 Very good. I assume you see the full screen. Matteo, can you confirm you see the slide, yes? Yes, we see the slide. Thank you. Should I start then? Please. Thank you. Well, first thanks to the organizers for the invitation and organizing these tutorials. I'm quite glad to see that there are so many participants. It's nice to have such a big audience from different places. I'm not going to say good afternoon or I guess it's morning for many people. So I'll be talking, I'll give a tutorial on large deviation theory and how it's, we can apply it to calculate distributions of observables and stochastic thermodynamics, such as work heat or entropy production. So I'll start with a bit of the context. The idea is that we were studying a system. So we take a stochastic system and then we observe it for some time. And then the quantity that we, we at the same time we measuring some quantity like the work done on the system or the heat exchange with the environment or the entropy production that's being produced by the system. And what we want to calculate is the probability distribution of that observable that quantity that physical quantity. While calculating the distribution is not an easy problem in general, not even for, for simple mark of processes. But what's known in many cases is that the probability distribution has an exponential form with the observation time. And so this is what we'll focus on. So this, this is an approximation that I'll make more precise later on, but this is the basis of large deviation theory. The theory itself is concerned with probabilities that will have this form, and then we'll get to understand what this form means. Now the form itself, the, the approximation involves the time, the observation time. And then this little function that we call the rate function I of a, and then the goal of the theory will be to give methods to calculate that rate function. And this can be done a different way. And then one of the way that we have is to look at the Fourier transform of the distribution or the Laplace transform of the distribution. And if the distribution has an exponential form, then also the Laplace transform, the generating function will also have an exponential form. And so the game will be to calculate this other function, the lambda k that we call the scale community generating function. And then we'll see that the two are related. So if we're able to calculate one function, we can extract the other one, but ultimately we want the rate function, which is going to give us some information about the distribution. So this will be the main problem that we'll focus on, but at the end I'll also discuss what we call the prediction problem. We want to understand how fluctuations are created. So if I look at the system and I look at different trajectories, then most of the time I'll see trajectories that gives me kind of a typical value of the observable, but I might be interested with a large fluctuation of the observable. And then I can relate that fluctuations to trajectories that are rare in the system. But, and then the problem will be to understand how those trajectories arise when they look like are they created by some modified force or modified current or density and then I'll be discussing this. So for this part we're trying to find an effective process that describes only fluctuations that get trajectories that give rise to fluctuation of the quantity. Okay, so the exponential form of the distribution actually arises in many different areas of physics, not just time dependent problem that relate to stochastic thermodynamics. I started studying the topic actually with equilibrium systems, so spin models or particle models that you study with ensembles. And the same probability arise there, but it also arises in many contexts in mathematics. So I'll present the theory in a more general way and then we'll apply it specifically to the case where we do an observation in time. But to begin with, we look, we have some random variables. So we have some random variable of interest. It could be an integrated quantity. It could be the magnetization of an n particle system. The point for the theory is that the random variable is indexed by some parameter and then at some point there's going to be a limit where n goes to infinity. But the precise nature of n depends on the application that you consider. So we have this random variable and again we're interested in calculating the probability distribution. In general this is very difficult, but in many cases we can show that it has an exponential form in some approximate way. And large deviation theory again is based on this approximation and the meaning of the approximation is that really the exponential is the dominant part of the distribution. So if you take the log of the probability, what you'll see is that the result is linear in that parameter n and then you have some corrections that will be sublinear with the parameter n. And so if you divide by n on the two sides and you take the limit, you kill the correction and then you recover this function that we call the rate function. In other words it also means that if I look at this exponential, I will have corrections that are exponential in front or polynomial in front, but even if they are exponential, the factor is smaller than n. And so really what's dominant is the decaying exponential with n. So the goal of the theory is to show that for a given random variable you do have the large deviation form, which we also call the large deviation principle in the theory, and then get the rate function. So we'll study methods to get that rate function. If you have the rate function, you