 I want to give a message to the online attendants. So as you see, you can make questions, raise hand. But please, when you make a question, ask it in the chat and to everyone. So don't send a private message to Andrea or to Sarah. Just send a chat to everyone so we can see it and make the question to the speaker. So with no more delays, I introduce Thomas Frans from Innsbruck. OK, so I think I have the microphone here. I hope this works. OK, excellent. So first of all, I'd like to thank the organizers for putting up this meeting here in Trieste. I'm very happy to be here. It's a very long time that I haven't been to Trieste. So great. Thanks for inviting me. So the conference is on non-Macovian dynamics and non-equilibrium setups. And this brought me, actually, into a little trouble since I have several research topics that show, so OK. No, I forgot what to do, OK? Now, maybe it works or maybe try another. OK, now it should work again. OK, thank you. So I have several research topics that I developed over the years. Many of them show nice facets of non-trivial Macovian dynamics. For example, here, this topic on cellular crowding and porous media that's connected to the percolation transition, where you have really persistent memory for very, very long times. But in a sense, it's equilibrium dynamics. Then I have a long history on the glass transition, which is kind of the paradigm where you have memory effects that separate from the micro-dynamics over orders of magnitude. But still, the theory that we are using is an equilibrium theory. So that's also not non-equilibrium. There's needles and bio-filaments where nice simulations have been performed and compared to theoretical predictions of Dore and Edwards. But also, this is equilibrium dynamics. We also did this for active needles. So this is the non-equilibrium. But in a sense, there are no interesting non-Macovian effects. And the basic equations are all Macovian. Then we have something on intercellular transport that could be classified as non-equilibrium, but also not real strong memory effect. And last, self-propelled particles, the active particles. But there we start from simple Laungema equations where there's also no memory. So the topic that I eventually picked is this on a dilute suspension that we drive out of equilibrium. And then this, before I actually start, I should make a remark on non-Macovian effects. So we learned yesterday from Letizia. She quoted from Kampen that most effects in physics are non-Macovian. Well, actually, there's a different viewpoint. There's a different viewpoint, namely in physics. We usually start with Newtonian equation or Laungema equations, where you have no memory whatsoever. And the memory or the non-Macovian effects appear once you start coarse-graining as we have seen. So on a fundamental level, there is no memory. Only once you start coarse-graining that's memory. This doesn't work only in physics. Actually, this works quite generally. So for any stochastic process, you can do a simple trick. Namely, you just extend the state space in such a way that your state contains the entire history. So if you do that, and that's perfectly legitimate, I talked to a mathematician. That's perfectly, so any non-Macovian process can be turned in a trivial way to a Macovian process. Unfortunately, this is purely formal and doesn't help you to solve any problem. So this is just a matter of notation. Anyway, so let me start my talk with a kinetic theory of a paradigmatic model named the hard spheres. And this model is kind of the origin when non-Macovian effects have been introduced. So the model is very simple. So you just have a hard sphere fluid. So it's Newtonian dynamics. The interaction is maximally simple. So you just have the exclusion principle that two spheres cannot overlap. And since this is so simple, there's only one structural control parameter, which is the packing fraction that is the fraction of the volume taken relative to the available model. So then people, then this is considered to be ballistic dynamics with specular scattering, energy conservation holds, momentum conservation holds. And the simplest quantity that you can look at is the self-dynamics of a single tracer. And this has been done by computer simulations by Alder and Wainwright many, many years ago. So they looked at the mean square displacement of a single particle and derived the velocity autocorrelation function as the second derivative of this. And they found here the original data. I don't know if the pointer does not work. So on the right-hand side, so you see the original data of Alder and Wainwright. And contrary to the expectations of these times that this velocity autocorrelation function should decay exponentially, basically in a memory-less fashion, they found a power law tail. So the velocity autocorrelation function in this system decays only like a power law. And this means now that there are really strong memory effects because even after coarse graining, you don't get rid of the memory. So even if you zoom out in the time scale, you still have a power law because this power law is scale-free. So it doesn't help you to zoom out, only you decrease the amplitude. How did they explain this? So they said, okay, so sorry, I shouldn't start an argument with okay because if you do that, then there's always something fishy. So here's nothing fishy about that, so there's no interjections. So they said, it's true that the momentum is not conserved of a single particle, but this doesn't mean that the surrounding fluid can just eat up the momentum. It can take it away