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


A group under the auspices of Kaduna State Alliance against terrorism has called on the People's Democratic Party PDP Kaduna State governorship candidate Issa Shiro to step down from the race and allow the candidates to tackle the insecurity confronting the state. Speaking at a press conference in Kaduna State, the chairman of the group says, despite the declaration of the criminal element as terrorists by President Mohammed the candidate continued to call them bandits. He said Kaduna State, like other Nigerian state, is still nursing the bruises caused by activities of terrorist elements, which have adversely affected numerous businesses. The demanded that security agencies and other relevant authorities investigate Hashiro for supporting negotiations with bandits publicly during a live TV program. A group African parliamentary advocacy and reform group advised Nigerians to take parliamentary elections seriously and ensure credible lawmakers are voted into Senate and the House of Representatives. The group observed that for some time parliamentary elections have always been conducted in the shadow of the executive during the period of elections. Now on the report. Observe that parliamentary elections, which are supposedly the symbol of democracy, is often times neglected during the build-up to the general elections. And this year is no exception. According to the recent representative of Konrad Adenova Stifstong in Nigeria, Maria Peter, the training is to ensure that the media is well prepared for the common elections. She said the election is a defining moment for the survival of the country, judging from Phyllis and the body language of the political actors. Her position was supported by the Director-General, National Institute for Legislative and Democratic Studies, Professor Boboka Suleiman, who said the training could not have come at a better time. He commanded the media for its role of holding the government accountable. Strong media coverage is needed to reflect the legislative work that's going to happen. And I just said this much is at stake because the next four years will shape, of course, the direction that the Nigerian democracy is leaning towards. So we all know that there is no proper democracy without proper functioning media. And one of the mandates of the Konrad Adenova Stifstong is to support all democratic institutions. The media is better placed to set the agenda for public discussion on critical issues, especially in the bill to the elections. Additionally, the media should facilitate community and nation building by helping people find common courses and work toward practical solution to the problems confronting Nigeria regardless of which part of the country they are from or which part of the country they live in. Facilitators too participate through historical perspective on salient issues about the democratic advancement in the country. There can be alternative views on alternative narratives called marginal narratives. Such marginal narratives may be representative of the majority of the people, but because they do not enjoy wide acceptability, if you like, or exposure in the media, they may be surprised. It is the duty of journalists of conscience that you are to ensure that marginal narratives are put impartially in the public space with a view to helping voters to do a good comparative analysis. Chairman of the House of Representatives Press Corps, Grace E.K., family organizers for the training and retraining of parliamentary reporters in different fields of reportage. As political and parliamentary correspondents, or if you like journalists, I believe this training will go a long way in preparing us ahead of this crucial national assignment of covering the 2023 general elections. Secondly, the redesign, or I will say the new Naira note, is a very topical issue today because it has affected Nigeria's across board and chosen it as one of the topics to be discussed here in Stanley and quite commendable. The workshop afforded journalists useful tips on how to ensure stability in the country, as well as a free, fair and credible election through their reportage.


What's going on everyone Sam here with late in sports cards, and I am ripping two boxes of 2021-22 Pinini obsidian soccer for Kyren G. So good luck Kyren. See what we got in here for you, buddy I would say I'm gonna say to stick around today for sure absolutely stick around today because We've got a very good personal coming up in like four or five breaks, so I would definitely I'm sure you guys already know It's a case of flawless that we get to open so Definitely exciting stuff there, but get to start with obsidian got a couple other products coming next which I'm excited for Come on. There we go. I don't know why soccer is so much harder to open, but What do we got here? We've got equinox on the eater Milatio that is four of 99 nice die cut right there Very nice. I don't believe so We've got Danilo or Danio for Ajax. That's four of 12 nice on the blue nice one there Got it illegal right on the dot man right on the dot got Roman parade for Southampton to 105 Nice one there Militow. Thank you Please simple ten if I mess up, please tell me Got this one's easy. I won't screw this one up to 149. We have Luke Shaw on the eruption jersey for England That one's cool Sick one there on the eruption Got Teo or Theo Hernandez to 50 on the cutting edge orange jersey for AC Milan That's pretty cool. I don't know Mike I'm not a hundred percent sure if forest You know forgetting select baseball first off the line. I don't know. Yeah forest doesn't know either See you