 from University of Bordeaux, and he's going to talk about first passage times of non-Markovian Gaussian random walkers. OK, thank you. So yeah, today I want to tell you a story about first passage times for non-Markovian Gaussian random walkers. So it's not the first time that we speak about first passage times. So basically, first passage problems characterize the first time needed by a stochastic variable to reach some target value. So we can define first passage time distribution and its moments, for example, mean first passage time. They are involved in a lot of interesting problems. So for example, transport influence reactions for the simple reasons that tractants have to meet before they react together, or search processes, for example, with this very nice historical example of searching a sequence on DNA. Today, I would like to speak about three paradigmatic first passage problems. So the first one is this one. So I have a random walker which diffuses in some confining volume and who is looking for some target region. In this case, the average first passage time is usually infinite and needs to be quantified. The second class of problem is the first passage in infinite space. So usually, in this kind of problem, the mean first passage time is infinite. And a third one would be the search for an energetically costly configuration. So in this case, it's a rare event kinetics. We obtain usually arenas kinetics. So the idea is that these problems are relatively well understood for Markovian random walks. We ask what happens when we add memory. So as you know, memory appears when the random walker interacts with other variables in its environment. So we have the choice between Markovian description with n degrees of freedom and can be large in statistical physics or some effective projected dynamics for the reactant only, which becomes non-Markovian. So as an example of models that we have in mind, motion is non-Markovian in all these cases. So if I take a reactant, which is attached to a polymer, or some bead moving with other beads in narrow channels, or tracer beads in complex fluids, such as pneumatics, viscoelastic fluids, or the examples that we saw yesterday with protein dynamics. And in all these examples, we have experiments or models that tell us that in some limits, at least, we have approximately Gaussian dynamics. So how could we predict mean first passage times for these kind of situations? Maybe so I will tell some existing results for these first passage times with memory. So from now on, I assume that I have a Gaussian unbiased random walk continuous at long times, which goes as a power low for the mean square displacement. So I can have sub-diffusion, sub-diffusion, or diffusion with a first exponent h here. And so now I assume that I put this random walker in confinement or in a potential in the ways that we respect statistical mechanics at a temperature t. So for example, with a generalized long-range equation. So there are some exact results to quantify first passage with memory. For example, in this model, so if I have a random walk with telegraphic noise or a random acceleration process, for the case without confinement, so here I also say that I have a stationary dynamic. So I don't care about coincides from height to from some temperature to another temperature. So I just look at a process with stationary increments. So in this case, we know what is the persistent exponent. So in infinite space, the survival probability will decay as a power low. But we don't know the pre-factor, which is hidden anyway here. So we will try to quantify that. And for relevant kinetics, in fact, there are a lot of theories. And some of them include memory because they give the results in the limit of weak noise for an arbitrarily large number of degrees of freedom. So this seems to be solved. But we will talk about this again. There are also general approaches. So for example, pseudo-Markovian approximations, which at some point assumes that the propagator for a non-Markovian process has the same properties as a Markovian one, but it can give wrong scaling. And also in the last decade, there have been a lot of works to obtain perturbative results by looking at an expansion about Brownian motion. So it's usually in 1D. And here our goal is to propose a non-exact, but also non-perturbative, formalism to catch memory effects, the impact of memory on first-passage kinetics, also in dimension, not only one, but higher. So the outline of this talk will be the following. So I will speak about these three problems one by one. So first, I look about the first passage time in confinement. So the hypotheses of this theory are the following. So I consider a Gaussian random worker, which is continuous in time, which is Gaussian with stationary increments. So this means that there is no aging, basically. So it is unbiased. So it's a symmetric work. I also assume that it is non-smooth. So at short times, basically, the trajectories are non-differentiable, as in Brownian motion. I assume that the MSD function is known. So I call it psi of t. That is a neat input from the model. I also assume that the MSD diverges at long times. So as I said, as t to the 2h. So this is to ensure that the random worker will explore the available space and not being trapped around some position. And I place it in a large confining volume. So now we want to predict the mean first passage time. So the first equation is a renewal equation. So it's a tautology. So I just consider the property to be on the target at t. And I said, if it is on the target, it has reached the target before at some time tau. So I obtain the integral of the first passage