and slowly transport it away, but the momentum cannot disappear. So there's no beaming of momentum or something like that. Which means the picture is the following. So you start your particle with some initial momentum or velocity, and after some time, this momentum is redistributed to a sphere of surrounding particles. So the velocity autocorrelation should be something like one over the volume of the sphere. So that's the memory that's in the system. And since momentum conservation is governed by the Navier-Stokes equation, or by the Stokes equation in this case, for a low Reynolds number, then you anticipate, well, the Stokes equation looks similar to a diffusion equation. So this radius should grow only diffusively. That is like square root over t of t. And indeed, this is the correct argument. So the velocity of autocorrelation function decays like one over r cubed. That is one over t to the minus three over two. So this is the correct argument, and you can even work out the prefactors at least for a colloidal system. So this is a paradigm that I want to use for this memory effects. This is equilibrium dynamics. Let me spend some time on equilibrium dynamics before we go to the true non-equilibrium case. So the question is now, what happens if you do Brownian dynamics rather than Newtonian dynamics? And with Brownian dynamics, I mean that I discard any hydrodynamic interaction. So the particles just undergo individual Brownian dynamics and from time to time they collide. And you can study this, say with event-driven Brownian dynamics. Then you expect that, well, at short times where the particles don't see each other, you just see the diffusion coefficient of the Brownian dynamics that you put in. But at long times, again, you will see diffusive dynamics, but the diffusion coefficient will be slower than the one that you put in due to the interaction with the other particles. Sorry, how do you just collide the brown-spec velocity? Well, if you're asking about the algorithm, this is how we do that. So you can write down the Smolakhovsky equation, many bodies of Smolakhovsky equation where you just have a boundary conditions. If you want to call this a collision or not, this is up to you. So anyway, there is a prediction from the, which is now 40 years old, from Hannah and Hess and similar people did this later that said that the velocity correlation function in this case should also display an algebraic long-time tail, although the exponent is different now. So there's no momentum conservation. There's only particle number conservation, but here also persistent correlation should appear due to repeated collisions with the same particles. So the exponent is different and actually the sign is different. So whereas in the momentum conserving case, the velocity articulation function gets a push. So here it's strictly anti-correlated. And this prediction has been kind of forgotten for many, many, many years. And the reason is this is what I called as the simulator's nightmare because this is a prediction that is made for low density. So the velocity articulation function here should be proportional to the packing fraction. But you see at low density, the particle basically just do random motion. So you have a huge noise. And under this huge noise, there's only a tiny, tiny small signal buried that shows this persistent anti-correlations. Okay, so here on the right hand you can see a figure of the mean squared displacement. So you see initially for different packing fractions, both on logarithmic time scales. So you see in short times, you see the Bayer diffusion where the particles don't see each other. And at long times, you see that upon increasing the packing fraction diffusion slows down. But the prediction is that underneath, so if you take second derivatives, there should be this long time tail. Okay, let me dive into the prediction of Hanna and Hess. So what they did is the prediction is for low density. And for low density, the problem simplifies a lot because you have to solve only for the dynamics of two particles. And then you can basically upgrade this. So you have to solve the motion of two particles only. And this they could do, so they derived actually the full propagator, but I focus here on the velocity or the correlation function and I show it to you in the frequency domain. So then you see the first term, the first term D is just a constant in the frequency domain, which means Brownian without interaction, just you just have white noise. So there's no frequency dependent whatsoever. The next term is proportional to phi, the packing fraction by construction. So by construction, this is what they calculated. And the next term, phi squared, we don't know anything about it because there you need more than two particles. And what you can see is now that due to the interaction, this becomes frequency dependent. So there is memory in the system. But it's more exciting than that because the frequency dependence is peculiar. It shows the square root of the frequency, which means that the frequency dependence is non-analytic. So it's a non-analytic function in the frequency. And now you can basically do the back transform. You can do it for all times, but more interesting is just long times. So if you go back in the temporal domain, you see that the velocity or the correlation function is this t to the minus five, five over two with some prefactor. I wrote this in this form that I have a B of phi. We will see that this prediction of this long time tail persists actually to all densities and it's not confined to low densities. And the