 First of all, I would like to thank the organizers for organizing this meeting because it looks like just coming back to normal conditions and I'm very happy about that such that we can communicate directly, which looks much, much easier and more, more efficient. So and second, yeah, thanks for giving me there just for having me here. So this work is a joint work with Felix Höfling, which we started just very recently and this is about confined colloidal motion and how and why would we consider that like memory effects. So in the context of non-marcovan dynamics or generalized Langevin equation. So basically confined colloidal motion, so colloidal means overdamped and this was covered in the second talk pretty well, confined in the previous talk, we also had a very simple example where you can see what happens and basically this is what I'm going to talk about. So generalized Langevin equations, they are efficient means just if you want to reduce the complexity of a system. If you have many, many particles, you can just encapsulate a big number of irrelevant degrees of freedom into memory kernel and then you have an efficient model. So the second point is that, and this is a formidable challenge, this is really a serious task, is how to parametrize the generalized Langevin equation and in other words, how to extract the memory from the forgiven system. So there are many studies which look at exactly the opposite problem when you have either a combination of memory kernels for instance, exponentials like in the disk elastic medium and how you can fit the given experimental data. So here I want to really focus on the question if you are given a set of experimental data or just a simulation data and the statistic can be limited. So how you can extract the memory kernel out of it. So on the basic ingredients of our system is just the passive or active colors which means that they are over that. Then periodic potential and this means that you have some periodicity along one direction or many directions and the approach is data driven. So data driven in the sense you can be limited to a statistic. So what can you do? So the first problem is that it's not hard. It's not easy to solve this equation. And the second is that if the statistic is limited you are really in trouble. So, and by answering the question why is it interesting? You can consider a number of systems where for instance if you can see the passive colors then you can have a number of system where you have periodic conditions. For instance, you have an array of traps. So this is a completely one dimensional geometry and you have already confinement by the periodic potential. You can drive the system like with a constant force and this was again very similar to the previous talk where you can even temporarily drive the so for instance with the temporal oscillations then this gives you more complicated effects like temporal mode locking. So on the other hand, if you go further the geometry can be two dimensional. The particles, there can be a lot of nanoparticles trapped in a potential there can be interacting on its own and additionally you can drive them and this gives you the possibility of having giant diffusion and also directional locking. When the particles deflect from the direction of the driving force they start to move because of the interaction in some other direction. So if you add activity to the system then it becomes more even more complicated and even more interesting. For instance, instead of the energetic landscape you can have purely geometric landscapes when you have a number of colloidal particles glued together and put an active particle on top of that so the particle fills the background and this is kind of not strict but still confined. So there is thermal noise and so on and due to activity you can have the possibility that the particle just from time to time changes the direction of motion. So this is another factor which makes the system interesting. If you additionally drive the system so if you just put, if you take the simplest system just for a single particle if you drive it additionally with an external constant force then you can see that for an active system the depending transition changes. So it looks very similar but still you can demonstrate that the behavior so the known, depending transition known for the passive system is completely different. So the nature changes and also the diffusion it's in some sense super giant. So it's also giant but you can increase due to the activity you can also tune and it becomes okay not others of magnitude larger but still yeah, five or six or 10 times larger and so on. So this is the motivation and if we start with the conventional Langevin equation just for a single particle so this is our generalized Langevin equation for a single particle for the velocity of mass m zeta is placed the role of friction and this is also the memory camel. So the fluctuation dissipation theorem is just to make sure that the thermal noises is modeled in a way that it doesn't break the physics and then if you proceed to the velocity of the correlation function and then if you do the Laplace Fourier transform basically this means that you just transform your system from the real time to the frequency space and then there is a relation between the velocity of the correlation function and this memory kernel written in the frequency space and this formula, this simple formula without going into the deep math you can consider as the definition of memory. So what you have here can be called memory for conventional under damp system and this is what we basically do and to solve this equation how to extract the memory so this is basically not very easy but still we have recently considered two situations which give you two rules how to extract the memory one is just through the frequency space and another one just in the in the time domain. So just to get the rough feeling how it works if we consider the trivial the simplest possible example of the Markovian friction if you just have just an equation for a particle moving in the viscous medium then with thermal noise then you can proceed so this is just an immediate calculation for the velocity of the correlation function then you proceed to the mobility because they're related directly and then here are close to this formula which defines the memory and if you compare so if you look at the previous slide here you can just read off this value and you see that there is no dependence on the