 Yes, I am. I think I'll just be sharing the PDF version of my talk instead. I think it will be much faster. So I hope this works now. No, we are waiting. So Angela, maybe you switch off your, I don't know, microphone, carry it. No, we are not seeing anything. What does it say? It says you are presenting. Does it say like that? I just clicked on the present now button. Now I'm waiting for the share button to light up. It's quite slow, actually. Strange, right? Because as soon as I press present now, I see the options. But you have to wait for that. Can I try uploading? And Neil can, I have a backup. Maybe, sure, sure. Sure, that would help. Thank you. So I'll just click on the cancel button instead. Yeah, yeah, you can also click it. Let's try this, OK? OK, and wait. I can present it, Neil. Will you be using the PowerPoint version, Angel, or the PBS? Yeah, the PBS. You had success with your case. Maybe, Angela can, yeah, you can proceed with presenting these slides, and Neil explains. I can see the presentation. Please do that. OK, OK. OK, great. So Neil, so we have to be strict about the time, because we are kind of over time. So I'm going to interrupt you at three minutes, and then at 10. OK, go on, please. OK, so again, I'm Neil Kaidik from the National Institute of Physics. I work on my thesis under the supervision of Dr. Perius Guerra. I will be presenting tonight some parts of my thesis actually entitled Stochastic Behavior of Two-Wall and Equally Bias Brownian Particles with Internal Resetting. Next slide, please. So before we start, maybe I just give you an overview of what to expect from this talk. So basically, for this paper, we study two particles diffusing in one dimension with an equal, so take note of the unequal part there and oppositely directed biases that drive them toward each other. So these biases actually would force the particles to eventually collide. But the system does not allow collision. So instead, just before colliding with each other, the particles go back to their initial position. That is where resetting happens. Next, please. So to study or to quantify the behavior of the system, first, we set up continuous mass or equation for the system via random walk formulation and then analytically obtain the time-dependent solution. By the way, this time-dependent solution is also the probability density function of the system. Next, please. And then from the probability density function, or we also call PDF, we determine the resetting rate of the system. And then we also take the steady-state probability density function and then analyze it. So basically, we have three quantities to show today, the probability density function at all times, the resetting rate at all times also, and then the steady-state behavior of the system via the steady-state probability density function. So next piece. So first, why Brownian motion with resetting? Separately, these two processes, Brownian motion and resetting, are actually very prevalent in nature. In fact, they have proven to be very effective in modeling various kinds of systems. That's why it actually came as a surprise, if I may say, that the seminal paper on Brownian motion with resetting was only published last 2010, I think. Although after that, there were so many papers on Brownian motion with resetting, which came pouring in. So what's common among these papers is, well, some of the quantities which were infinite or which they verge in Brownian systems without resetting, they become finite now because of resetting. And so one of the examples there would be the mean first passage time. With resetting, they become finite. Well, of course, for some system, not all though. And then also for the first passage time, and also it has been found that for many systems, the resetting actually would force the system to generate or to reach non-equilibrium steady-state. And then what is also common among the system is that the type of resetting are external. So that means an external force decides if the system resets or not. So what's not so common is the internal resetting, and that is why we are interested in the topic. So it was actually in 2017 when, I think, one of the few, the second, maybe, paper on Brownian motion with internal resetting was published. It was by Evans and Falsao. So as we go along the discussion, I will also be presenting some of the results that they presented in their paper for comparison. A next slide, please. So to be able to get the master equation, I think it's better that we discuss the system first. Next piece. So our system is actually a system of two particles which are confined to move in one dimension, from negative infinity to positive infinity. Next piece. So these particles originally can be found at positions negative L and L. Next piece. There. So that's their position. So the particle on the left would move to the right, a probability of 1 half plus alpha, and then to the left, next piece, at 1 half minus alpha. And the particle on the right would have 1 half plus beta to the left and then 1 half minus beta to the right. Now take note, this is actually where the modification of our paper from the paper of Falsao and Evans came into the picture. So in their paper, the alpha and the beta here are equal. But for our paper, we relaxed it a bit and we would allow any value for alpha and beta. Of course, it should range from 0 to 1 half. Next piece. Then to continue the master equation, it is good to introduce a new set of variables. That is y is equal to xr plus xl over 2. So basically, you can picture it the location of the center mass of the system and then z as 1 half of the distance between the particles. Next piece. So just go on. So just keep on clicking. OK, just keep on clicking. So I'll just discuss the whole thing. So sorry. There. For the master equation, you would want actually to relate the probability of finding the particles at certain locations at time t. And then we connect it or we relate it to the probability of finding them at certain locations at t minus 1 or 1 time step back. So one example, for example, the green colored arrows there. So for the particles to be found at the present location now, it could be that t minus 1, the yellow dot, is from one step to the left. And then the red dot is from one step to the right. And then the probability of these particles going to the location right now. Let's just focus on the green arrows first. It's actually equal to the probability that they can be found at the location of the tail of the green arrow. And then the yellow dot would go to the right. The probability is 1 half plus alpha. And then the red dot would go to the left. The probability is 1 half plus beta. And same thing with the other possible configurations. So all we need to do now is just to add these