later Loki Shag Got a redemption we'll save that to the end. We've got nice Harry Kane on the patch autograph to 149 nice one there for Tottenham That's sick Nice card Yeah selects so awesome like so awesome Yeah, King's solid. I like Harry Kane a lot. I do actually know Harry Kane is and then our redemption is Volcanic material signatures electric edge of Dijon Kulisevski or Dahan Kulisevski Pretty cool right there not gonna pretend that I know where he plays but Nice redemption and box to good luck take the money What do you mean, Osprey got another die cut here? That's pretty cool. Oh Spurs. Gotcha We've got Luca Modric or Modric for Real Madrid equinox. That is one of ten nice hit there I gotcha. I just don't know if Very nice. I just JB. I would be all for telling you I just don't know if Chris would be comfortable with that one with me disclosing that information Marcus Lorenti or Lorenti that is nine of 25 for Atletico Madrid practicing true there Camille got Jeremy Doku on The base to 105 nice one there. I believe there is actually simpleton I think there's one more that Diogo de Lowe That's cutting-edge Jersey Manchester United or Delot I in a 50 early Holland on the atomic Jersey for Borussia Dortmund. That's awesome That's really awesome. I will say I did not pay It's under a thousand dollars. I did not pay a thousand dollars for that It's below a thousand early Holland on the atomic We've got autograph to 50 of Robin Gossens for Germany nice one there. That's cool Yeah, the Holland was sick. I like the Holland and then we've got for US Aurora of Tirna Davidson That's pretty awesome right there Cut a US. Thanks. Thanks illegal appreciate that And that'll do it for your boxes there Kyren Congrats on all the hits and we'll get those right out to you. Thanks again, man


So what the magnets do is you can now physically feel the vibration because you hold the magnet to next to the coil, it'll shake in your hand violently because of the energy coming from it. Also, you can, it's kind of acts as a focusing element of the energy, of the energy field as it comes out of the coil. I put the two q-coals yin and yang in front of me. I play my biohacking frequencies, methylene blue, bpc 157.  I don't have to meditate. I'm just sitting there working on my laptop and I'm getting this daily energetic therapy every day and every day I'm getting stronger. I'm 43. I do boxing and I'm basically outpacing all these 20 and 30 year olds. So scalar is another kind. I mentioned that there's different types of chi energy. So scalar is a different kind of energy. So scalar waves is a little bit more complex, but basically when two electromagnetic fields of equal, let's say frequency or equal intensity, when they meet in the middle, let's say you have one coil on the left, one coil on the right and they're facing each other, they're both creating this magnetic field and when they meet in the middle it doesn't cancel out. It cancels itself out. You know how noise cancelling headphones work? It basically uses a mic to pick up what sounds it's coming in and then it plays a sound that's opposite of that. That's why it cancels out the sound. But according to a lot of conservation, energy never cancels out. It goes somewhere. It has to go somewhere. So when you have two electromagnetic fields and they are equal to each other and they face each other, in the point of middle they don't cancel out. They create another type of energy called scalar waves in the 90 degrees from it perpendicular to the energy field. So long story short, basically it's like this. It can't really detect it because it's cancelling out, but you can measure the effects of it on plant life or on humans or on biology.


In the related developments, now a member of the PDP Board of Trustees, Bodejad, says it is time to chart a new direction for Nigeria. He made the statement while Griffin Newsman and the PDP local government leaders are head of the local council election in Lagos state. Plus TV Africa's correspondent, Ungozika, on hair chasing, Hasmore. For keen political observers, this statement is not strange as the host from time to time expresses his opinion to the media and PDP followers. This time around, he argues that it is time to consider restructuring as panacea to the merits of challenges confronting the nation. The PDP chief then said that Nigeria federation is skewed and distorted. The challenges before us are far beyond the confines of partisan politics. The turmoil within the Nigerian entity is about the twisted present situation. Very twisted indeed. Nobody is at peace. It is about who we are. It is about what we ought to be. It's about the greater tomorrow. He says leadership requires wisdom and according to him, this is what is lacking. Leadership requires a lot of patience, cultivated wisdom and demonstrated intelligence. You do not aspire to position of leadership by mere women pantheists. Like the Boy Scout motto says, you must be prepared. You must be honorable. You must look beyond personal advantages and be willing to make sacrifices on behalf of others. Addressing the spate of insecurity in the country, Badejad says that the country is in a terrible state and it is not beyond redemption. The insecurity was only in the southern part of Nigeria for a long time until they saw and had also some MAPC governors talking about insecurity in their states. So it is a national matter. We cannot continue like this. The first responsibility of any government is to protect lives and properties of the citizens. The PDP leader says his intervention, it's not partisan, therefore it should be taken seriously. Go Zika or HSE for Plus TV Africa.