interpretation is that we have this algebraic memory. So again, this does not fade out upon coarse graining because it's self-similar. So there are strong memory effects. And the interpretation is basically that the particle remembers that it's where it already collided with another particle. Okay, so I have a long story in these long time tails. So we touched these in the so-called Lorentz model. So in the Lorentz model, you basically have one tracer particle that explores in this ordered array of particles. And it is known already in the late 1960s that for the Lorentz model tails like this appear. But actually this prediction was for ballistic dynamics and the calculation is horrible. So it's 40 pages long and I never really followed the entire argument. For Brownian dynamics, so and the problem is that for ballistic dynamics, even if you treat the two-body problem, this is not good enough because in ballistic dynamics, once you scatter, you never come back. So you need kind of to solve the simplest way in the Boltzmann equation for your particle to come back. So this is already difficult. In Brownian dynamics, it's much easier. So we decided to do Brownian dynamics in the Lorentz model and we indeed found these predicted tails. And what my PhD student at the time did is she divided also by the density. So note the densities are not low here. Nevertheless, she found that there's data collapse and I said, wow, this is great. If you find data collapse, this means that I need to calculate this expression only to lowest order in the density. And this is something that I can do. So in two times, this is in two dimensions. So the exponent of the tail is different, but everything can be worked out. Okay, now only recently we came back then to this full problem where all particles moved. So this dilute colloidal suspension and my collaborator, my former postdoc managed to generate data and look at the quality here. So we have something like orders of magnitude in the velocity articulation function, which is already buried below the Brownian noise. And we also divided by the packing fraction. So the packing fraction here varies by a factor of 10. So if you take this into account, you even have one decade more. And what he sees is indeed that for all density, you get this power law tail for the lowest density, the black curve here is just a theory of Hanna and Hess. This fits perfectly to the theory. The amplitude of this tail now depends sensitively on the density and there's no theoretical prediction at least not within this Hanna and Hess theory to first order in the density. There's a peculiarity, namely that also at short times you have a power law tail. This has something to do with, which I call the skin effect. So there's also a non-analytic dependence of the skin depth on the frequency here and that's the origin of that. Okay, so how did he actually do that? So how did he do that? I told you basically you have only noise at low densities. And the idea was the following. So we developed a noise suppression algorithm and the idea was the following. So when you simulate, you draw random numbers for your noise. And the idea was to use the very same noise history to generate two trajectories rather than one. So in the one trajectory, this is where all spheres are interacting just like we want, they collide from time to time and we generate a second trajectory where the spheres just go through each other so they don't see each other. And then we take the difference between these two and generate a genuinely interaction-induced part. And since our system is dilute, most of the time they just will go in parallel which means that this delta, this interaction-induced part will just remain constant. And every time there's a collision something will happen. And then you can just rearrange and say, so basically what you do is you put this, you put delta R zero to the left and this small delta R to the right and then square and take averages. Then you find, well, this delta R zero squared on average, this we know because this is Brownian motion, this is just 60 zero T. So we have three non-trivial terms and after rearranging the one on the left-hand side, this is the one that we want. Then we have the interaction-induced term by itself and we have a cross term. And what Sir Vendor Mandal did is he plotted each of these three terms. So here's the bare noise term and then you already see that the term of the noise-induced correlation is by orders of magnitude smaller. So where is it? This one is the term that is smaller by orders of magnitude. So this is a signal that is non-trivial and the cross-correlation term is again smaller by orders of magnitude. So this led to the suggestion that this cross-correlation term actually is exactly zero and we have some toy models where we could show that indeed for height interaction, this should be zero. And if this is true, then you can just calculate the velocity autocorrelation function by looking only at the interaction-induced part because on taking second derivatives, you get rid of the noise term here, the bare noise. And by doing this, he could generate this velocity autocorrelation function. This is not confined to height sphere, so you can do this also for soft sphere. So he reproduced everything for soft sphere and indeed you find this a long-time tail, similar density dependence, but this feature that you have this, also this short-time tail, this is lost for soft interactions. Okay. Okay, so I told you there's no theory for beyond Hannah and Hess. This is not really true, so I come from the mode coupling