulsion indicated by U, then there's the diffusion constant, D, and these pairwise interactions mediated by the potentials. So you can see that the dock is interacting with all of them, sorry, with all the ship, because it has this sum, so it can interact with each of them, and then the ship only interacts one to one with the dock. And equivalently, this is, we can write the Fokker-Planck equation, so you can see that the interactions enter in the drift of these particles. And the reason why I show this Fokker-Planck equation is because this is what we need in order to derive the Doipaliti field theory. So because we have two species of particles, then we have two types of fields. One is Phi for the dock and one is Psi for the ship. And within each of these fields, there is an annihilation field, for example Phi, and a doyshifted creation field, which is Phi tilde. So I don't have time to go into the details of how to derive this, but you can see that from the Fokker-Planck equation. We can use that to write this action functional so that the part governing the free motion of the particles is in the A0, so the Gaussian model, and then anything else is in the perturbation part of the action. So yeah, without the perturbation, there's no interactions, and yeah, interactions are modeled with this extra perturbation part. So that also gives us the bare propagators. In this case, the Phi, or dock, is indicated by this blue line, and the wriggly pink indicates the field for a ship. So at the top, you can see the bare propagators governing free motion, and at the bottom, we have the two interaction vertices that we are going to use. There are more, but we are not going to use them, so I don't show that. Ah, and then there's also this dashed line which indicates the potential. So the potential acts on one field due to the presence of the other field. So the first thing we wanted to calculate was the two-point correlation function. To the structure, but if you do power counting, what do you understand out of this? So some part of interaction would be relevant, do you have any idea? In everything is, well, actually I didn't look into this particularly, but we are not going to do any renormalization groups, so. I think it's an nightmare, but it's fine. So at this moment, we didn't wonder about this. Okay, so the two-point correlation function. So in this system, there is one dock and one ship. Later on, I will talk about more particles, but for now, there's only these two particles. So the system is initialized with some, yeah, with some, well, yeah, at X naught and Y naught. And at a later time, we want to know what's the probability that the dock is at X and the ship is at Y. And so that's the perturbative way of describing this observable. Well, what we found out is not very surprising. So these two-point correlation function at stationarity can be mapped to the distribution of a particle in an external potential on a ring. So that's not super surprising. And in fact, if you don't give any, so if the self-propulsion of the particles are zero, this two-point correlation function is a Boltzmann type of distribution. So to visualize these results, we run some simulations and we chose this type of potential. So both particles have obeyed the same potential, but we may choose a different amplitude. So in this case, A is the amplitude and if A is positive, this potential is repulsive and if A is negative, the potential is attractive. So here's an example where both particles have a repulsive potential. And what happens is that they tend to stay away from each other as much as possible. When both potentials are attractive, they tend to stay close together. They're also fluctuations, so sometimes they also get a bit separate. And maybe more interesting is the case where one particle is chasing the other one. So here, the red cross has an attractive potential, so it's attracted to the black dot which has a repulsive potential. And well, so not very surprising. Again, if we calculate the two-point correlation function, the simulations match the theory and yeah. So what happens is that when both potentials are attractive, particles tend to stay very close. We can see that bump at the bottom or if they are both repulsive, they tend to be very far away. And if one is chasing the other one, essentially we have a uniform distribution of the distance between these two particles. So other observables that we're interesting to look into are the mean square distance. So you can see, oh here, at the bottom, there's the amplitude of the potential of the dog and on the vertical axis, there's the amplitude of the potential of the ship. So you can see on the left bottom corner, it's both attractive and the opposite side is both repulsive and then on the other ends is one attractive, one repulsive. So the mean square distance increases as the potentials get more and more repulsive and also we observed, well, we calculated the velocity-velocity correlation. So even though we don't have alignment interactions because of how they interact, effectively there is velocity correlations. Whenever these two particles are attractive or repulsive, their velocities tend to be anti-correlated. Whereas if one is chasing the other one, they tend to run in the same direction and their velocities are positively correlated. And we also calculated the entropy production. So here, everything is Markovian, there's nothing hidden and so we can calculate the entropy production in close form and what's most interesting is that we found this condition for detailed balance which, well, it's also consistent with numerics and what it gives us is a, what it tells us that the case where one particle is chasing the other one is the farthest scenario from equilibrium. Okay, so yeah, now very quickly but we also want to know what happens if we have many particles. And in this case, well, having access to the analytical expressions is very hard and that's something we are currently working on. Maybe in some perturbation expansion. So here, I just wanted to show that the two-point correlation function between one dock and one of the N-sheep gets more and more, it accentuates the type of feature. So here