Hey guys, how are you today? So we're here with another vlog and whoever you are out there that just does not like any of the vlogs on my channel or the Periscope videos, just go ahead and stop watching now and give it a thumbs down because you know you're going to anyway. So yeah, every single one of my vlogs and Periscope videos lately, they all get at least one thumbs down. I don't see, I never give thumbs down, FYI. If I don't care for a video that I've watched of somebody, I just stop watching. If I like the video, I give it a thumbs up and I make sure to make comments. But you know, I just, anyway, I don't know. It's puzzling. What's going on right now? I'm doing a few different collaborations. I guess they're collaborations technically. Two YouTube video hops and one Periscope TV thing and with a bunch of other artists. I have coming up on the 26th of September past the brush on Periscope. If you don't have Periscope, go to periscope.tv and download the app for your appropriate smartphone tablet device. It won't work on a desktop. At least not, right now I don't think. I'm doing a DigiHop for Poet Spice here on YouTube with her digital images on October 3rd, which I haven't filmed yet, but I'm prepping for that today. Hopefully I can film that tomorrow because I'd like to get these all filmed and that I can get filmed and loaded and scheduled and yeah, we'll see.  I'm one of the biggies on the list. When did that happen? When? And I'm approaching 4,000 very quickly. I just, when I started this channel, had no idea that was gonna happen. It would be really cool if I could get to 5,000 before the end of the year. I don't know if that's gonna happen, but you guys all out there, don't forget to like, share, and subscribe and share with your friends. Promote my channel for me. That would be great. If you have a small channel, you'd like me to take a look at it and maybe add to my featured channel list on my channel's main page, by the way. I have a featured channel list, featured channels list. Holy cow, I can't speak today. And they're all small channels that are at the time of me adding them to the list where 500 subscribers or less. And really some really great channels and I'm gonna keep adding to the list. So the list may get a little long. And every once in a while, I will be probably editing the list and as these channels get to be over 500 removing them, not cause they're not still good channels, but the idea is to promote the little guys and, you know. So if you have a little channel or you know of a little channel that you'd like me to take a look at, send me the link. I'm happy to do that. What else? Lists of stuff that I've got video editing to do and... Always stuff, right? Always stuff. We had an incident, very strange incident with the sugar bowl this morning. All three of us went to bed last. There's only three people in my house. My husband, myself and our daughter went to bed last night. My sugar bowl was intact and one piece was sugar in it. Nobody is claiming responsibility for it, but when my husband got up this morning, the sugar bowl was broken. It was very weird. I only have three people in my house. It's very weird. I had to order a new one and I had to go to replacements.com to get it because you can't just buy one sugar bowl at the store. Yes, anyway, it's gonna cost me like 20 bucks to replace the sugar bowl. I don't know, it's very weird. I got some happy mail though on a less weird note. I got some more happy mail from Stephanie Parks in Auckland, New Zealand. And I love when she sends me these happy mail envelopes because not only do they come with all these really cool stamps, but the bags have this aboriginal design on them. So, so cool. So, let's get this opened. Let me get some scissors. I have no idea what this is and I wasn't expecting anything. She sends me some really great handmade stuff and she always wraps it up so nicely. Nice piece of chipboard. I know that it's just the packaging, Stephanie, but nice piece of thick chipboard. I don't have to say that.  You know, we mixed media artists who never have enough bubble wrap, right? And then she said, look at this. I mean, this is, she just, everything is just wrapped to perfection. Another really great piece of really thick. It looks like a matte board. Fabulous, love it. Nice piece of corrugated cardboard. And I love, look at that, love that string. Here's an envelope with my name on it. Look at this paper. But you wish it was Feel-A-Vision TV where I wish you could feel this. It feels like a cotton fabric, the paper, like cotton rag. There is some book text in here. It feels like it's painted book text. Oh, they got a little stuck together. Probably because I bet you it got hot. Let me see if I can unstick it. So she's got her business card paper clipped on here and there's a nice note. And I'm going to read this off camera and I'll be right back. Okay, she sent me a really nice note on a piece of book text that she covered with Gesso and then kind of did a painting background, just plain painting background. And then she did her letter on top of it. That's a really great idea.  