theory of the glass transition. And I don't have time to explain that theory here, so this has been developed by my PhD advisor, Wolfgang Götze, who passed away only last year. So there the theory focuses on the intermediate scattering function of collective density fluctuations by the projection operator of Zwanzig and Mori. You can derive exact relations between this quantity and, oh, there's a typo here actually, and so-called memory kernels. And the question is, of course, then what is the memory kernel? So in a sense, this is empty, unless you have some physical intuition to model this memory kernel. Here the idea is, so I put a picture here on the right, which is probably my most important contribution to science. This picture has been copied many, many times. I did this when I was a PhD student. So you see here this red particle is kind of trapped by the surrounding particles. This is called a cage, and the Zwanzig-Mori tells you that this memory function is essentially stress relaxation. But you see by this picture, stresses can only relax when particle relax, so there is kind of feedback mechanism. And that motivated this idea to write the memory kernel as a functional of the density correlation function itself. And that gave very nice predictions for the class transition, non-trivial prediction that have been tested in experiments and so on. So this is a very successful theory, which I think it's fair to say is still the basis for theoretical development even some 30 years later. Anyway, this theory also makes a prediction for the velocity autocorrelation function if you do this for Brownian dynamics. And indeed it gives the correct power law. It's also anti-correlated, it's anti-persistent, but it makes a prediction for the density dependence. And in these two figures I show again, right are the simulation data and to the left are this mode coupling theory prediction and you see on a correlative level this all works. This also predicts a strong density dependence here. The density dependence in mode coupling theory is stronger, significantly stronger than in the simulations. As you can see here by this pre-factor. So this couples more decades than the left-hand side. It was a technical challenge to actually implement this because you have to resolve long wavelength fluctuations which we usually don't do in mode coupling theory. So that was not easy but we managed to do that. And then we see, we can explain now the effect of the density depends due to two things. So one is that, so you see that the diffusion coefficient itself slows down. So that's encoded in the pre-factor but more importantly it's the compressibility of the system. So the structure factor depends or significantly on density at the long wavelengths. So the compressibility of the system plays an important role as it's highlighted here in this red formula. Okay, so we were very happy that basically, well actually I can tell you that long time tails and glass transition was the topic of my PhD thesis. I kind of managed to bring up a theory that does both but only many years later we managed to do that which is basic, maybe that's probably the question that my PhD advisor really had in mind. Can you build a theory that has both? And this is now the result here. Okay, let me make this resume for this equilibrium part. So we have this Brownian dynamic simulation. We found the tail and have an explanation for the amplitude. Within mode coupling theory we produce qualitatively the density dependence and have an explanation for this in terms of the compressibility and the slowing down of transfer. So now let me go one step further. So yesterday we learned already about microbiology. There are different ways to do that. So you trap a particle and look either at the fluctuation or you pull on the particle actively. This has become quite popular in the early 2000s. So people basically trap a particle and pull at it or just look at the fluctuation. The idea was you measure kind of the frequency dependent diffusion coefficient and connect it to the frequency dependent viscosity so a material parameter in terms of the so called generalized Stokes-Einstein relation which is not really true but approximately. So you can measure on a microscopic scale rheological properties. That's why it's called a microbiology. So coming back to our colloidal solution what I want to do now is I want to drive this out of equilibrium and again, so and here I'm really confined this to low density. So low density. So the setup is the following. So we start in equilibrium and at a certain time I start pulling with a strong force on one particle. So I drive the system out of equilibrium as shown here in this sketch. So the force is constant throughout and the question that I want to answer is so there will be a steady state at long times. So the particle will move on with a constant velocity so average velocity and I'm also asking so I'm asking how do you approach the stationary state and so what is the time dependent drifting velocity and later I will also ask about the fluctuations. Why is this question interesting? Well we learned earlier that by the fluctuation dissipation theorem to linear response the velocity that is the deterministic response is encoded just in the velocity autocollation function. So this is linear response and by taking this formula you can make special cases. For example, you go to infinite time then you see okay linear response is just the mobility is just the diffusion over KBT so that's the linear response coefficient that has been predicted by Einstein and the diffusion