 So just a quick thing, yeah, this is the first slide. So I noticed that the title of my talk has changed in the program. So originally, one thing was accepted for the talk. But then the other thing that I submitted for the poster appeared in the program. So now I have two talks, and yeah. So I have two talks, and the one you, OK. So the one you like the most are we going to tell about. So as you wish. So either talking about dissipation time uncertainty relation, or normal Corvian, Brownian dynamics, and non-equilibrium bath. So let's vote for the dissipation time. OK. Yes, let's vote for the second one. Yes, it was slightly more for the second one. Yeah, OK. And indeed, I guess the right thing, because you see, the first is the one that you wanted. Amazing, OK. Yeah, then. OK, that is my pleasure to renounce Jean-Marie Falasco, who is going to talk about non-Macovian Brownian dynamics and non-equilibrium bath. Thank you. Thanks a lot. So this is some joint work that has been going on for a few years with Marco Bayezzi and Klaus Kroij, when I was back in Leipzig, and other collaborators. So this is a story that I'm going to touch on things that we heard today already. So for example, long time days. And I want to start from essentially the from normal Brownian motion. The one that we are used to talk about actually is using simple Long-German equations. But actually, if you talk to experimentalists and they have enough time resolutions, well, even simple Brownian motion in a one component fluid has to be described with a generalized launch of n equations. And this is, as we heard before from the talk by Thomas, is because of momentum conservation in the fluid. Whenever you move a particle in the fluid, you are displacing a huge amount of fluid around. And it takes some time to rearrange the fluid around so that you will always have some long memory. Long memory, of course, on time scales that for a micron particle is a border of milliseconds. So we'll have this kind of friction coefficient that is now time-dependent with exponential days. And if you are in a bath, which is at equilibrium, so the temperature is fixed and is t, so we know that the fluctuation dissipation at equilibrium must hold such that the correlation of this Gaussian noise is given by the same symmetrized function that appears here. So this is essentially, this is from experiments. This is what you observe. And this is what would be predicted by a simple Long-German equation without memory. So all these ingredients, still, because of fluctuation dissipation relation, you end up with equilibrium distributions and you end up with essentially fluctuation dissipation on the level of the observables. So mobility of a particle dragged in this fluid and the natural, the fluctuations in the mean square displacement will be proportional to each other. So now, my point here is what happens if we put the environment out of equilibrium? And one simple way to do that is essentially to induce a thermal gradient in the fluid around the particle. And this is actually something that happens very, very often in experiments whenever you have a particle and you do some tracking or tracing with laser light, such that part of the laser light is absorbed by the particle itself, the particle radiates around, and so it creates a temperature gradient that moves with the particle. And so now we want to see what happens to this Long-German equation, generalized Long-German equations, if we can still write it down and what are the properties of this. The point is that we cannot directly write it down because the environment is out of equilibrium. So what we can do is to step back, go to underlying this level of description, and this description is the one, for example, including the degrees of freedom of the bath in terms, for example, of hydrodynamic quantities. So temperature, velocity, the local velocity of the fluid, and mass, mass density. So in general, since this particle moves slowly with respect to the heat diffusion, essentially the temperature gradient, you can show moves statically with the particle so that this is already, you can find this temperature gradient, the mean temperature gradient around the particle in this way. Then what you can do is to consider fluctuating hydrodynamics. So on the hydrodynamic, on the Stokes equation, this is stocked because we are at low Reynolds number, on the Stokes equation you can mount Gaussian fluctuations. Gaussian fluctuations with the assumption that locally these are equilibrium fluctuations. So at each and any point, you have this fluctuation dissipation relation with the local temperature. So the fluid is in local equilibrium. We want to see what happens to this particle that now samples and probes this fluid. So what you can do, essentially, you can coarse-grain these things. You can get rid of the hydrodynamic fields. And so you can go, again, you can find out that still there's a Langevin equation holding. Sorry, you need to know eta of t. Eta of t, yes. This is a property of the material property. So you need to know that your fluid is water and I need to put a guess here. Usually, yeah, you parameterize this thing with a Vogel-Fulker function there. But in general, the dependence here, it's not so strong, it's so important, okay. And for the theoretical results that I showed you after, to get analytical results, we actually assume a constant eta and you see that they will fit nicely with the numerics. So the main modification here is that you still have a Langevin equation. The difference is that you have a different noise autocorrelation function that in frequency space you can rationalize in terms, if you want, of an effective frequency dependent temperature, which you can calculate analytically and it has a very specific physical meaning that you can explain very well. So this temperature is a local, essentially, is an average of the temperature on the fluid around the particle sampled with the amount of energy that you locally dissipate in the fluid when the particle