Yes. Thank you very much, and I want to start by thanking the organizers for letting me give this talk to present this work that was published last year. It's about signatures of irreversibility microscopic models of blocking. And the idea is very basic. We just realized that there was a gap in the literature of the beach that model surprisingly that was about the measurement of the entropy production in medical simulations using microscopic models. So we basically did it, and these allowed us to study the equilibrium limits the behavior of the face transition of the enterprise production rate and to detect signatures of irreversibility and steady state distributions. So that's basically all I will be talking about in the next 10-15 minutes. So let's start with a little bit of motivation. The motivation comes from the fact that flocking, as you know, is a general non-equilibrium phenomenon, especially in two dimensions, because the same type of face transition is not allowed in passive systems in the passive counterparts of flocking systems which are active paramagnets where we have particles which are equipped with a directional degree of freedom, something that is vector like, and it comes with a symmetry that is the continuous rotational symmetry and paramagnetic short range interactions. If we add to this ingredient self-propulsion, then we have active paramagnets, beach-like like models, and in this case through long range order can be observed even in two dimensions. So this model apparently violates the Merman-Wagner theorem, and this was known since the beginning, since the introduction of the beach-like model, but it is not self-propulsion per se that causes this effect. And this has been shown very clearly in 2020 by Hal Tosaki, who showed that if one builds a model that where the particles are more tiled, self-propelled, but the dynamics fulfills detailed balance, there is no spontaneous symmetry breaking of the rotational symmetry in two dimensions. So this means that irreversibility is the key ingredient that is needed to observe this flocking transition. So these motivates are interesting in trying to quantify this irreversibility in flocking models. And we decided to study the active XY model, which is a continuous time counterpart of the beach-like model that is just more analytically tractable, and it consists in describing the system as a collection of active branch particles, which have a constant self-propulsion speed P0. And the dynamics of the orientational degrees of freedom follows this stochastic differential equation where the deterministic part of the deterministic torque comes from an XY Hamiltonian. And in this XY Hamiltonian, J is the strength of the interaction, we will set it to one, and NIJ is the connectivity matrix. It tells us whether bird I and bird J interact or not. And since we're considering short-range thermo-magnetic interactions, which are also reciprocal because they are derived from Hamiltonian in this case, we have that they depend on some notion of distance between the particles. They depend on the positions, on the special configurations of our birds, and as particles are motile, this can change over time. So this creates an effective time dependence of the n-parameter that appears in this XY Hamiltonian. And this is what brings the system out of equilibrium. It's the coupling between the external degrees of freedom that are the positions and the internal degrees of freedom that are these orientational variables. So if we want to compute the entropy production rate, we need to specify the time reversal parity of the coordinates. And we assume that the positions, we interpret X as the position, so we assume that they transform as even variables under time reversal. Whereas we interpret V0 times E of theta as physical velocities. So they transform as odd variables under time reversal. So theta is mapped onto theta plus pi. And with this prescription, we could compute to the entropy production rate. We found two formulas starting from these equations for the active X-ray model, which are equivalent in the steady state. The first formula can be interpreted as the rate of dissipation of the stochastic heat into the thermal bath. You can recognize this N-A-J sine theta i minus theta j that is proportional to the torque, the alignment torque. So this is the basically the work that the alignment torque makes on the orientational degrees of freedom. And the second formula can be obtained again from the equations assuming local detail balance, for example, if you want. Sorry. Can you repeat, please? I think that was an accident. I've muted the person. Okay, sorry. Okay, the second formula is, can also be obtained by integration by parts from the first formula, neglecting the total variation of the X-ray energy. And the second formula can be interpreted as the work of some fictitious reshuffling forces that rewire the interaction network. And as a sanity check, we just, I mean, it is plausible to assume that at the steady state, the X-ray energy is constant, at least grows sublinearly with the time length of the trajectory. And indeed, this was realized in our simulations at the steady state. So having these two formulas, we could compute the entropy production rate from simulations and we consider two variants of the model. A metric variant where the particles interact only if they are within a distance r, and a topological variant of the model where we let each particle interact only with the first shallow Boronoi neighbors in 2D. And these two models are known to exhibit a very different phenomenology. Notably, the first model, the metric model has a first order phase transition. You can see that there's space coexistence at these intermediate values of the density between a polarized and very dense cluster and dilute and disordered phase. And this is also reflected by the bimodality of the distribution of the number of neighbors. Whereas, in contrast in the topological model, the configurations, the spatial configurations look much more homogeneous independently of D, and the distribution of the number of neighbors is a uni model. Correspondingly, we find a different shape of the entropy production curves. The second one is in log scale. So it's hard to compare but we observe a key singularity for the topological model, whereas in the metric model, at least in the, I mean, as we increase the size of the clock, there seems to be a flattening in this region here where we have phase coexistence. Okay, so another thing that we can observe at a more coarse grain level is that the entropy production rate roughly peaks at the transition point in both cases. And it goes to zero in, as the temperature goes down, temperature is D here, or it can also be interpreted as the rotational diffusion coefficient of the active random particles, or SD goes to infinity. We can also replot these curves as a function of the rescaled distance from the mean field transition point for the metric model and see what is the residual dependency on the density. It seems to, it seems that the entropy production rate gets lower as the density is increased. In the logical case we have the opposite trend with density. Namely, the entropy production rate increases as the density is increased and