coefficient is then just given by the Green-Cubo relation. It's interesting if you now look so the linear response prediction is interesting for the time dependence because we said that this velocity autocollation function displays power loss so also this linear response so you should basically approach the long-time limit only algebraically that is there are long-time persistent memories in this. So this is what linear response says. Okay, what is also known is for the stationary velocity so there's a paper by Squires and Brady who looked at the stationary case so you pull hard at this one particle and they defined a mobility that depends on the force unfortunately the point doesn't work so the velocity, I write the velocity in the stationary case as a mobility times the force, okay? Mobility times the force and the mobility depends now on the force itself it's made here in this dimensionless fashion and it's called then a Pichli number. There are several cases that are familiar so if you do linear response that is in Pichli number zero and say at infinite dilution the mobility is just a Stokes mobility and it has been worked out many, many years at the beginning of the 20th century what is the first order density correction of this linear response coefficient that's essentially the suppression of diffusion due to packing and Squires and Brady worked out the stationary mobility for as a power series of the Pichli number so you see for Pichli zero it reproduces this well-known result but they have corrections then to the mobility in powers of the Pichli number and what I did is I wrote their formula a little bit different I put here absolute magnitudes to highlight that the result should not depend whether you pull up to the right or to the left but now you see this is now a non-analytic function here in the Pichli number so the figure of merit so this response curve is non-analytic in the driving so the questions that I want to ask is now where does this non-analytic contribution come from? Second, how do you do numerics because if you blindly plot this asymptotic series you see the radius of convergence is at about one whereas if you do the numerics correctly then you can go to arbitrarily high Pichli numbers the simulation is here compared for liquids so there were truly all particles move but you can do it also for a Lorentz model where only one particle moves to lowest order in the density this doesn't really matter because you have to consider only a single pair and then you put yourself in the center of mass or whatever okay so where do these non-analytic contributions come from? Okay so again to lowest order in the density we have to look at the two particles the Smolowowski equation and I show the equation directly here for the relative motion of this pair the different terms that show up is a diffusive term a diffusive term here and here's the relative diffusion if everybody moves the relative diffusion is just twice it works on R it's on R no R prime is the initial condition so this is the forward so it's not the this is Fokker-Planck equation if you want to there should be other way around this no we are acting really on no I mean there is a where the force is constant so the force is constant yeah so the force is constant you can write this either way so this term here says now this is the relative diffusion of the pair and the force basically wants to tear the pair apart so we start our system in equilibrium that is everything is allowed except overlap so the minimal distance is just a sigma then you have some no flux boundary conditions so the particles cannot penetrate each other and what we are interested in is say the generating functions of the displacement relative displacements which you can get by integrating the solution so there are some technical details that I don't want to go too much into the point is if you go in the Fourier domain and make some tricks this equation here looks like this there is a nice trick that has been introduced by Brady and Squires they say let's make something like I call this a gauge transformation so if you factor this thing out and concentrate on the residual you get a very easy equation which is just the Helmholtz equation so the Helmholtz equation can be solved in oh okay there is a price to it this parameter kappa now depends on both frequency and the force that is on the Peclet number and then you can solve this equation in fault beauty for with some undetermined coefficients in terms of this spherical Bessel function blah blah blah these are just Legendre polynomials and what you need to determine is these unknown constants here and although this equation is simple the boundary condition is not so now you kind of couple different angular channels but it's not spectacular so you just get a tri diagonal matrix for these unknown coefficients and you see in particular if the Peclet number is zero that is in equilibrium this matrix is diagonal and you get the solution immediately so you can do now a perturbative scheme and so on and calculate everything in particular the propagator in terms of self energy so this is all not really interesting at the end we have a formal expression of the mobility as a function of the frequency and the force as a series expansion in terms of these coefficients that we determined so fine so what you find out now is that this expression has only even powers in the Peclet number as it should so it doesn't matter if you pull to the right or to the left and there's no reason to believe that this RL is a non-analytic function of as the result of the equation so at