basically have a lot of information about the distribution. You don't get the distribution exactly, but you get the distribution to the dominant form. And then we'll see that we can extract lots of information about the fluctuations of the quantity. For instance, we'll see that the rate function is always positive. It has often a zero, and the zero has to be where the density does not decay exponentially, and so this is where it has to concentrate exponentially. So a zero rate function will give you a stationary point or an equilibrium state, and the shape of the rate function gives you basically the shape of the density, but just on the log scale. Okay, so now the question is how do we get the rate function? So there are many different techniques to do this, and I'll mention only one which is referred to as the Gartner-Illis theorem. So this is a result in a mathematical theory that came in the 70s, 80s, and the idea again is to get information about the distribution, not from the distribution itself, but from the Laplace transform of the distribution. So what we consider is the generating function. So we have the observable again or the random variable, and then we calculate the generating function. So that's an expectation of the exponential, and k is a parameter that will be dual to the values of the random variable. We take the log of this, and that defines a cumulant, and then we divide by n, and then take the limit, and that defines what we call the scale cumulant generating function that is CGF there. The theorem says that if you can calculate this function and this function is differentiable, then you have the two things that you're after. You know that the distribution of the observable decays exponentially with n, so you have a large deviation approximation, and the rate function, this I of A is the LaJean transform with the lambda k. And here I'm writing the LaJean transform in a more mathematical way, but this is really a LaJean transform of the function lambda k there. So I of A, the function that we're after, is dual to lambda k in a LaJean transform sense, and then you can relate, if you know the shape of the rate function then you have information about the distribution. In particular, again, as I mentioned, the zero of the rate function will give you the typical value of the observable because that's where the probability will concentrate with n. You can also show that this will be given by the derivative of the scale cumulant generating function. So again, you have this relationship between the two functions that will give you information about typical values and fluctuations, and then one function encapsulates the same information on the other side. So that's the main result we'll actually use, but I want to stress that there are other results and then, depending on the application, you might be able to use the Gartner realist theorem with other methods to get the rate function, but that's the main one we'll use. Okay, so now I'll go back again to stochastic thermodynamics and I need to define now the two objects that we need to do a large deviation calculation. The first is the system. What is the model that we're using to describe the system, the stochastic system that we're studying, and then the second one will be the definition of the observable. For the system, we can model the system, we'll model the system as a Markov process and then we have to make a choice. Some systems will be modeled by a John process or Markov John process and some other systems you can model using a continuous space so kind of a diffusion model for the purpose of the system will focus on diffusion. So the basic model that we have is a stochastic differential equation we have a state for the system that's xt. This can be the state of a many particle system it could be one particle. So this could be a vector state but also a many particle state but for the, to simplify the notation I'll just write this as a, as a, as a kind of a simple random variable X of T and so this is the evolution it's a differential evolution that has Gaussian white noise and so there's a deterministic part that we call the drift, this kind of the forces acting on the system, and then the reservoirs or the noise will be introduced with mathematically with the Gaussian white noise and so the noise also can be a vector noise and so you can have different heat paths at different temperature, but mathematically this will all be included in the sigma that you have there. And in terms of probability the system is described by a probability density and the probability that you reach the state X at time t and the evolution for diffusion is given by the Fokker-Planck equation which is a linear PDE and so you have a linear operator which I'm going to write later on. You can also write the Fokker-Planck equation I think it's well known as a conservation equation using some probability current and so that defines the probability current. So this is the way that we describe the evolution, the stochastic evolution of a diffusion or in physics, but in mathematics is also a dual picture of this where instead of looking at the evolution of the density you look at the evolution of expectation. And so this is this will be useful for us. If I take any test function of the state and I calculate the mean or the expectation this will change with time and so I can take the time derivative. There's an equation that's very similar to the Fokker-Planck equation where there's an operator acting on the test function that will give you the expectation. The generator actually for the expectation evolution is this. So this is the drift, this is the