It's very cute. So she saw a video that I did recently where I talked about in-cheese and I had that in-cheese book that's full. I love in-cheese. And she got an idea for making an in-cheese book that she made for me. She also sent me some collage papers, some origami papers. These are all feather patterns. Okay, I have to share with you guys. These are origami papers, she says. Look at this. These are so cool, Stephanie. She also said she watched my lavender watercolor Wednesday painting recently where I did the lavender flowers and she enjoyed that very much. These are so cool. I love these. It sends me the best collage papers and tissue. These were wrapped in this piece of hand-painted, or handmade, I should say, paper that has real leaves in it. So cool. And this must be the book she's talking about. She said she went to the local cobbler and he helped her do some rivet something down on it, rivet the clip to the folder for me. I don't know what that means exactly, but we'll find out. Oh, I feel what it means. So it's wrapped in a piece of, this is the kind of paper I use for drop paper. It's like packing paper. And it's made out of a mucilix bar box, mucily, mucily bar box. More bubble wrap. And I love that that it's made out of a food box. And she, basically it's a little two-ring binder and look at this cover. I love the cover. And then inside, she's got a Tim Holtz. This is a Tim Holtz, I think, two-ring, or similar two-ring binder thing. And she had her cobbler rivet it to the cover for her. And then she created a inchy sheet protector. Look at that. She created an inchy sheet protector so I can just fill it up with the inches. And I could, it's a good idea. And I could make more of these and put more pages in. And then I have a binder full of inches. I love that. I love it. It's a great idea, fabulous idea. Thank you so much. You always send me the great gifts. It's wonderful. And I love the inches she put in here, by the way. Did you guys notice that? Look at the inches. Love that. All right. And yes, that is a great idea. I think that was brilliant. What else is happening today?  I gotta work on Poet Spice's DigiHot video. And I've gotta edit some stuff on YouTube that has, oh, I bumped the camera, sorry, that has some problems with the descriptions underneath the video. I think that's it right now. I can't think of anything else.  I think that's all right now. Anyway, having fun, just making art. You know, getting into a routine now that I am home full-time and just doing art full-time. Finding a routine in my schedule that takes care of myself, has time for art, has time for my family and all of that stuff. It's a big adjustment, but I'm working on it. And I think that's it right now. All right. You guys go out and have a great day, have a great week. Do something nice for yourself because you deserve it.  My family has the hardest time eating leafy greens. They're good for you, though. All right. I'll talk to you all later.  All right. I'll talk to you all later. Bye.


8 Best Places To Teach English Abroad For Food Lovers One of the many benefits of teaching English abroad is that you can combine your new career with your own hobbies and passions in countries all over the world. If sampling new and exciting cuisine is your thing, then you will have plenty of great food-loving destinations to choose from. Here is our pick of the best destinations where teaching jobs and great food options are both in abundance. 1. Spain Spain has historically had one of the biggest markets in Europe for ESL teachers and it is also home to some of the best cuisine on the continent. Spanish food is generally divided into three distinct areas. The Mediterranean region which is famed for its seafood, gaspacho soup and paella, the inner region where you can indulge in hearty soups and stews, and the Atlantic region where you will find all manner of fish dishes. Wherever you are in the country, you can also enjoy a huge range of bite-sized tapas with your after-work drinks. Food lovers' highlights. Tortilla Espanola or Spanish omelet can be eaten for breakfast, lunch or dinner. Every cook has their own method but it is often said that you can tell the quality of a bar or restaurant by their version of this local favorite. For a great start to your teaching career in Spain, why not take a four-week-end class TEFL certification course at one of our four Spanish locations. 2. Thailand If you are looking for a long-established TEFL market with an internationally loved cuisine in Asia, then Thailand certainly fits the bill. Here you will find local restaurants and street food of such high quality and value that the majority of locals rarely bother to cook for themselves. Thai food is famous for blending five key tastes – sweet, spicy, sour, bitter, and salty – which all combine to make some of the most distinctive flavors in the world. Crucially, TEFL jobs are just as plentiful as the local cuisine in most parts of the country. Food lovers' highlights. Two well-known dishes that you won't be able to get enough of are Pad Thai, stir-fried noodles, and Tom Yum, hot and sour soup, both of which are utterly delicious, widely available and exceptionally cheap. For a great start to your teaching career in Thailand, why not take a four-week-end class TEFL certification course at one of our four Thai locations. 