 in the Fick Jacobs equation for diffusion in narrow channels, please. The hybrid floor is yours. Yeah. Thank you. And thank you to the organizer for, I mean, letting me give this short talk. I would talk about this long-range effect of Fick Jacobs equation. Fick Jacobs equation is an approximate approach to deal with diffusion in narrow channels. That is not exactly the single pi diffusion, because we are not dealing with the size of the particles, but just with the diffusion in a macroscopic way. Then I will talk about this, maybe this side. I mean, I will talk to you about this Fick Jacobs approach. Then I will motivate some of these long-range effects. I mean, indeed, what I mean by long-range effects is an extra term in the diffusion equation that goes to the, with the fourth derivative of the concentration. And then we calculate this correction, I mean, this extra term for within this Fick Jacobs approximation, and then some conclusions. Then what is this Fick Jacobs approximation? We know that when there is diffusion in a channel, the particle, I mean, starts with the ballistic diffusion. It behaves like free diffusion, for instance, in this plot for the mean square displacement. And as the particle starts to fill the boundaries, it fills these entropic forces, and then for large times, for large times, then a particle again defuses like free diffusion, but with a different slope, that is, that we have an effective diffusion coefficient. Then this Fick Jacobs approach, what we want to do with these things is that we project over the transversal variable to have just the effective movement on the longitudinal variable that is, I mean, along the channels, for instance, like it could be in two or three additional cases. Then, well, I want to motivate that there are some experiments where this approximation can be applied, like, for instance, these transmembrane channels made of this DNA origami, or some tracking of particles of these microfluidics when, I mean, the people put these particles inside of these channels with optical tweezers, then indeed they manage how to have a different diffusion coefficient in the X and Y direction, that that's why something needed. Also, it can be applied to this ATP-ASA, right, in this part where there are the active sites. And recently, there was some approximation by Miguel Rubí from Barcelona to put some of these nanoparticles that carry some drugs inside of these blood vessels to treat some cancer tumors. Then, I mean, this is like the blood current, and then this cylinder here is maximized here, and these little spheres are the cancer cells. And then from the center, the particles need to diffuse to the cancer cells in order to, I mean, deliver the drugs from there. And then there's some simulation that says that if one puts some ultrasonic pulses there that can modify the shape of the channel, then the particle can go through the channel, and otherwise it cannot reach the target, for instance, right? And then all these can be modelled within this entropic approximation for the diffusion equation. Okay. What is done so far is, I mean, I just told you here the first approximation that is considering the two-dimensional diffusion equation. One can project over this transverse coordinate by defining the marginal distribution here. And what one kind of thing, if we assume some reflecting boundary condition, I mean, new boundary condition on the large boundary of the channel, one kind of thing, this equation that is called the Finke-Koff's equation with this extra function here that is just the width function, I mean, this the width of the channel that depends, I mean, as this channel fluctuates over the longitudinal coordinate, it depends on the position, right? Then in some experiments, we need to consider higher order correction to this equation. I mean, for instance, here we just put the constant diffusivity, right? But to match the experiment, this can be modified, given this effective position-dependent coefficient. And then people in the literature, there's a lot of, I mean, options and ways to how to obtain the right generalization for this coefficient. I mean, well, I return to this later, but here are different possibilities for the shape of this diffusion coefficient that here depends on how fast the width of the channel changes. And it could be done in a 2D challenge or 3D challenge. And there's one of the most common famous approximations in this given by Kalinayi Perkus in 2005 and 2006, something like that. And this could be done for, I mean, symmetrical straight channels, but also for channels that has different shapes. Indeed, we done some work about how to generalize this coefficient for a general 3D channel. Let me go back to slides here. Where can we interpret this thing if we rewrite the figure of the equation, like this Moldukovsky equation of, I mean, general speaking, a Fokker-Plan equation with this drift term here that is due to the shape of the channel. Then we have this entropic potential here. Okay, I was so slow, sorry. But then we want to add this long-range term to this equation. This long-range term can be modeled, for instance, by adding this Laplacian flux here. That is, I mean, the Laplacian can be understood as the upper edge of nearest neighbors for the particle. And then in a crowded environment, this is one of the simplest approximations to this. Also, we can start with this memory kernel. And I mean, for instance, makes an expansion for this kernel and then consider up to this order to have this kind of correction. Also, this appeared in a different equation, like this kind here. But let me just tell you about this model that is the Shabby-Waddy model for the taxes based on a walker, a random walker that is, I mean, persistent, but also have this census distance that can be, I mean, influenced by these neighbors. And they found this equation with these fourth-order terms and also this non-linear interesting term that takes into account this stuff. Okay, well, also