nabla, and this is the kind of the Laplacian, the diffusion operator. And the Fokker-Planck operator is just a dual of this in the usual scalar sense. D is the sigma-sigma transpose, so that's the diffusion matrix. This doesn't have to be diagonal. It can include correlation between the north sources, it can include different temperatures, and then we'll see some examples. For us, the setup will be that we consider a garlic process. So we look at steady state non-equilibrium processes that are described by in terms of these diffusions. And so the density with time will converge to a stationary density and the current with time will converge to a stationary current. And so we're in a steady state. And then we want to calculate the probability of observables in the steady state that will be the context. So this is for the system. I'll focus on diffusions, but everything that I'll say actually will be applicable also to jump processes with minor modification. What changes is the form of the generator and what you do in terms of calculation. Now, there are many examples in physics that are relevant. So I'll mention three. The first one is the under-damped Langevin equation, which is the noisy Newton's equation in a way. So you have here the, yeah, this is the momentum equation. Here you have the force equation for the momentum. So this can include conservative forces, non-conservative forces, some friction, and then you would have some noise. If you take the over-damped version of this, then you end up with the over-damped Langevin equation where now it's just a first-order equation for the position. And then there are many other equations and some of them will actually will be derived from a potential, and so we call these gradient stochastic differential equations. And these are interesting because you can solve them exactly. In particular, the stationary distribution is a Gibbs state involving the potential. So this would actually describe equilibrium systems, reversible systems. And if you have forces acting on the system that are not gradient, then it's just more difficult to calculate the stationary state. But we'll assume again that there is a stationary state. It's just that in those cases we won't have a Gibbs state. Okay, so that's for the system. Now for the observable, the definition of the observable will depend also if you consider a jump process or diffusion. And again, I'll focus on diffusion. So in general, you can write down an observable using two contributions. Many observables will have the form of an integral of some function of the state, and we call those density type observables. And many observables in physics will have a different form. There will actually be stochastic integrals of some function of the state, so g of xt, but multiply by the increments. And I give here some example. For instance, if you look at the change of the potential energy in a system, this will be the integral of the gradient of the potential multiplied by the increment. And the multiplication has to be done in probability theory using what we call the Stratonovich convention in order to follow the normal rules of calculus where if you integrate the gradient of a function, you get the function at the end point minus the function at the initial point. So this will be the result if you use the Stratonovich convention. So this is this is the convention we're using normally and stochastic thermodynamics. Another quantity could be the work. In this case, this is the force times the displacement. So this is obviously of this form here. So there's no density part. The entropy production is also of this form here. So this is slightly it's a bit like the work. So it's the force times the displacement, but you define it also with the diffusion matrix. There are other observables that will have more of a density form. For instance, the occupation. So here, this is what we call the empirical density. This counts the fraction of time that the state actually will reside in the state X or in the value X. So this is a random variable. It looks at the occupancy, the local occupancy of the of the process. We know it converges to the stationary distribution, but there will be fluctuations around that value. The same with the other quantities. What we'll see is that the quantities in the long time limit converges to values. These are the steady state values. For instance, the stationary potential energy that is exchanged or the stationary work that you do. But we're interested in calculating or studying fluctuations around those steady values. Okay, so how do we do this? So again, we assume or we know that the distribution will be exponential with the integration time. So you see all these observables are integrated in time. So the parameter that will go to infinity is this integration time capital T. So all the large deviations will be in that limit for a large time. We're going to have this dominant form or this dominant contribution to the density. And so we want to get the rate function, which will give us information about the density itself. So to do this, we'll follow the Gartner's theorem. I need to calculate the scale community generating function. This function now involves the observable a of T. And so everything has to be defined with the integration time. So now the generating function is this with the time. I take the log, I take one over T and I take the long time limit. If I am able to calculate this function and this function is differentiable, then I can apply the Gartner's theorem to say that I do have a large deviation approximation for the density of the observable. So I know this and the rate function will