3. Mexico If Latin America is more your scene, you should take a look at Mexico as it has the largest demand for ESL teachers in the region and one of the most widely known cuisines in the world. Developed over thousands of years, Mexican food is as varied as it is delicious, from simple staples such as tacos, nachos, and burritos to more elaborate offerings that include spit-roasted meats and whole-marinated fish baked in a mouthwatering sauce. After a day in the classroom, a cold local beer or classic margarita are also a treat to look forward to. Food lovers' highlights. One must try dishes pozo which is a traditional super stew that is often served up on special occasions such as birthdays, Christmas and New Year. You should also try chilaquiles for a whole new take on breakfast, Mexican style. For a great start to your teaching career in Mexico, why not take a four-week-end class TEFL certification course at our training center in Chiapas, 4. Vietnam The popularity of this gem of a country in Southeast Asia is rapidly growing, both in terms of ESL teaching and its fabulous food. Jobs are abundant and well-paid in relation to the cost of living and the street food culture is second to none. One fact that appeals to many people is that Vietnamese cuisine relies heavily on fresh herbs and vegetables and uses little oil or dairy which makes it one of the healthiest in the world. Due to its French colonial past, you can also expect good coffee and warm begets during your stay in Vietnam. Food lovers' highlights. One of the most famous dishes in Vietnamese cuisine is the noodle soup known as pho. Every region has its own variation and it can be found at street stalls and restaurants throughout the country. For a great start to your teaching career in Vietnam, why not take a four-week-end class TEFL certification course at our training center in Ho Chi Minh City. 5. Japan When you mention Japanese food many people will only think of raw fish. However, Japanese cuisine is actually much broader than you might realize. Across the country you will find a wide variety of dishes developed over centuries, as well as many that have been adopted from other cultures in recent years. Japan actually holds the record for the most Michelin-starred restaurants in the world. However, you don't have to break the bank to eat well as there are plenty of affordable noodle shops and street stalls to try if you are on a budget. Food lovers' highlights. Among the many mouthwatering street food dishes available, takoyaki is one of the most popular.  For a great start to your teaching career in Japan, why not take a four-week-end class TEFL certification course at our training center in Tokyo? 6. Italy No list of popular cuisines or TEFL destinations would be complete without mentioning Italy. Although pasta and pizza dishes are hugely popular all over the world, there are many other more unique delights to enjoy in every region of the country. Wherever you find yourself in Italy, you will find simple fresh ingredients expertly put together to produce exquisite dishes that have been passed on from generation to generation. The country's status as one of the best wine producers in the world also offers plenty of opportunities for connoisseurs and enthusiastic amateurs alike. Food lovers' highlights. One treat that you can enjoy at any time of the day or night is gelato, the world's most delicious ice cream. Which of these traditional flavors will be your favorite vanilla, chocolate, hazelnut or pistachio? For a great start to your teaching career in Italy, why not take a four-week-end class TEFL certification course at one of our two Italian locations? 7. South Korea Although less well-known internationally than many of its neighbors, South Korean cuisine offers some of the tastiest food in Asia. It also has one of the largest and best-paid markets for foreign teachers in the world. This great combination makes it a top pick destination for thousands of food-loving teachers every year. Local dishes are largely based around rice, meat and vegetables, which are famously served up with a variety of side dishes. The most famous side dish of them all is kimchi, a fermented vegetable dish that is served with virtually every meal. Food lovers' highlights. The bibimbap is a popular Korean dish that typically consists of a bowl of warm rice topped with vegetables, chili paste, soy sauce, an egg, and sliced meat. This local favorite is equally popular with many foreign teachers in the country. 8. China Although many of us have enjoyed eating Chinese food in our home countries for many years, those dishes are unlikely to bear a close resemblance to the authentic cuisine you will get to enjoy on a daily basis while teaching English in China. As it is such a huge country, the local food you will come across will very greatly in style, ingredients and flavor from one region to the next. For example, the province of Sichuan is famous for its spicy dishes while Shandong province is known for its excellent seafood. Food lovers' highlights. If you like spicy food you will love kung pao chicken. Wonton soup is another crowd-pleaser that comes in either pork or shrimp varieties. Peking duck is also a favorite with many visitors and is considered one of China's national dishes. For a great start to your teaching career in China, why not take a 4-week-in-class TEFL certification course at one of our four Chinese locations? Are you ready to teach English and eat great food all over the world? Apply now for your TEFL TESOL certification course and start teaching in a matter of months. Speak with an ITTT advisor today to put together your personal plan for teaching English abroad. Send us an email or call us toll-free at 1-800-490-0531 to speak with an ITTT advisor today.