 A, ja. OK, so, ja. The next speaker is Ivan, the Telitci, and is talking about generalized range of mind equation, thermodynamics, and applications. So, thank you. Now. Does it work? Can you hear me? OK. Oh, god. Not a good start at all. OK. Let's suppose there is a K here. And... Hello, everyone. OK. So, I'm Ivan, the Telitci. I currently am a postdoc at Max Planck. We see at the institute for the physics of complex systems in Dresden. And first of all, I would thank the organizers for letting me talk about this work, which I have been... I hope not. About this work, which I carried out during my... I can't believe it. I mean... OK. Let's hope for the best. OK. Which I carried out during my time as a PhD student in Padova under the supervision of professor Marco Baghesi. And my work is about the generalized range of equations, and mostly about its thermodynamic properties. So, everybody here knows about the range of equations, which have been introduced at the beginning of the 20th century by Langeven to describe the stochastic motion of a Brownian particle. And furthermore, everybody knows that, associated to this stochastic process, there is a set of partial differential equations for the probability density function of this process, which are called the Fokker Planck. We see this time. OK. Which come in the form of a continuity equation. In the particular, we also know that we have some stationary solutions associated to this equation. And in particular, we have equilibrium solutions, which correspond to zero currents, while some stationary solutions, which are associated to a current, which has zero divergence. In particular, for force, which comes from conservative potential and diffusive matrix, which is constant, we know that the equilibrium probability density function takes the form of a Boltzmann distribution. And here are some experimental examples of a realization of this equilibrium distribution. But because we don't like Markovian dynamics in this conference, we consider the generalized Langeven equation, where there appear some memory kernel. This memory kernel is linked to the correlation of these colored noise. And in this case, non-equilibrium is achieved by considering a force, which may be either non-conservative or explicitly time-dependent. This is the case that we are going to consider. But the problem with these equations is that we have no associated Fokker-Planck equations. And here is an example of memory kernel, which we have already seen a lot. And there should be a minus. But we have no associated Fokker-Planck equations to these generalized Langeven equations. And in order to circumnavigate this issue, we came up with a modification of the Laplace transform, which is defined as follows. It is nothing too fancy. It's just a very deeply linked to the property of frequency shifting of the Laplace transform. And we apply this, in first instance, to the linear generalized Langeven equation, which is defined as follows. So I know that linear equations are not very spicy, but we are also applying this technique to a nonlinear setting. Maybe I will talk about it at the next conference next year. But for the moment, we stick to the linear equation. And what we have done is to solve these equations by using this new technique. And one thing that I forgot to mention is that in the equation, we keep these initial time unfixed. And by applying this modified Laplace transform to this equation, we are indeed able to keep track explicitly of these parameters, which may be afterwards sent to minus infinity to consider, for example, stationary states. In particular explicitly, we find the solution of the generalized Langeven equation as follows. The solution is given in terms of these susceptibilities. This is something that has already been done at the beginning of the 70s by Fox. And the nice thing of this equation is that, again, this parameter tm is explicitly present in the solution. And furthermore, because the susceptibilities have well-defined limits for large times, regardless the form of the memory kernel, we are indeed able to consider many kind of initial conditions. And among all of those, we consider initial conditions, which are in equilibrium corresponding to tm equal to zero. And initial position and velocity may be distributed as a Gaussian. This may also be obtained from the study of this solution, but it's actually trivial, so I won't go deeper in it. And the second situation that we are going to consider, which is more new, maybe, is tm that goes to minus infinity. So as you can see, when tm goes to minus infinity, the argument of these susceptibilities goes to plus infinity. The limits are well-defined. We see that, for example, these terms here go away, so we lose memory of the initial conditions. We like to consider this as a generalized steady state, where, indeed, memory of initial conditions is lost. By using the Gaussian properties of the noise, furthermore, we are also able to infer that the distribution of position and velocity at time t is also Gaussian. So, just to recap everything, we have, more or less, shown that the probability distribution associated to these initial conditions as a form of a Gaussian with these covariance matrix, in particular for these initial conditions, the covariance matrix has this form. And this form of the covariance matrix cannot be changed by any deterministic protocol, which may be the center of the trap, which is moving, or some external