be the Legendre transform of the lambda k. What's interesting for Markov processes is that we can actually calculate the lambda k explicitly as an eigenvalue. So this is a result that I'm not going to show how to derive, but it's quite interesting. The lambda k is actually the dominant eigenvalue of some operator that happens to be a kind of modification of the generator of the process. So for the diffusion, I'm showing that generator explicitly. So the LK, so that modified operator that we call the tilted operator or the tilted generator or the twisted generator involves the drifts of the process, but it also involves the little functions that will define the observable. So if I go back to the observable, a general observable is defined with two functions, that small f or that small g, and these will be actually determined according to the observable, the specific observable that you're looking at. So it could be the gradient of the potential or the drift that comes there. So if you know these functions, then you'll know what the twisted operator, the tilted operator is. So you have the g function here that's pushing the nabla, and then you have the f function here that's just an additive perturbation on the generator. If you set k equals zero, actually that's the generator of the process itself, that's the generator of the evolution. So you see that this LK is a kind of perturbation of the original generator. Okay, so the upshot of this result is that calculating the rate function obtaining the large deviation from observable is really reduced to calculating a dominant eigenvalue. So it's a spectral calculation. So it looks a bit like quantum mechanics, but the difference now here is that in this setting, if you look at the LK, it's not a emission operator. This part actually makes it non-emission. So this part is a bit like the diffusion part. This is a diffusion part. This is emission. This part is also emission, but the first part is non-emission. So we have to be just a bit careful in the way that we're going to do the spectral problem. This is not a conventional quantum mechanical spectral problem. So because it's not emission, you need to find eigenvalues by considering the direct problem, but also the dual problem. So you have right eigenfunctions and left eigenfunctions, so this RK and LK. These two will have the same spectrum, so the eigenvalues will be the same, and in particular the dominant eigenvalue will be the same, but the eigenfunctions will be different. And you need the two eigenfunctions because the boundary conditions are defined in terms of the product, and not just individually for RK and LK. And so this actually comes from the spectral theory of non-emission operators. And then here we have a normalization that we're going to use for the spectral problem, which is the following one. The product has to be normalized to one, and then LK also is normalized to one. This is a convention. This is not really important. But this means that the problem, the spectral problem is just slightly more complicated than what you would encounter for quantum mechanics, but essentially it's a problem about finding a dominant eigenvalue. So it's a bit like finding a ground state. Up to a sign it's like finding a ground state, but for an operator which is not emission. In other cases, we can show that even if LK is not emission, the spectrum is real, and so LK can be conjugate to some emission operator. In that case, it's equivalent to a quantum problem, like finding the ground state of a system. And then I'll show an example where we have this. And then we know that this is actually applicable whenever you deal with a reversible system. But the interest, of course, is for non-reversible system, non-equilibrium systems. Right. So this is the theory. So we have the system, the observable that defines the tilted generator. And now the game is to find the dominant eigenvalue to get the lambda k. And then I will transform that lambda k and then we get the rate function. And with the rate function, then we can say something about the fluctuations of the observable, how likely they are, whether the distribution is symmetric or not symmetric, where the stationary value is, so the typical value. And to illustrate this, I will focus on a very simple example, which has been studied in the past quite a lot. And it's still studied, actually, it's an example with laser tweezers. So we have a pool of brown particles. So here you have to imagine that you have a small bead of glass and water. And then you can focus some lasers on it in order to create a force. And then you can drag the particle around. And these are called laser tweezers. Because the particle is very small and in its water, it has a stochastic motion. We can follow the Brownian motion, actually, of the particle. And then as we start shining the lasers, then we can apply forces to the particle. And the model in this case will be a lingerie dynamic. So it's a noisy Newton equation. So here we have the force here. There's a drag force coming from the viscous drag of the water. And then the spring force will be the force created by the laser tweezers. So for low power laser tweezers, you can show that this is the force that are created are basically linear forces. So they're like spring forces. And of course there's a noise that's also coming from the surrounding environment that has a certain temperature, namely the water. So here we can work at different observables quantities that are relevant to stochastic dynamics like, for instance, the work done by the laser tweezers onto the particle in time. So this will be WT the work. And from the first