moves at frequency omega. So this function phi here is really, the function is the energy dissipating the fluid at a given point R, when the particle oscillates at frequency omega. So the one is the same quantity that comes out of linear reversible thermodynamics, essentially. So what's the meaning of that? This means that essentially these quantities, so the thermal fluctuations felt by the particles, they change with the frequency. So they change with the time scale of our observation. And this is because there's a dynamical coupling between far points in the fluid and the particle itself. And this is mediated by, essentially, the shear modes of our fluid. So essentially, if you imagine that your particle is oscillating at high frequency, so we look at the high frequency part of the spectrum, you have back and forth oscillations. The spreading of the hydrodynamics mode is then confined because they annihilate because of the short oscillations. And so this sample is, this average is only around the particle where the fluid is very hot. So at high frequency, the temperature is hot. Then if you go to low frequencies, essentially these average goes to larger volume that are farther apart and so they are cooler. So this temperature will go down. And that's essentially what you can calculate analytically, assuming an effect of viscosity that is constant. And here I showed you two curves because what I calculated is for two different motions, is for translation, but also you can do that for rotation, Brownian rotation of the particle. And since you display fluid in a different way, so the shear modes are different, you have two different curves. So you can already imagine that different degrees of freedom will feel different kind of effective temperatures. How actually would you calculate? How do you sample this temperature? Because this is a property of the noise. We rationalize the noise correlation in this way, but how do I measure this? Well, one way to measure that is essentially, is for example to take this particle, confine it in a harmonic potential and make this potential very strong. So what I, the mean square displacement, now the particle, would be like a filter, we will filter this effective temperature with the response function of the position, but if the confinement is strong enough, the only frequency that we will see will be the external frequency of our oscillator. And that's what we did in MD simulations where we were trapping this hot particle, confining it, then changing the confinement, and so we were sampling these effective frequency dependent temperature at various frequencies. And so you see the blue curve, which is what you want to observe and what you want to sample, and the red dots that are the mean square displacement are different for different frequencies. The other point that I want to mention is that you have, if you now look at the dynamics, you take this particle, you drag the particle, you measure the response, the linear response, so you get this mobility, so when this force is very small, and you measure the spontaneous fluctuations, integral velocity, velocity fluctuation functions, now they are no longer proportional to each other and actually, if you plot them in a parametric plot as a function of time, you will see that the slope of this line is changing and is interpolating between two different values. One is the late time, the zero frequency of that function that I showed at the beginning, and the early time, temperature is the one related to the kinetic energy. So it's the one associated to velocity fluctuations, which is something that is reminiscent of what happens in glasses. But here, everything can be analytically calculated and in a way, kept under control. So let me just finish because here I just showed you an example of one way to drive the environment out of equilibrium where everything is under control, but one thing that is nice here is that the environment, because of the Lorentz number, because the system is water, is a very simple system, everything is essentially linear, and so I get a linear-long-driven equation. But in general, if my environment, if I have strong interactions, I will have no linearities. So one way to deal with that, and this is what we did here, was actually to now couple my system with the environment. The environment now is no linear, is out of equilibrium because there are active particles because it's a steel fluid or something like that, and because still there's a scale separation, so the system is very large, is these big particles, they are large, they are much slower than the environment, essentially I can describe this coupling via a response theory. So motion of the blue particles are just small perturbation on the background, on the environment that is out of equilibrium. And so using linear response, non-equilibrium linear response, you can essentially derive equations for long-driven equation, generalized long-driven equation, for these big probe particles in the non-equilibrium environment. And I just want to flesh some properties before concluding. So the average force on each, felt by each of these blue particles is because of the linear response structure is the average response with a steel environment, and then there is a thermal due to friction, and this counts also interactions between the probes, the big blue probes, mediated by the non-equilibrium environment, and statistical forces that are there and they do not depend on the velocity. So the first thing we are used to see it, and in equilibrium it's just the gradient of the free energy of the environment. When we are out of equilibrium, this is no longer true, it's no longer a gradient force, and since because of this thing, these statistical forces are not reciprocal, so the force in one probe exerts on the other, it's different from the reverse, can be different from the reverse. And a similar thing happens for the friction, so this memory kernel is no longer given by the green cubo, but there are modifications in this