field, which may be, for example, an electrical field. So, in this case, no need for Fokker-Plank equations. But the most important part that I would like to talk to you about are the thermodynamic properties of these generalized Langeven equation, and in the linear case always. And to do so, we first consider the first theorem of thermodynamic in the context of stochastic thermodynamics by starting from the heat. And to do so, we use Sekimoto's definition of heat, which is this stochastic integral of involving the forces that the particle is exerting on the bath times the velocity of the particle. These forces can be easily be obtained from the original generalized Langeven equation. Then we can consider also the variation of energy, which in this case is just the sum of a kinetic and the potential energy, and it takes this simple form. But most notably, the work, which is the sum of these two quantities, has the same form as for Markovian dynamics, which means an integral of this protocol, which is changing the center of the trap times the force, which is exerted on the particle itself. So this is something that already five minutes. So the thermodynamic work has this form here. And because we have the solution of the position for the position of the particle, it is easy to calculate quantities. One thing that we find is that the variance and average work are proportional and have this simple form here. It is just dependence on the particular properties of the system. And because we have this proportionality and because the work is Gaussian, we have that an integral fluctuation theorem holds. Furthermore, for initial conditions in the infinite past, we find that the variance of the work has the same form as from starting from equilibrium. Finally, what we find is that average work in the stationary state corresponding to a dragging, which is constant, this form of lambda, and tm going to minus infinity, we find this simple form, which is equal to the structure that we find in form Markovian dynamics, but instead of the usual coefficient, we have this integral of the memory kernel. Here we could talk about the fact. While studying these issues, there are some arguments saying that these integral should be finite, otherwise there are some problems associated with it. And in this case, this is maybe another clue that indeed this quantity here should be finite, otherwise we would have some thermodynamic inconsistencies. Here is an example, but I don't have time. It wouldn't be a discussion about stochastic thermodynamics without entropy production, of course, which can be split into contribution and environmental contribution proportional to the heat, a systems contribution proportional to the Shannon entropy and for stochastic thermodynamics the second law holds on average. What we find in this linear case is that the rate of entropy production, which is easier to calculate, takes the following form. These two terms are usually equal to zero in the initial conditions that we are interested in and we don't care about it. The most important part is this one, where we introduce this new dynamical quantity which is called retarded velocity which involves a convolution between the average velocity and the memory kernel. Again, we identify a steady state for a constant dragging and tm that goes to minus infinity where we see that the average rate has the same structure as for Markovian dynamics with the substitution of gamma zero that becomes gamma, the stoch's coefficient that becomes this integral from zero to infinity of the memory kernel. Again, this may be a clue that indeed the limit of this integral should be finite. How much time do I have? Nothing, basically. We have some result. We consider an exponential memory kernel to show what is the effect of memory on the quantities that we have studied in our paper. Starting from equilibrium, you see that as memory becomes stronger and stronger there appear some oscillations in the averages and in the variances of all these quantities. This is a more or less typical property of systems with memory and also inertia. In the case of the steady state, instead, we have that all the averages do not depend on the strength of this memory while the variances are affected by it. To conclude, we have introduced a modification of the Laplace transform which is particularly useful to address the issue of steady states. We explicitly solved the linear-Langeven equation with time-dependent driving and by doing so we are able to calculate variances, averages of many relevant quantities such as the thermodynamic work. We also provide an explicit formula for the average rate of entropy production by introducing this new dynamical quantity called retarded velocity and finally we show that variances of memory are the variances of integrated quantities. To conclude, I would like to thank my former PhD supervisor Marco Baghesi, professor Felix Ritor, which helped in the writing of this paper, my new group at the Max Planck Institute and of course you for your attention. Thank you. Thank you very much for the interesting talk. We have time for a couple of questions. Priority given to students first. There is one question. Are you a student? Hi. I'm not sure how dub this question can be, but it's just because I'm not familiarized with the technique of that modified Laplace transform. Why does the usual one doesn't work? Because I imagine that it doesn't work. Why do you need this modification? Because... So