law, we know that the work will be transferred in some part into potential energy for the particle, but it could be also some energy exchange between the particle and the surrounding environment and this will be the heat. The first law, which is valid still at a random variable level. So at a stochastic level, the law still holds. But what we can see, we're not going to do that calculation, but if you do the calculation actually these quantities can have different rate functions, because they can have different distributions. So although they're related by the first law. So the quantity that we'll look at is the work. So for the laser system, the work will be the work done by the spring force. And so we have, if we pull with a constant velocity, this will be the work done per unit time. And what we want to calculate is the work distribution. So the distribution of this random variable as an observable. This again, is not so easy to calculate but we know that it has a large deviation form so we can use the machinery of large deviation to find the distribution. To do the calculation, you can, you can change all the variables. You can put yourself in the co-moving frame. You can redimensionalize all the variable. And what you end up at the end is a non-stannual limbic process and also in the over-damp limit. So I'm not going to look at the under-damp dynamics. I'm going to just for simplicity. You can do the calculations for the under-damp dynamics, but I'm going to look at the over-damp dynamics so there's no inertia. And then change all the variables to make them non-dimensional. And so what we end up is, is an under-damp limit process. So this is a linear stochastic differential equation. The gamma is basically the friction and sigma is basically the square root of the temperature for the surrounding environment. So here actually the drift is linear and the observable, the work observable also if you change the variables correctly, then it's just the integral of the state. So this is a linear observable. In this case, the small f function will be f, x, and there's no g function in this case. So for this system, this is actually a gradient system in one dimension. It's a reversible system. There's a stationary distribution. It's a Gibbs state. It's a Gaussian distribution for the state, but that distribution is not sufficient to get the distribution of the observable. And that's just the whole point of large deviation theories that just knowing the stationary distribution or the stationary current is not enough to get the distribution of the integrated observable. And so for this we have to do the spectral calculation. So here the tilted generator is the original generator plus kf is no g function. So if I go back to the full tilted generator that I had before, this one here, I'm putting g equals zero and I have f, which is x. So if I put this in my generator, you see I have the generator of the diffusion, the Einstein and Levy process plus kx here. This is non-emission because of this time here actually, the space derivative is skew symmetric, it's anti-emission, but this is the emission part. So the whole thing is non-emission, but we know actually from some theory and stochastic processes that the spectrum of Lk is real. Even though it's not emission, the spectrum is real. So you can actually make a kind of transformation, a symmetry transformation in order to bring this operator to an emission operator that will have the same spectrum as Lk. So this HK is what we call the symmetrize generator. It's our emission now in this case. This is an emission. This is just a constant. These are all constants. So this is an emission operator. And what I'm saying is that the spectrum of this emission operator is the same as the spectrum of Lk. So it's actually easier than to work with HK because this is really a quantum operator. In fact, if you look at it, this is the amintonian of a quantum oscillator just written in a funny way by pushing the oscillator or translating the oscillator in some ways. But this is the Laplacian. So this is the usual term that you have in Schrodinger's equation. And this will be the potential applied to the quantum system. So this is a quadratic potential. So this is the quantum oscillator. So then finding the dominant eigenvalue here is finding the ground state of this quantum oscillator. You can just open any textbook on quantum mechanics and then you're going to have the value. And then we end up with the scale coming generating function as the dominant eigenvalue of Lk or the minimum eigenvalue of HK. And this is the answer here. So epsilon I've changed, unfortunately, my notation. The epsilon here is actually the sigma for the noise. So this is an error. So this is sigma square K square and two gamma square. And this is differentiable. So I can use the Legendre's transform of this from the Gardner-Listerium to conclude that the rate function is also a parabola. So this is gamma square A square over two sigma square. Again, the epsilon is the sigma there. So I get a rate function, which is a parabola. So that means automatically that here the fluctuations of the work will be Gaussian. And what the rate function will tell us again is that so everything decays exponentially. So if I look at the observable for increasing observation time, then I should see that the distribution concentrates. Where does it concentrate? It concentrates on the point of the rate function that is zero because it doesn't decay exponentially there. It decays exponentially everywhere else. At that point, I'm going to have concentrations of probability. And so that's the typical value of the work. In this