these fourth-order terms appear in some approach to switching diffusion, but I skip this. And what we've done is exactly to consider this fourth-order term in the diffusion equation. I mean, this is how it looks like, I mean, when we expand it, and it then projected to one dimension. I mean, let me skip some slides here. And we found this equation after some standard procedure. This can be related to the fatigue of the standard term, but then appears this other terms. One interesting term is this here that, I mean, depends on the derivative of Y that usually that's not happened in the standard approach. I mean, that's why I put it there. But after some regression of the terms, one can arrive at this equation. And then we found that there's also a kind of flux due to this fourth-order term, but applied to this approximation that is, I mean, this is the first, where is this, here? The first approximation to the density of particles that, I mean, one can done for expansion. Then one can correct this equation through the, through the, this cali-niper coefficient or a simplified version of it to have the correction to this diffusion coefficient. Let me show you just here how correction looks like. Then here we have some correction to the diffusion coefficient. This I mean the coefficient for the second derivative and a correction for the drift term. I mean, the red terms here are the standard terms in this one-dimensional approach. And here appears this new scale that is the coefficient of the fourth-order correction to the diffusion, to the standard diffusion. And for instance, I mean, these are the corrections. And for instance, we test this, we test the expression for an error channel. And we saw that also in this lowest-order approximation, I mean, usually in the lowest order, Fick Jacobs has constant diffusivity, but here also have some corrections, position-dependent corrections due to this new scale. And then when we go further and try to apply this generalized, I mean, projection method, we need to consider all these three, well, I mean, these scales, that is the diffusivity ratio. And here this lambda plus sigma is like the new escape plus the diffusivity ratio, essentially. And what we find so far, I mean, is that the projection method gives you some extra terms that can be written like this. It's not just a simple polynomial term, but an operator on this marginal probability, I mean, this concentration of the particles. And then we find this correction, right? This is operators here. And finally, just let me show you the, I mean, no, sorry, this here, that this is the correction for the, I mean, diffusion coefficients that depends on both the scales on the bi-harmonic term and also the scale in the diffusivity ratio. And also appears some correction with the same shape for the trig term and so on. So this can be interesting. I mean, this is how we specced in this narrow channel approximation to find these large long range effects. And well, I think that's it. Sorry for the delay. Thank you Guillermo. Any questions? So essentially you, how can I say, so if you have an underlying stochastic process, which is Markovian, if I'm not wrong, when you write the, the, the, the, the Kramers-Weil expansion. So if you, if you, if it truncates, you get the diffusion. But now if you add the higher order derivatives, in principle, I think you should add infinitely many. Yes. So there is a theorem, no? So the fact that now you are adding by hand some of them is not going to spoil some fundamental property. Because, you know, there is this Paulast theorem tells you that if your process is a bona fide, the stochastic process, blah, blah, blah, blah, then if you add the fourth derivative, then you have to add infinitely many, so the 68 and so on and so forth. So that's why I was a bit, you know, are you sure you are not going to describe something which is not a stochastic process? Okay. Yeah. Well, maybe this also did with an hydrodynamic effect or something there. I don't, I'm not sure. I mean, you can make a, you can say, well, I mean, maybe it's quantitatively less relevant, but the problem is that you are spoiling a fundamental property, you know, so I would be very, very cautious. Okay. There is another one. Thanks for the talk. I have a more broader question about the boundary. Are there results known for disordered boundaries, if the boundary WX itself is a random? Oh, yeah. I mean, this is not, I mean, we are not thinking in that case, but there's a couple of papers about it. God, we have some one paper on this. And then, well, in that paper, they not also put this random stuff on the boundary, but also a probability of stick to the boundary. That is also important for instance, biological cases, but here we are just dealing with reflective simple purposes. Yeah, we're indeed working on some defusing boundary condition, but yeah, it's a different story. Thank you. Thanks. Any other questions? Yes. My first question is that I didn't understand completely where the long range effect comes from. It is more mean field effect. Yes, by, yeah. Well, as I said, we need to be careful, but the main idea is like this second derivative in the flux can appear due to the upper edge of the nearest neighbors of some walker, for instance, right? Yes. And then you put it by hand on the flux. Okay. And my other question is that the terms you add to your Fokker Planck equation was due to the non-linearity of non-Morocovian or was due to the non-Morocovian effect or was due to the mean field effect? I mean, the last, the very last effect was due to the adding of this large long range effect. But in the first part, when we add this width function, it appears naturally due to the, I mean, let me call it the entropic effects of the boundary on the equation. Yes. But in the second case, yes, it's due to the same flux. Okay. Thank you. Any further questions? No questions in the chat, I guess. If not, you can send an email to Sarah. Let's thank you again.