probabilities. That is actually the probability of finding the particles at the current location. Next piece, Angel, sorry. So we just have to be careful, though, with this configuration, wherein we find the particles at xl equal to negative l and then xr at positive l. Because when we find them at these positions, it could mean that they come from the configurations that we presented, or it could be that they just came from a collision, which is not allowed in the system. So take note that the collision can happen from negative infinity to positive infinity. That is why our probability for this case, the contribution, would be a summation in the form of a summation k from negative infinity to positive infinity. Take note that we need to insert the conical delta there, delta y0, delta zl, so that this possible configuration will only have a contribution to the probability if y equals 0 and c equals l, or when the particles are at their resetting position. Otherwise, this whole thing will be 0, and we will go back to the four configurations that we just discussed. Next piece. So the master equation is just actually the sum of these probabilities. To transform this discrete master equation into continuous master equation, all we have to do is just to let delta z, delta y, delta t approach 0 and then take this Taylor series, and then eventually we get, next slide please, this. This is now our continuous master equation. Then we now have a chronicle delta y, chronicle delta x minus l f of t. The f of t there is defined at the bottom most part of the slide. We also call it the resetting rate. Take note that this equation is actually a non-homogenous differential equation, but the non-homogenous part, which is the direct delta y, direct delta x minus l f of t, that part there is actually 0 if y is not simultaneously y is equal to 0 and x equal to l. So the rest of the values there, if y is not equal to 0 and x equal to l, they just become 0. So they actually correspond to the homogeneous solution to the equation. But if y equals 0 and. Neil, I'm sorry for interrupting. I didn't want to interrupt you earlier, but this is the 10 minute mark, so we have five minutes. But let's leave some time for questions. OK, go on. OK, OK. All right. But if y equals 0 and x minus l, then the non-homogenous part will just be f of d. So that we can actually get the complete solution to the master equation given in the next slide. Next slide please, Angel. Sorry. OK, so that. So again, the first term of the right side of the solution is corresponds to the homogeneous equation. And then the next term is the, no, correspond to the homogeneous equation. The next term is non-homogenous equation, which is actually solved, well, at least for our case, using the Green's function method. And then we also solve for g, y, z of d. And it is given in the most part of the slide there. And then all we have to do is just to insert g, y of z of d to the equation. It's just that we don't know f of d prime yet, and that's what we're going to solve. To solve f of d, all we have to do is to insert the p, y of z of d to the equation and then get the equation at the bottom and then take the Laplace transform of the whole equation. And then we get this, correct, and the next, OK. And then finally, we get the resetting rate, which is given at the bottom most part of the slide. This resetting rate is actually equal to the resetting rate obtained by Falsao and Evans in their paper. What's interesting about it here in the context of our paper is that the press, well, u, which is alpha minus beta, the additional part because of the non-equal bias, the additional parameter resulting from what you call is an equal bias is absent in this resetting rate. So what we have here is just the presence of v, which is the drift of the system, which is equal to alpha plus beta times delta x over delta t. So what does it mean? Since v only v is present, sorry, could you go back to the previous slide first? OK, since only v is present, it means that even if the system or the particles have an equal biases, as long as the sum of their biases are equal, then the resetting rate is the same. So that means for a particle with, say, alpha equal to 1 fourth and beta equal to 1 fourth, so alpha plus beta equals 1 half, it will have the same resetting rate as a system with particle, say, with alpha equal to 1 third and then alpha equal to 1 sixth, because the sum of that would also be 1 half. So there, next slide, please. OK, so we tried to plot the resetting rate. These two plots here are actually recreation of the plot obtained by Evans and Falsal. So what they observed, actually, was that the resetting rate would eventually approach a certain constant as time with sufficiently long enough time. Then for this case, they plotted, say, P, E, V. This is actually picklet numbers. So that is small v, big d, 0.03, and then the other one is high picklet number. Now we added a mid value, so P, E, V equal to 3. And then for the sake of illustration, we chose v equal to 3 and v equal to 1.5, because that would mean that v over L will be 1. And it's obvious here that, in fact, the plot approaches 1 as t approaches infinity. And that is actually the constant that the resetting rate approaches 2. That is v over L as t approaches infinity. So Neil, let's wrap it up, because we have 30 seconds only. All right, OK, so next, please. And then I think we can just show the plots, the probability density function plots. So basically, we got this. So next slide, please. And the next slide. Next slide, plus, sorry. OK, here, if you've noticed, for the third and fourth plot, so that is the top most. So we can see already, because in the original, when u is equal to 0, or when alpha and beta are equal. So Neil, over time, we have to really cut this, because maybe 10 seconds, can you do that? OK, so maybe you can just proceed to the conclusion. So basically, this is just proof. All right, so what did we find out? The resetting rate of the system was found to be independent of the parameter u. So we just talked about it a while ago, which is the drift of the position of the center of mass of the system, and only depends on v. As time approaches infinity, the resetting rate approaches a constant value, which is equal to v over L. Now, here is where our another result surfaces. So the probability of the system having z less than 3, so 3 is the resetting position. Those higher SV increases. Moreover, when u equals 0, the probability density is symmetric with respect to y equals 0. But when u is not equal to 0, meaning if the biases are not equal, the probability density shifts towards the direction of the particle with lower directional bias. So I think that's the most important. Thank you very much. All right, thank you. Thank you very much.