case, I'm going to have zero as a typical value of the work because I'm in the co-moving frame. I've done like a change of reference frame there. And then I'm going to have symmetric fluctuations around that zero, which happens to be Gaussian because the rate function has the simple form of a parabola. In other cases, if you look at other observables, then you will get a more complicated rate function. And so it will describe a distribution that's not necessarily Gaussian. For instance, if you look at the heat, the heat has a rate function which is not Gaussian and so the heat doesn't have Gaussian fluctuations that has different tails actually an exponential tail and a tail which is non-Gaussian. So I'm not showing the result here, but it's something that you can find I can give you some references. The other part of the talk that I'm going to now I need also the eigenfunctions. So as you calculate the ground state of the Newtonian or the dominant eigenvalue of this operator Lk, it's also useful to get the Rk or the Lk. So the Rk will be the right eigenfunctions of Lk and Lk will be the dual eigenfunction. Or in the emission problem actually we have a psi just like in quantum mechanics because the problem is emission and so there's a single psi. All these eigenfunctions are related so these will describe the ground states or the dominant eigenvalue so that the eigenstate related to the dominant eigenvalue. And again, for this problem you have explicit solutions because you're dealing with the quantum oscillator. So we know the wave function for that system and then I can transfer the result up to Rk and Lk. So I have explicit solutions for this. And the Rk actually will be very useful for us for what's coming next. So at this point I can calculate the rate function, I get information about the distribution of the observable and this is very useful. I can test for instance whether there's a fluctuation symmetry like the fluctuation theorem. I can see what's the likelihood that I'm going to have some fluctuations away from the steady state and so on. The other problem I want to describe and then I'll finish with this is this problem of fluctuation prediction. How is the fluctuation and are fluctuations being created in time? The physical point of view that the problem is that you have a system, again you look at some observable and then you're looking at some fluctuation of the observable and then you say, okay, this is a rare event, this is a fluctuation for the observable. What are the trajectories that create that fluctuation? So in all the possible trajectory of the system, I can identify here, I'm just coloring them, I can identify the trajectories that creates that fluctuation. The problem physically is, can I describe those trajectories so that subset of trajectory as a new physical process, ideally as a Markov process. So mathematically then what I have is initially I have a Markov process and here I'm conditioning the Markov process on the observation of a fluctuation. So what I'm going to do is I take the process and I conditioned the process really like in a Bayesian way on observing some fluctuation of the observable and the fluctuation will be the event AT equals A. So I'm just conditioning everything on seeing this. Now physically this is the same, the conditioning is just the same as selecting trajectories, I'm just selecting a subset of all possible trajectory and that's the conditioning. So we worked to this for quite some time and then what we can show is that the conditioning itself is Markov in the long time limit and it's described by a modification of the original process. So you get a modified Markov process that will describe those red trajectory. The process has to be modified because you're actually describing trajectories that are not the typical trajectories of the process and so you need something that will change an original process to look at those trajectories. So this process is very useful because it really gives a physical understanding of how fluctuations arise in time and then I'll give you some examples. So for diffusion actually we know what that fluctuation process is and it comes from the Eigen function. So the theory for diffusion will say that what we call the fluctuation process or the effective process which is the process describing only those red trajectories is given by another diffusion with the same noise, the same diffusion matrix, but the drift is modified. And the drift is modified explicitly by this formula you take the original drift and then you add this term which depends on the g functions defining the observable and then this gradient of the log of the Eigen function related to the large deviation. And then depending on the fluctuation that you consider the fluctuation a you're going to set the K parameter of the large deviation using this formula. And this is a formula that's very close to the formula that fixes temperature and the derivative of the entropy so inverse temperature is the derivative of the entropy for a fixed energy. So here, if you take the derivative of the rate function for fixed value of the observable, that sets the value of K. And mathematically what we have is an equivalence between the subset of trajectories that have been conditioned so the process condition on seeing some observation in the long time limit is equivalent to this modified diffusion. So the modified this diffusion in the long time limit really describes those trajectory and that's what we call the effective process. The strategy with equilibrium statistical mechanics is that when you condition when you select those trajectory you're really defining a micro