frequency it's just static friction. So this is trivial but it just gives you a feeling how it works. So this is how we proceed in the frequency space of course there are some subtleties but still this is the method which seems to work for real data and if you want to do the same in the time domain then I just don't want to discuss the details just it's just a single paragraph go around the formula 23 in this publication and try what is important is that there are two things first of all if you start from the same equation you don't integrate you can integrate it but it's much more efficient and less numerical errors if you go for the time integral so not directly for the core but for the time integral this is one thing so this is doable and the second one if you discretize that and solve the solution it looks a bit awful but if you do the predictor character scheme so if you take into account on the flight if you take a weighted average of your result so it doesn't look so awful and oscillating it gets more and more reasonable if you also compare with the analytical solution so I cannot prove mathematically that this is something the most reasonable in the world but still it's just a scheme which works it's finally perfect. So now if we proceed to the case of confined colloidal motion so confined and colloidal first of all confined it can be trapped in an optical potential somewhere else and the particle if you just put and formulate an equation of motion then we can ask a question can we put the complexity given by the potential or the strapping force into the memory kernel so this is how we core strain the system and the second point is that this is a colloid and they are usually slowly moving so just colloidal particles are one micrometer so pretty large and they move with also micrometer per second maybe at maximum 10 micrometer per second so if you evaluate the radiance number so you see that it's an incredibly low value so this means that so it basically measures the ratio of inertia to viscous forces and this means that here you can neglect the inertia term and proceed to the reduce system so here you have to make the decomposition so this is not mathematically rigorous but still you split off the Markovian contribution and then put the rest if you do the same procedure as before you have to modify the definition of your memory so the formula looks similar but it's not identical and just to make sure that it works as it should so we just have considered a simple example where you trapped the particle you didn't drive it so everything is, you can calculate and it works well I don't go into the details the only thing is that if you compare with numerics with analytics all the information about the constants they are known and this means that previously for instance in this system which I mentioned we did it ad hoc in the sense that we have assumed something and now we know exactly why the diffusion and how it depends so all the parametric dependencies are captured with that approach so this means that physically it's not it's very transparent so if the particle is freely diffusing you have some background diffusivity if it's trapped then the diffusivity is reduced so just the intensity of diffusive motion is suppressed and in between you have just the crossover from one region to another one and now we know all the dependencies in terms of original parameters within for this system so this brings me to the conclusion and basically in typical MD, MD simulations or underdamped systems you can see and pretty much capture long-term tails and so on so in contrast for colloidal system you have to go to the overdamped limit that it seems helpful and from the point of view of complexity reduction so the memory function approach can give you some insight some details about the parametric dependencies of course probably you can get them in the standard way but this is just for free and it looks like it makes sense and if you add activity especially if you combine activity with the confinement this makes the system really interesting so this is an interplay of counteracting tendencies and this is just in progress thanks thank you very much for this nice talk any questions? Yes, Medidja? You mentioned that there was a difference in the depening of an active particle so can you say a little bit more? Is it the beta exponent or is it the creep? The difference between both and what? The pinning at a certain point you mentioned the pinning at the beginning there, yeah here? Go back No, this one No, maybe later four I guess this no, one less the active pinning Ah, I mean you mean active the pinning Yeah, so can you say a little bit more about the difference with the normal one? Well, you have to just consider the exponents of the transition because if you have no noise it's just one half the square root behavior if you go for the active noise first the degree is different and also if you switch if you keep the active noise so there is thermal noise and there is active noise and they act differently and if you completely switch off the thermal noise then the exponent in this transition will be dependent on the activity so it really it works like a parameter so this means that it's no longer one But you mean the exponent in the creep? I mean it's no longer one half just the short answer Any further questions? Yes So, I'm not sure about everything but it seems to me that you are saying okay, so give me the experimental data and we'll get your memory function or effective equation essentially Yeah, this is the dream This is the dream So my question is what happens if your equation is both non-linear and non-marcovian? That's hard That's even not easy in the simplest case when you have just a linear equation because well, it's not just Yeah, you don't know if it works If it works, okay if it doesn't work you have to think what to modify I don't know Okay Pragmatic answer Any other questions? If not, then let's thank... Oh, there is... I think that's Andrea's comment I'm not sure but apparently we have several chats and we only see one Yes, please Those following online send the questions in the chat so that they are visible to everyone Yeah, if anybody is interested you can just send me an email Or you can send an email to Sarah Or preferably all of the mentioned In particular, the email is... Apparently she has a big inbox, so... Okay, then if there are no further questions then let's thank the speakers of the entire session and enjoy lunch So maybe for the...