the... I asked myself the same question when I started studying these issues. The fact is that most of the times to study the generalized LaJava equations, people take these initial time to be arbitrary and then they set it to zero. But we know that steady states most of the times correspond to initial time that a system that is going for an infinite time then it equilibrated some kind of steady state and then you see indeed a steady state. In this case, because we don't have Fokker-Plank equations, there is no at least from my point of view intuitive way of letting these initial conditions going to minus infinity. And furthermore, the Laplace transforms are very useful and easy to use. So my idea was indeed to introduce this generalization of the Laplace transform where you can just track explicitly of these initial conditions in the past and then you just send it to minus infinity avoiding let's say singularities or anything else. So I hope I addressed your question. Just a comment. I can provide you an example which this gamma hat so the integral is finite but you can make it arbitrarily large. Actually it becomes infinite critical point. You have an example. Of course, if you pay, I can give it to you. Please tell me. I also have a question about this generalized Laplace transform. I'm afraid it's going to be a stupid question. It just caught my attention. So you're taking tm to minus infinity but this must very severely constrain the functions g of t, you can do that. Because then what you have is for negative times an exponential explosion by the exponential factor. I agree with you. So need to have even functions that are of exponential order in both directions or this must constrain just for the mathematical existence. Of course, of course. Let's say that most of the times you apply this Laplace transform to functions which are kausion for example, susceptibilities or the memory kernel. So this issue doesn't come. And furthermore I honestly tried to avoid these issues by just explicitly solving the equation hoping that everything went well. So in this case, for example, you can see that the only dependence on this initial time is in the susceptibilities. Because typically if you have a function that has support on an entire real line you take a Fourier transform. And that is also analytically nicely defined. So I guess that also doesn't work here for some reason which I just couldn't get. Fourier transform doesn't take track of the initial conditions. So if you take the Fourier transform of the equation you have the convolution theorem and everything, but you don't have explicitly the initial conditions here. Which is the thing which I tried to have here because at the beginning of my PHD I wasn't so... Is it still a linear transform? Does it have all the mathematical properties in the Laplace transform? Yeah, of course. But in this case I think it's more easy to see the dependence on initial conditions and it's more straightforward to consider steady states. So it's like a sum of two Laplace transforms when you flip the time? Well, actually again you can see it as consequence of the property of frequency shifting because if you do change of variable you just obtain e to the minus tn... Yeah, this I understand, but then you also have a delta function there. If you do that you are not supposed to consider times where this gets negative as far as I understand. So this is really a mathematical question. I understand it works because you got the result, I just cannot understand how. We can discuss after that. Thank you. Since you mentioned several times that there is no Fokker-Plank equation so I'm wondering can you make a Markovian embedding for the memory kernel that you suggest by just adding one auxiliary variable and then you do have a Fokker-Plank equation. In the simple case of an exponential memory kernel you can do Markovian embedding and everything works fine. But all the results that I present are also valid for example if you don't have any straightforward or trivial way of doing a Markovian embedding. Here, to be totally honest I used exponential memory kernel because where do I have it here maybe? Ok. In order to find the solution you basically have to calculate these susceptibilities which are defined in the Laplace space. So here you have the Laplace transform of the memory kernel. If you put something too much complicated basically Mathematica cannot compute these susceptibilities. So in order to make everything analytical and to get these all the plots that I showed you at the end I take it to be to be exponential. But for example as the case of entropy production at the end where you have this integral to infinity of it doesn't work anymore at least it's at the end that it doesn't work anymore. So the rate of entropy production has this quantity here that is the integrated from zero to infinity of the memory kernel. In that case you can think of some polynomial function with exponential dumping which is absolutely not trivial to calculate the dynamics with arbitrary initial conditions but still you are able to figure out the rate of entropy production in the steady state and without any particular assumption on the memory kernel. So I hope I addressed your question. Ok, I see that there is a lot of there was a lot of interest. You can postpone the discussion to the next coffee break. And I think we should move on in the interest of time to the next talk. Be careful. I hope so. I hope I can.