density at tau times the probability to come back on the target at t, given that the first passage was equal to tau. So if it is Markovian, then we can solve this equation, which becomes a convolution equation. But in the non-Markovian case, it's more difficult. But we can still manipulate that to obtain an equation for the average first passage time. So it goes as a volume multiplied by this integral. And here qp of t is a probability density to be on the target at a time t after the first passage. So we should focus on what happens in the future on the first of the first passage. So to do that, for example, let's consider several trajectories, so several runs. For each trajectory, we look at the value of the first passage time. We take this value as the origin of the times. So now I have rescaled times. And for example, if I take the average of these green trajectories, what I see is a shift, which I call mu of t, and which is not 0. Whereas the random walker at the beginning was a symmetric random walk. So this means, basically, that the first passage moment at the first passage, the system is not at equilibrium. And we also observe that the probability distribution functions of these future trajectories are almost Gaussian and that the variance is almost exactly the same as the variance of the initial process. So assuming that these properties are true, we derive the following results for the mean first passage time. So in the large-volume limit, it can be calculated with this integral, where there is psi of t, so the msd, mu of t, which is the average future trajectory, which we don't know yet, and the initial position, x0. And we can derive a self-consistent equation to calculate mu of t. So it's a little bit ugly, but it's quite compact also. So just by solving this equation here, we can predict mu of t as a function of psi, report here, and then have predictions for the mean first passage time. An important thing also is that if I do a pseudo-marquivir approximation, so I will just say that at the first passage, the system is at equilibrium, so it gives me mu of t equals 0. And I will obtain some transition at h equals 1 over 3, which is not seen in simulations. We can derive the asymptotic behavior of these trajectories for large time, so it goes like this. So it's a very simple formula. x0 minus some power law with the exponent which changes at h equals 1 half. So basically, we should see these qualitative things. If the work is sub-diffusive, then after the first passage, it will come back slowly to the initial position of the random worker. If it is diffusive at long times, but not at all times, then it will tend to a constant. And if it is sub-diffusive, it will cross the target and go away. So the initial condition is never forgotten, and mu is not small, so we can expect a large memory effect. And that's what we see. For example, I take the fractional Bernoulli motion. Here I plot the first passage time divided by volume as function of the initial distance to the target. Blue is simulation. Red is a theory. And green is the pseudo-Marcovian approximation, which quantitatively is not quantitatively correct. And we can also check that the value of these average trajectories are correctly predicted by the theory. And we can extend the theory. Here I have explained the theory in 1D. We can extend it to several dimensions. And I show the results also for non-scale invariant processes. So this is a result in 2D for motion of a bead looking for a target in some Maxwell fluid. Here a fractional Bernoulli motion in 2D. And here an example in 3D again with a Maxwell fluid. So we have quantitative and non-perturbative results in these cases. So now I will speak about this second problem of what happens when we have no confinement. So then the mean first passage time is infinite. But then we realized that in free space, the survival probability goes as S0 over some power low. So we know the exponent. And in confinement, we have a mean first passage time, which is finite. And what we have derived is the relation between this pre-factor S0 in this case and the limit of the mean first passage time over the volume in the other case under some decoupling approximation, which tells you what happens in the future of the first passage time, does not depend on the value of the first passage time. But it's not a very important approximation. So then we have this quantity from our formalism. So we have the pre-factor also in unconfined random works. So we have tested that our theory gives correct results in a lot of cases, again, in one or several dimensions for scale invariant or non-scale invariant processes. And now I want to spend a few minutes that are left to the case of rare events. So just to be specific, I want to take this very simple example. So I have a polymer chain. Let's take it as a stupid bead spring polymer model without a hydrodynamic interaction so that all correlation functions are known in absence of reaction. And I ask now, what's the time that the extension in some direction reaches the value z, which is much larger than the duration radius? So this is obviously a relevant problem because to reach a large extension, then you have to overcome an energy barrier. So I have a spring in series. And at the end, we see that the beta times the energy is z over the duration radius squared. But as I told you, we know how to solve this kind of problems in the weak noise limit for any number of degrees of freedom in the system. So in principle, the problem is already solved. So here I show the results of simulations, which I have taken from this reference. So this is the time rescaled by the exponential factor, which