you can see that as we increase the number of ship if the two potentials are repulsive then they tend to be even farther apart. So calculating the entropy production here is more difficult because we need not only the two-point correlation function, we also need the three-point correlation function and we have N plus one particles. So that's at the moment something we want to calculate and we are working on this. Again, we obtain the same condition for detailed balance. So the explanation is quite simple. If the system is a satisfying detailed balance then introducing a new particle that satisfies also this condition will keep the system at equilibrium. And so finally, maybe the take home message is that in general this system is out of equilibrium because of these non-reciprocal interactions except in some particular conditions. And the farthest case from equilibrium is the picture where one particle is chasing the other ones. And well, so this is an example where we can calculate things maybe perturbatively but in close form and we would like to extend this system to maybe other types of motion more complicated like run and tumble or active Brownian particles, et cetera. And thank you for listening. Yeah, thank you very much for this very nice talk. Ah, very good first question. Yeah, thank you a lot for this. So rather like a conceptual kind of a question which is like, you know, I got really interested in when you said, I mean, the highest interpret production is seen in the chaser being chased like scenario which is like very, yeah, intriguing. Did you have any chance to look at like, but it's like some biological settings where you could map actually this chasing, you know, being chased scenario. And then, you know, there are some papers out there in like that are done by some biophysicists and stochastic thermodynamics that show that high centropy production isn't always biologically super meaningful and so on and so forth. So how can your results be linked to these results? Did you have the chance to take a look at that? So there are examples in biology where there are these type of chasing interactions. I have read some paper about some cells that whenever they come in contact then they tend to separate or they slow down their motion or it's much more complex. But yeah, and of course it's something that, I mean, it's something I want to know more. So I mean, if people have more examples, that's something that, yeah, I would like to see. I will find you. Okay, thank you. Thank you very much for your nice talk. I was wondering in the case when you have many particles since, yeah, for example, this dog is interacting with all the background chip particles, right? So maybe quite problematic to treat that some over the whole particle system. And you said that you're working on that and was just curious on whether you have considered like doing some kind of heart default, like relate to the density peaks of chips since those density peaks are the ones that are attracting the doctor most or something like that. Are you? Well, are you asking how we look into this and... Yeah, like which are your ideas to deal with this many particle? Well, so the diagrammatics, so the first approach was to look into the diagrammatics. So this is only for two particles. When you have more chip, then, so if this is the dog, then you start having, okay, this is the chip, and then you start having another one and so on. And so you have one dog and end chip. And the interactions are, oh, so you know, like you can have any interactions and it's really hard to keep track of. So we were puzzled about this for a long time. We know about how the system behaves when it's at equilibrium. So maybe that's an interesting point to look at. But yeah, this is still, we are working on this, so maybe we are wrong. Yeah, I think it's super cool because reminding me to this classical picture that people does when you introduce superconductors and everyone that has this picture of no, you have the ions in the lattice and then the electron comes in when there's some kind of back-and-see. And there's this chasing also and runaway. So maybe with some kind of mean field, I don't know. I think it's super cool. Like, I really like the talk, thank you. Thank you. I'm curious if you have looked at the case in which do you have many dogs and one sheep or many dogs and many sheeps? No, but it's quite symmetrical. So maybe having many dogs and many sheep would be interesting. We have many dogs and one sheep, right? I mean, if they... Well, so actually because of this, we started looking at many videos about dog and sheep and you can find some videos of sheep that start chasing the dog as well. So, yeah. So we also have a question in the chat. Do you encounter some face transitions by varying the interactions? No. So what we found is that the... Well, there are different phenomena, but the transition between these different regimes is very smooth and there's nothing sad on... No, yeah, okay. Benjamin, maybe the last question. Thanks for the talk. Also, to continue with this discussion of the many sheep, one dog, is there a hydrodynamic limit that you can take because you have already these fields that are somehow a coarse-grain description. Can you replace it by a sheep liquid in the limit of n going to infinity? Well... I mean, would it simplify? Because of course, you showed the plots for 100 sheep and I imagine the coin at work is being terrible. But if you take n to infinity, is there... So that's something that... I would like to know, because, for example, we were wondering about having an effective distribution, effective n-particle correlation function. That's something... So I don't know about it. But in this case, if you have a perfect symmetry of sheeps around you, you will get stuck again, right? Because you will be equally attracted to all sides. Yeah, there are fluctuating sheep liquid. Okay, and if you do that, can you please draw that for us in the next talk? Because I would like to see how this looks like. Okay, I think on that note, we should go to the coffee break and end the session and thank you again, Rosalba, and all the speakers of this session for them. Nice talk.