correlation function due to the probability currents induced by the driving, and so we will break this reciprocal kind of interactions. Yeah, so, well, if you want to ask me a question about a specific example, and no, with this I see that I'm going over time, so. It's fine, you can go ahead. So it's just a simple example where we were using this kind of linear response theory, so we were considering a one-dimensional system, soft sphere as an environment that is driven by a constant force, and there are two big probes that are confined. This is kind of two-point microbiology, but in an active, kind of active driven environment. And what we were measuring here with several things, one in particular is this memory kernel that encodes the information about the force that one probe exerts on another probe when it moves with a certain velocity. And because in general, this thing is asymmetric, as I told you before, and this is because we essentially, we break this symmetry imposing this force F, and the more we drive the system, the more we change the equilibrium properties, and the more we make these two functions different, even to a point where the entire integral of this function can change sign. So this means that we have essentially a negative response. We move one particle in one direction, and the other is going not in the same direction, but in the other, and this is just a screening effect because we bring one probe closer to the other and we screen so the second from the original flow created by the external force. That's an example of what we can do and we can see with this theory. So with this I conclude and thanks a lot for the attention. Yeah, thank you very much for the super interesting talk and you're already also lead it over to the next talk, which was super nice at the end. Yeah, so questions from the audience. Yeah. Hi, so first of all, thank you for the talk. My question is a bit weird probably. So if the answer is really trivial, just tell me and forget about my question. So it's related to the fact that you mentioned the effective temperature felt by different degrees of freedom might be different. So for example, a translational degree of freedom might feel that effective temperature is different from a rotational one. So my question is if I engineered a system in such a way like to have both a rotational and translational degree of freedom because maybe this particle has an internal structure or can this be exploited from the point of view of having a separation of two different thermalizations for separate degrees of freedom? Like the rotational degree of freedom will thermalize, I don't know, to contact with a thermal bath to a different temperature than a translational one. Even without engineering anything here, they will do that. Unless you have specific cases, for example, in the case I showed you before, where you have a very strong confinement, in that case you will select only one frequency and that is the same for the two, but in general they will be different without any engineering for whatever parameter. But can this create, so is this a temperature difference that can be explored, can this be created, can this be created a heat engine? In this case, in this case what you, so what you can see is that there are correlations between position and velocity, so this will not be there. So they're not independent. But what I'm not saying that this temperature, they have a thermodynamic meaning in the sense that it's not that looking at them, you can infer in general what kind of fluxes of heat you will have. I see, okay, perfect. Thank you very much. There are people in the audience that know much more than me about nothing. There's a question in the chat, maybe you can just read it yourself. Do you take specific functional forms for the coupling to derive the probe equation? So no, the coupling, so it's referring this equation, this question is referring to these. So in this general theory, no, this is completely, it is just general. I have like Newtonian equations for the big particles. I have in this case overdone launch event motion for the small particles with some driving. And the forces, these are the forces on the big probes I, these are general, is just that they have to come from a potential, such that the force on a probe is minus the force on the environment. But in general, they are not specified in this theory. In the specific problem here, these are like soft spheres, so they can penetrate each other because we are in one dimension, otherwise we have problems. Yeah. And there's another. Yeah, yeah, go ahead. Do you need to take a large number of QS red particles to derive the probe equations? One can derive it for finite, you can derive it for finite, wait for the red particles. Yes, this is, there's an assumption that there's a scale separation in terms of time between the dynamics of the small red particles and the blue particles. For the number, no, there's no assumption that has been made in this theory about the number of degrees of freedom in this environment for the theory to hold. I hope this answers the question. Question for you, Mika. A quick question about the simulations. So you're doing water and which and... Which in the first day, yeah. Molecular dynamics. This. Yeah, so which and some are using MBT and VE or do you remember? So this simulation is MBT. We were using like, this is Lenard-Jones interactions for the fluid. This is Lenard-Jones plus a Fene interaction. So this is confining for the particles combining the colloid. And there was a velocity rescaling algorithm to keep the velocity of the colloid heating up. There are very few molecules in the bath, so did you check... Here, this is just a snapshot. It's bigger. It's way, way, way bigger. So Landjeven is a good approximation. You check things work in a time scale and okay. Yes. I see. Yeah, thank you. I think in the interest of time, we should move on to the last speaker of the session. And thank you again, Jean-Marie for this very interesting talk. Thank you.