this is just due to the fact that increasing density is as the same effect as increasing the self propulsion speed so these increases the rate of reshuffling of the interaction network. Before saying we can also observe from these curves that there are two equilibrium limits that correspond to two well known reference systems the positive magnet in the commuting reference frame of the flock SD goes to zero, and the ideal gas of active random particles which is also an equilibrium system. In this case there are no interactions to probe the non equilibrium nature of motion. And we can rationalize these two equilibrium limits if we recall that in the active XY model, all non equilibrium effects come from the reshuffling of the interaction network, and this is suppressed in the low temperature limit, because the motion is suppressed, and in the high temperature limit, because in this case, the motion as low persistence and these suppresses reshuffling particles are just moving around their own positions. So the situation where a shopping is most efficient is actually at the transition point because the motion is persistent enough to cause reshuffling, and this persistence is not wasted into collective motion. This can also be observed by looking at the autocorrelation function of the interaction network. Another thing that we could observe close to I mean in equilibrium the two equilibrium limits was the scaling with D, the rotational diffusion coefficient and I will not go through the details of this, but with a simple approximation, we could predict the scaling in our low scaling in these two regimes as D to the one half when D goes to zero and as D to the minus two, where when D goes to infinity, and this is in good agreement with the numerical simulations, especially in the topological case. In the metric case, this agreement is still good in the high temperature phase, but the prediction is failing in the low temperature phase where we have phase coexistence. In the remaining time, I want just to mention how we detected signatures of irreversibility in steady state distributions. Again, we restarted from this expression of the entropy production rate for the active x-ray model that is interpreted as the work of fictitious reshuffling forces. And here we need to compute the derivative with respect to time of the NIJ matrix. This is easy to do if we use this parameterization for NIJ that we used in the metric model. And this gives, when we compute the time derivative, a delta function that only selects pairs of particles at a distance equal to the interaction radius and multiplies this by this factor that is proportional to the is equal to the projection of the velocity vectors onto the distance vector that connects bird i with bird j. And if we call q alpha phi, the probability distribution of pairs of particles at distance are parameterized by these two angles into dimensions and we massage the first equation, we obtain something that looks like this. So it's an average over the distribution of pairs of particles of this function epsilon alpha phi. And epsilon alpha phi is a function that does not depend on the parameters of the model. We can plot it. It looks like this. And of course, as you can see, it's completely anti-symmetric under the transformation alpha plus pi, which is the fact that the time reversal transformation has on the coordinate alpha. So in order to have a positive entropy production, we need to have an asymmetry in the distribution of pairs of particles, which was actually detected in our numerical simulations. In this plot, I'm showing the anti-symmetric part of the log of this distribution. And as you can see, there is a strong positive correlation with epsilon alpha phi. The last thing I want to mention is that this asymmetry is in fact a low dimensional manifestation of a more general asymmetry, which can be derived for the x-ray model by imposing the condition of irreversibility. It can be easily shown that if we want the entropy production rate to be positive, then the steady-state distribution must satisfy this inequality. It cannot be symmetric under the change of sign of the P, which means that the irreversibility induces an explicit symmetry in the steady-state distribution. And more generally, the same type of inequality can be derived for stochastic differential equations, which have a completely irreversible drift term, and which have in their state variables some coordinates which are odd under time reversal. And again, I don't go through the details because I don't have time, but the condition of irreversibility implies an asymmetry in the steady-state distribution, and this is quite general. So with this, I conclude, I want to thank the institutions that funded and hosted me during my PhD, when I realized this work, MIT for current fundings, and my collaborators on this project that started at the time being called their summer school on theoretical biophysics. And thank you all for your attention. I'm ready to take questions. Okay, thank you, Federica. We have time for one question, I think. David. Okay. All right, yes. Very nice talk. One thing that struck me is if you've looked at ways of generalizing it, because the behavior of EP as you go through what in the equilibrium regime would be a phase transition is a very underdeveloped issue, and in particular, it strikes me that you could discretize your space and generalize. So it actually rather than just a two-dimensional continuous space, you had a network, where as individual quote particles change, among other things, they would be changing the local network topology. And in particular, it struck me that this might be related to various models of social networks in the literature, where people, which are very often modeled where people's opinions change with time, and also who they are friends with or anti-friends with enemies with, changes with time based upon locality in some sense. Obviously, I would not expect you already thought about this, but does that make any kind of a sense? Do you think one could go that way? It would be discretizing, it wouldn't be an SDE anymore, but in many ways that might make things simpler. Yeah, indeed. I mean, I needed to discretize to compute this from numerical simulations. So, there are some caveats related to the how you discretize the strato and the convention and all that, but if one pays attention to this, I mean, in the end, all the formulas that I wrote here are continuous time writings of the discrete time dynamics and the discrete time formulas that I used to compute actually the anthropological rate and to simulate the dynamics. But yeah, I think it's, one can do the same for, for instance, staying in the regime of active matter, can do the same for the original beach deck model, which is defining at discrete times, it's a completely discrete system. Yeah, I'm just wondering more generally where we don't have like a two dimensional continuous space underlying things, a simple grid, but an arbitrary network. So it's not even necessarily a planar graph or things like that, which is all over many, many real world situations. I think, I think in view of the fact that the session is running on, let me move this discussion to the discussion period. Thank you again.