appears in the Arrhenius law, versus the number of degrees of monomers. And I fix here that delta E is 18 kT. So it's already pretty large. So very, very rare events. And what I see is that there is a discrepancy between simulations and predictions of the rigorous weak noise theory, which increases with n. So the weak noise results will not predict the correct behavior for large n. And in fact, you have a problem of limitation of limit. Basically, the weak noise limit kT equals to 0 cannot be inverted with a large n limit in the polymer model. So we want to catch collective effects. So we adapted the theory by saying that, OK. Now we have to compute the trajectory in the future of a first passage event to this elongation set. And then we will see, after the first passage, we will see some retraction dynamics with a trajectory mu of t. And so we adapted the equation. So we have an expression of the mean first passage time as this reactive trajectory. And mu of t is, again, determined by some equation. So in this equation, there is the MSD. And the initial position as it disappears appears because it's a relevant problem. It doesn't matter. But here, what is important is the force. So this is the slope of the potential at the target configuration. M was done. And this is the result for a long change. So this is the dynamics in the future of the first passage. And this is the equilibrium dynamics, which is not vanishing this time. Because if I release an equilibrium polymer at some fixed elongation set, I will see it retract. But much more slowly than after a first passage. And what we see is that we have predictions for scaling laws in the non-Markovian theory, in the Sodomarkovian theory. So we have a factor of 10 of difference. And in the rec noise, we have even different scaling laws with the number of monomers. And we can check our results with simulations. So this is the prediction. And it works perfectly. And here are also existing theories that attempted to go beyond the rec noise limit. And we see that those are still discrepancies with them. And that's it. And the conclusion is that the state of a system at first passage is not a stationary state. So basically, even if we look at equilibrium processes, which are non-Markovian, we get a non-equilibrium state at the first passage. And if we analyze the trajectory in the future, we get information for the kinetics of first passage times for these three situations. We can generalize it to imperfect reactions. So if you want to have information of that, go to the poster by Tony Mendez. And I didn't speak about any stationary initial conditions, in which case, where the persistent exponents that appear. And Nicola told you about it yesterday. So thanks for your attention. Are there questions? Students. Nobody is up. Benjamin, you have a question? Yes. In principle, yes. But then you have to do all the theory again. And I think the second moment does not just depend on you. So you have to compute other. Yeah, yeah. Somehow other things have been up here. I chose this. I was very much told, by the way, very challenging. So when you showed those fractional Brownian motions, so how do I understand that? You put a fractional Brownian motion in a confiner. Yeah. When the FPM hits the boundary and continues, it doesn't know that it hit the boundary. Or how do I make it? No, no. So what we did, in fact, so we have done two things. Because at the time, we didn't know how to put confinement in for the FPM. It was done by Metzler two years ago. Yeah, so what we did is that we have done the same as in 1D. So we said, if we have a reflecting boundary here, maybe it's just the same as hitting a second target here. So we analyzed this first-passage problem. And then we did another thing that we produced an algorithm which is with the Husking algorithm. Then we generate x8 plus 1 from the previous positions. And then we rejected the positions that went beyond this wall. But what we saw is that with these two algorithms, if the volume is large, we got more or less the same results. So basically what you were saying, that the problem with this boundary is basically fully contained in the ensemble of dots. It's just a change of measure algorithm. I don't know. Otherwise, we cannot have the same. No, this is a variance. In fact, I mean, it don't be a fractional Brownian motion anymore if you have a boundary. Yeah, of course. That was the basis, the ultimate question. That would be, in fact, we also have done simulations with a generalized range of equations with the power of the kernel inside some harmonic potential. And harmonic potential can be seen as a confinement, too. And we obtained the same results as here. Thank you. Andrea, you have a question? Yes. I mean, in the past, a long ago, there was exactly for this kind of processes, and Klaus and Omarkov, and there was this perturbative approach to calculate the persistence exponent by Klaus and Omarkov. Yes. So I don't know what they mean now. You're thinking of others. So I was wondering whether you checked out this. Yeah, yeah. In fact, so all the theory, so the theory is definitely not exact in the general case. But we think it's, we have strong arguments to say that it's exact at first order of perturbation. And then we recover all exact results that are known in the return. Actually, that's one of my questions. Actually, there's one model we know that's a random acceleration process for which you can actually calculate analytically. It's a non-Markov process. OK, so is this one we cannot? Because now it's not in the initial hypothesis. OK, all right. If there is no other question, let's thank Tomah again. I mean, this is good.