the end you ask where do the non-analytic terms come from well let's have a look back at our complex wave number due to this case transformation now I put explicitly the square root so you see this kappa sigma encodes two things namely the frequency and the driving and you can do now two things this function is of course perfectly smooth except if the argument becomes zero and this happens precisely at two cases namely you can put now the frequency equal to zero and then you see here we have the absolute magnitude of the Peclet number and this is actually the case that we produce is Squires and Brady so here there are the non-analytic driving in the response curve if you put the driving to zero you get the square root of a frequency which gives the long time tails so you see the long time tails and this non-analytic driving are kind of two sides of the same coin and this I think is a really beautiful result and maybe in this sense it fits nicely into this workshop so we have really two non-equilibrium system with non-analytic everything and we have true memory effects here and actually we solved this earlier for a similar system on a lattice where you can work out everything without perturbation theory you can solve for everything okay so let me just show what happens here so there's a pile up of probability so once you start pulling on the particle there will be probability particles will pile up in front and there will be depleted behind this happens both in simulation as well it's in our analytic solution okay this I want to show you but then I stop so now you can ask so I said how do you approach the steady state so we said in equilibrium in the equilibrium or linear response you expect that this should be a power law and here you see now this whenever you do a driving a non-trivial driving then you follow this power law for some time but then you deviate from it so this power law is cut off and you have an exponential approach to the true state and actually there's also a sign change for which I don't have an explanation so what you see now and this is really fascinating that there's a divergent time scale emerging that separates two regimes so one is an intrinsic linear response regime so up to a time so tor is the natural time scale for a problem and peckley so if peckley becomes small this diverges so if you are below this time scale essentially you recover this linear response prediction so essentially you follow this tail here but there's a second regime so if you go at larger time scale you drive your system to a true non-equilibrium state and these are separated by this diverging time scale and this actually makes a lot of sense because you smoothly connect now non-equilibrium steady state with an equilibrium relaxation in the system okay so this is I think one of the major findings of this project okay so I have more on fluctuations so you can calculate not only the mean motion but also the fluctuations around this this I skip and go directly here to these conclusions so what we worked out is the response to a step force and we were able to calculate completely this time dependent mobility to lowest order in the density for all peckley numbers for all forces and for all times and the major finding is this non-analytic in the frequency and in the driving if you have any finite driving then this long time anomaly will disappear so basically linear response is wrong at this long times even for arbitrarily small forces so that's qualitatively wrong and the origin is again that you have this repeated encounter with particles you can do everything on the lattice where you can work out everything not only the velocity articulation function and we found similar things on the lattice here what we want to do next there's a proposal pending we want to extend this for the viscosity so rather than pulling out a single particle we want to shear the suspension this is way more difficult because my nice trick with this gauge transformation doesn't really work anymore so what is really fascinating what I would like to do I don't know if this is possible so I would like to know are there some general mathematical constraints on correlation or response functions even beyond equilibrium so we know in equilibrium there are correlation functions and there are rather strict restrictions how a correlation function should look so basically if I draw a function at the blackboard and ask is this a correlation function of or not you can give a very easy answer take the Fourier transform the cosine Fourier transform this has to be positive if this is not positive then this is absurd and the question is is there anything similar to that even beyond equilibrium that would be really nice and for this particular system it would be nice kind of to have a marriage between mode coupling theory and these low density expansions kind of to extend this to arbitrary density thank you very much this is what I wanted to tell you today thank you very much for the nice talk so I give the boys two questions and I'd like to emphasize that there has been almost no questions from students so I encourage them to ask questions and I give you the boys first so if there's any student with a question please is your chance nobody I think that we didn't have conference in two years in person so it's a good chance okay no questions it's online so I thought everything is clear or it's too difficult thanks Thomas very nice talk so I know you are you did not yet do that but since you already announced that you plan to do that what do you expect in terms of these constraints on the response