 We are more relaxed with the time now, so because we managed to basically make up for some of them this time. Okay, let's proceed with your talk. Okay. Okay, good afternoon everyone. So today, I will present a part of a larger work which is called Controlling Particle Currents with Evaporation and Resetting. And I've done this work with the supervision of Edgar and my other supervisor, Andrea Gambassi and collaboration of Shamik Gupta. So I go straight forward to the model. And so I consider a Brownian particle which diffuses on interval minus LL. It starts from a position X0. And then we add the periodic boundary condition at the edges of this interval. So minus L and L become topologically equivalent. This means that each time the particle crosses the L edge, it comes back to the other side and whichever side. On top of this motion, we add another mechanism which is a resetting. In particular, we consider the region minus AA which is marked by yellow color. And you can see that in this region, there is a constant rate R in which the particle can be taken and brought to the position X of R, that is the resetting position. And so in general, this type of problem are well-described by a master equation. And which of course describes the time evolution of a probability density of particle. In particular, for our problem, we have three contribution to this master equation. The first one is the usual diffusion. The second is a loss term which keeps track of the particle that leave the yellow area and because of resetting. And then we have the last term which is again termed to the position X of R and it is proportional to the fraction of a particle that are reset at time T. So this equation in principle, you can use it for any potential, I mean for any resetting rate space dependent. And of course, then you add the constraint of a periodic boundary condition which is for our problem. Then there is let's say a natural question that arises from this type of treatment in the case of a finite size system such as periodic boundary conditions in which we ask ourselves, how does resetting occur? In particular, we know that let's say as the problem is structured that resetting is instantaneous. This means that each time a particle in the interval minus a a is reset to the position X of R, this phenomenon happens instantaneously. Then we might have two possible way of reaching this point X of R which is clockwise or counterclockwise. This allows us to define a protocol that is a resetting protocols which tells us for each time a particle reset which is the probability to go left or to go right. In particular, here I present an example of a resetting protocol. And we can see this is the shortest path protocol which means that in general, each time a particle we can see for example, the first picture on the top. We have that a particle resetting at the point X can reach the point X of R into possible paths. The first one has length X minus X of R while the other is just the complementary and which is the protocol which induces the particle to reach X of R with the shortest path. So in the first example, so the first picture on the top we have the particle moves clockwise and in the second picture we have counterclockwise movement of the particle. So this is, let's say, it's an important property of the system. It can be basically reassured in what is called caging virus. In particular, as we have already said, this type of resetting is instantaneous. This means that the probability density of particle will be completely independent, insensitive to how particle is set because it just depends on the position of the particle and not how particle is set. On the other side, we have infinite possible protocols that is we can generate infinite possible currents in our system. And this converges to a unique property that is a gaging variance for our probability distribution. In particular, we can have for the same probability distribution infinity many possible currents. This can be also understood mathematically by just employing the master equation. In particular, since the probability is always conserved at this time, we can define a total conserved current, which is called j tot, and which is the sum of two currents. One is the usual diffusive current given by fixed load and is given like third equation. And then there is the last the second contribution to the total current is the resetting current. This resetting current can be analytically computed by integrating the right hand side of the master equation. And this leads immediately to the question at the bottom of the page. So we have that the resetting current at position X at time t is equal to a space independent constant, which depends only on time plus a contribution which comes from integrating the probability density. So as we can see, this space independent constant is what generates these infinitely many possible currents because once you fix the protocol, it automatically fixes this constant. Of course, this is just, so the fixing a protocol is equivalent to fixing the resetting current in a point on the ring. Of course, you can change minus L with another point on the ring, but then you have just to modify the extreme of integration in this formula. So recapitulating, you have that for any protocol, you can define a resetting current, which is different and vice versa for any fixed constant in this resetting current you can generate, you can associate a protocol. And here I show you the stationary current in the case of the short path protocol. This means that we are taking the limit of really large time when the quantities do not depend anymore on time. And of course, since we are on a one dimensional system, the current will be also space independent. And this means that we can plot the value of the total current in the figure A, the quantity J, which is our J tot as a function of different parameters of our problem. In this case, we have the resetting point and the different values of the resetting grade. So as we can see, this current is completely symmetric with respect to the origin. So one can just stick to the, let's say, right part of this plot and see that. So in this case, we have a particle that resets according to shortest path protocol. So each time a particle resets inside this gray area, it will stick inside this gray area. Whatever, the shortest path is always inside this gray area. So this means that particle are confined inside the resetting region. On the other side, so this means that whenever you increase your resetting grade, the particle will be more constrained inside this resetting area. This is why this green line is completely flat because there is no current and which means that the particle are basically stuck in that region. Then whenever you move, let's say in a plot, always in plot A from XR over A equal from one to three. So right outside the gray area to the first peak, you have that the current increases because the resetting pushes away particle along the shortest path, which still is the one that connects the gray area to the resetting point. When the resetting point reaches this peak in correspondence to XR over A equal three approximately, you see that there is a second contribution to the current which comes from particles that reset not along the positive side but also from the other side making a tool. This means that we'll have two source of current which we'll have negative contribution and this will reduce what will suppress the current. So here we present also the comparison with the simulation in figure B for a particular choice of a resetting rate. Then I also show here the probability distribution. Let's say these are two snapshots of the probability density of particle and for the same parameter except the resetting point which is marked by the vertical blue line. In particular in the first picture, we have the resetting point is outside the gray area that is the resetting region. And this means that each time there is a complete depletion of particles inside the resetting region. So the particles are pushed away from the resetting region and we have a peak outside the resetting region correspondence of the resetting point. Moreover, this is a feature of problems with resetting and the probability density as a casp and correspondence of the resetting point. Then there is in the second figure, we have instead the case in which the resetting point is inside the gray area, the resetting region. And in this case, the probability density is actually picked inside this region. And yes? Generally at about minute 12, are you wrapping up? Yes, yes, I'm wrapping up. Okay, great. Okay, so finally, like conclusion, I, what is nice about this problem is that we can see how topological constraints, so the fact that we don't have an open geometry but a ring, modifies the features of a system. And in particular, we see how it is possible to generate infinitely many currents in which we have complete control. And we see how all these currents, let's say, converge to the same probability distribution. Well, thank you very much for the nice talk and observing the time. Since questions are fun, let's have one quick question. Okay. I have a question. Okay, Alina, go on, please. Thank you for your talk. So I had a question about the resetting. Is it possible to envision a resetting which happens in an oscillatory manner? So it goes forward, then back, then more forward, then back, then eventually hits the original position. Oh, okay. Yes, resetting. I think maybe if you want, you can generalize the idea by adding a noise to the resetting position. Like generalizing not resetting position as a delta pointed in X bar, but as a distribution that you can do. Yes. Because I'm thinking of, let's say, applications that I have in mind where that's more likely to be realistically happening when you have like a particle moving in a bath, in a media. Yes. It's very unlikely that you'd have instantaneous resetting without collisions, right? Yes, yes. So anyway, Joseph's curious. Thanks. Yes, yes, you can do that, of course. Great. Thank you very much for the question too. So the next question is,