canonical ensemble. Micro canonical will be like selecting all the micro states that have a fixed energy and so here we're just selecting all the trajectories that have a fixed value of the observable so that's a micro canonical ensemble of trajectories. So the idea of this effective process is really equivalent to a canonical process where there's no conditioning but you just fix a kind of temperature or if it's inverse temperature to reweight all the trajectories so it has to match the micro canonical ensemble and really mathematically that's what this is about it's really a generalization to trajectories of the notion of micro canonical and canonical ensemble. So this is more of a technical point what what what's to be remembered really is that this modified process is an effective process for understanding how fluctuations are created. And to illustrate this I'll just go back to the example that we studied before with the driven particle the Brown and particle with the laser tweezers so I go back again with the original diffusion that I have it's an manual and big process again with the correct normalizations of the state over damp limit you're going into the cool moving frame so this is the system that's the model. If you look at the linear observable which for us was the work, you can do the calculation. I have the dominant eigenvalue I have the corresponding eigenfunction that R K so I can put the R K in this formula. The G is zero the R K I have the explicit expression so I take the number of the log. And what I realize is that for a given value of a the effect of the on the drift is only to change the, the, it translates the drift so it changes the fixed point of the process, which makes sense because on average, I don't have any work fluctuations so the process goes positive or negative with the same probability so the typical value of the observable is zero. And now I can say well, what kind of trajectory is actually give me a positive value of the observable. In this case it has to be trajectories that stay longer on the positive side. And this is predicted by the effective process because the drift of the effective process is a drift that's been translated on the positive side. On average, the system will stay more positive and so we'll create a positive value of the observable. So the calculation actually tells us this so we have an explanation as to how fluctuations are created. If we take a different observable for instance if we look at the integral of x square, then if we do the calculation which can be done also exactly in this case we see that the drift is not translated what's changes that the friction coefficient of the change as we look at different fluctuations. So again here I'm showing a kind of typical trajectory in blue. It has, on average, zero value of the observable and now I can say our fluctuations created for a high value of this observable now you don't translate you just squeeze the system just a bit less and so it has more fluctuations positive and negative in order to have a bigger variance because this is basically the second moment as an observable. So overall we see so for this simple example that the effective process will explain physically how fluctuations are created by extra effective forces that are not true physical forces they you can view them in a way as entropic forces that come from the fact that you're looking at just a typical trajectories that are not typical in the system. You would have to wait a long time for to see them but if you see such a fluctuations they come from a process that has this drift effectively. Okay, so I think I'm going to stop there I think I've been a bit quick but I just wanted to give you an overview of how we apply large deviations there are many directions of applications and stochastic thermodynamics or in physics. So I mentioned that we can apply large deviation theory for equilibrium system like particle system we can also apply this it's been applied in to study stochastic models of turbulence or temperature anomalies and climate models. And then it's been used a lot to study non equilibrium system and then in that context, large deviation theory provides us with the theory of describing the fluctuations of quantities of observables, but also the study states for those observable it's a theory of fluctuations and study states for non equilibrium system it gives us methods, analytical methods to get information about distributions of observables like work heat and reproduction and so on. This notion of effective process has been studied a lot recently and so if you're interested in this I can give you some references. You can also look at density and current fluctuations. And then what's interesting is that for linear system so if you look at Einstein Olympic processes in any dimension, we have explicit results for this so we have exact results for linear system. And then I've worked on this quite recently. And it's used to look at phase transitions and fluctuations and we call those dynamical phase transitions, and it's used to study fluctuation symmetry so you might know about fluctuation relations or fluctuation theorems, and these will be encoded as some symmetry and the rate function so if you know the rate function then you can extract really fluctuation relations. So here everything is classical. I want to just mention in relation to Massimiliano's talk that you can apply this also to quantum systems so it's not, it's not a theory per se that is classical. My own application or my own driver is on the classical side but there's been some work also on applying this for quantum systems. I want more information. I