function out of equilibrium because I mean typically you know the stuff that constrains the mathematical properties of response function in the equilibrium it's very intuitive now what can happen most likely that the time dependence will not be monotonic so what are you expecting so far we are just collecting fingerprints of true non-equilibrium behavior so what can actually happen is that so for this colloidal system in equilibrium all correlation functions should be completely monotone functions that is not only the decay is monotone but also the derivative decay is monotone in the secondary blah blah blah blah and this is clearly violated so here we see fingerprints that these functions are no longer completely monotone functions and the frequency dependence I don't know if you can manage to have a spectral density that becomes negative because I don't really know so I mean in the time dependence not the spectral density not I don't dare not but I think I know that if you would have a non so non-monotonic response in time so no oscillations but strictly positive that happens yeah so then what you may wish for I have no idea if it happens that there is a bound on the ratio of the imaginary and real parts of the eigen-modes or the eigen-values that actually enter there for example that could cut because this one would expect from algebra in terms of what typically happens just that this system is a manifold system so it's a very ambitious project so this is a long time goal and I don't have any clue how to do that impressive thank you okay so my question might be very trivial but it goes back to the first part of the talk when you were talking about the role of momentum conservation in the Hartzphere models so my question is like it seems like imposing like a conservation or symmetry affects and causes memory effects to arise so of course maybe it's not precisely momentum conservation as you said if you replace it with particle number conservation you still have this effect but my question was like do you have any intuition on how memory affects and these tails are affected by removing perfect momentum conservation for example allowing like some like small non-conservation or something like that well I guess then you end up similar to this that you will see persistent correlation for some time but at the end they will just get destroyed and depending how you play with the parameters so there should be a smooth crossover and yeah so but you see this is what any computer simulator would tell you anyway say yeah I have a finite box and of course if I run my simulation too long you will see artifacts of the box size so you have to make sure that you are in the window where you actually want the physics to see All questions? There's one online and it's asking for driving a single particle in a static potential landscape do you expect that the same non-analyticity in the clay number shows up? Yes, this is well so we did this for so on the level of lowest order in the density it doesn't matter if all particles move or one tracer because so you can as in classical mechanics you just put yourself in the center of mass or center of diffusion in this case and you have the same equation so what happens is so there are differences at higher orders in the density because at higher order density what can happen is that you have clusters and then your particle bumps into such a cluster and it will take a very long time to get out so it's already hitting a single particle it takes a long time to kind of rub around that particle but if you want to have particles that kind of build a shape like that a wedge shape this will take excessively long and this will happen at any density so it's even more interesting there is a prediction by the mathematicians that say well I looked at at the limit basically density going to zero and then you can do with the force whatever you like the mathematicians also consider the opposite limit where they say well I have a small but finite density and I have some force and then let the density go to zero very strange things happen so the system can fall out of equilibrium if you really pull hard enough so there is a hierarchy of time scales where basically particles bump into clusters so I threw out all clusters from the very beginning in the calculation because I consider only first order in density but all these limits are not commuting so in literature you sometime find these long time tails by order in vain they have they have an extension which is called Dorfmann's Lemma all transport coefficients are non-analytic functions of all quantities frequency, wave number density at least in kinetic case I showed you here in the external driving so basically this linear response point that we are typically looking is the most fragile of all okay just one question otherwise so you looked at just one single tracer so now imagine that you put two tracers would there be an effective interaction mediated by the surrounding fluid the simulations are for in a dilute suspension and qualitatively everything is the same I cannot do the calculation for more than a pair so the simulation corroborate that the scenario remains the same at least for reasonable forces if you pull really really hard and you have a low run system where particles are frozen then you will fall out of equilibrium at very large times that is kind of the picture so it's not that these things happen only to lowest order in density you can go to higher densities and even more interesting things if you pull really really hard okay let's thank Thomas again for the great talk