have the review paper. It's quite old now 2009 and I have some lecture notes that date from 2018. I have other lecture notes or the tutorial information on my website if you're interested. And, and also some some research directions if you're also interested. So I think I'll stop here. Thank you for your attention. Thank you. Thank you very much. We open to discussion so maybe I can start because I don't see questions as of now. So, I understand say that this is a theory for Markov processes. Now, say, but I can imagine it can be applied also to know Markov processes but one thing that I would expect there is that if you have systems with a long range memory, then you would have a phase transition in large deviations and then the Gardner-Ehrlich's theorem does not tell you all the information, right? So what you're going to lose, yeah, for non-Markovian system, you'll lose the connection with the spectral theory. So you're not expecting the scalcumin generating function to be given as the dominant eigenvalue. So this actually, this result really comes from the Markov nature of the process that you're considering. But for non-Markovian, you can still in principle calculate the generating function of the absorbable. Markov assumption in this result. If you can calculate this for Markov and non-Markov and then it has the right properties, then you can extract a large deviation. But as you point out, it might not be an exponential of T. It might be an exponential of some T to some power, but this is also covered by a large deviation theory. So everything I've said is actually with the T there, but the theory is not stated with T. But the Gardner-Ehrlich's theorem would only give you the convex envelope of a large deviation that you may not be able to describe. Yes, that's a point that I haven't mentioned, but it comes if we if we look at the Gardner-Ehrlich's theorem necessarily because the rate function is given by a Legendre transform, it's necessarily convex, but the rate function by definition doesn't have to be convex. So this means that Gardner-Ehrlich's theorem is kind of limited to cases where the rate function is convex. If you expect the rate function to be non-convex, then you have to use other methods to get it. And there are other methods actually to get the rate function and rate functions that are not convex. So yes, this arises when you look at an equilibrium system with long range interacting systems, then the rate function in many cases is not convex, so you have to use other methods. So there's one method called the contraction principle that we use in that case to get the non-convex rate function. Yeah, so but say in another setting, so this convexification of the rate function corresponds to essentially the Maxwell construction, right? So that essentially you are looking at large deviation which are realized as mixture of two different distributions, right? So I was wondering what would this mean in a dynamical system? So I mean, if you know of any examples where this arises. Well, so in the context of stochastic processes, a non-convex rate function will have to come from a non-Markov process. Because Markov then you could imagine that you have also for some time you're doing this fluctuation and then for some of the time you're doing another fluctuation, so you'd have a Maxwell construction. So people actually are quite convinced that although it's not proved, but any rate function for Markov process if it exists is convex. No, that's clear, I mean. Yeah, that's clear I think. So for non-convex, you would have to have some breaking, it cannot be Markov and so you need then long range memory and time maybe two or two. It means then that the fluctuation is really not created as a mixture. But I don't have any example of any example of this. I did study with Rosemary Harris some large deviations for non-Markovian. So in process where you have long range memory and time, but even then the rate function is convex. Okay, see. Okay, any other question? I was very quick actually and I think I see the time now. I think there's one in the chat. I can read the question actually. Can you let us know about the large deviation theory in quantum fluctuation processes? Yes, it's not something I work on myself, but there's lots of nice work by the group of Juan Garan at the University of Nothing I'm looking at quantum processes. So you can describe some quantum processes using Limblad equation or using also stochastic differential equations and you can define also observables in that case. So for instance, like a simple example is a two level system that's driven by some laser and so you can look at the occupation in the excited state or in the non-excited state so you can look at the occupation and time. And so the system will be kind of by stable system and it will have a fluctuating occupation and you can look at the large deviation of that occupation. So this will be an observable that is as the type of just an integral of some function of the state. So you can look into this. It's not actually covered in the two references I have there, although I think I give some references in the lecture notes about quantum systems so you can have a look there. Otherwise you can send me an email and then I can give you some references in that direction. Okay, so we don't seem to have further questions. Okay, so then then I think maybe we can take a break until coffee break until 6pm Central European time. And I reconvene in, yes, 25 minutes, 35 minutes. Is this okay? I see David is also there. Hi David. Yeah, sounds good. Sure, because I'm still half asleep. More coffee. I can agree to that more coffee.