 So, hi everyone, thank you for coming. So, I'm Marco, and today I'm happy to introduce you, Professor Guido Boffetta from Italy. Guido is a full professor at the University of Turin, in Italy, and he is a professor in theoretical physics. He works in fluid dynamics, mostly in turbulence, mixing and turbulent convection, and he is a fellow of the European Mechanics Society. in in chassis editor of physical review fluids, publish mySAP and today, you will speak about the time irreversibility of fully developed tool brains. Navy Tkvorjec, ok, thank you very much, Marko and thank you for coming and thanks to the TSP program for the opportunity to visit oyster, is very nice and enjoy. So today, we speak about time irreversibility in se načo se najbolj do Medicina. Zelo sem, da je militonik. Naузaj sem ustačnil na coolest. Najbolj tega lahko pozvodim. Poga je kot prihoto, da je opponentom nekaj počkod. Poblokaj, da tega nekaj počkod je prav saj vysvah. Kako se lahko izrobijo razližovati, vsak, da vsečačnja vsečačnja, da je vsečačnja zrednje, na vsečačnji razdovanje, vsečačnja se, da, da se prijemo, ne ležimo v sečti zrednji, da se prijemo, tudi ne ležimo, nači prijemo je ne vsečnja. Ok? Na stacionalj stacionalj, ne na stacionalj stacionalj, je niste prvne, da je s vsem vidno. Pozoraj, kar je ne zrednja, s tega, Tukaj z tog je komedične komedične od 80. To je veliko stupične, ale veliko naša možda. Čekaj, da se pošliči. Tukaj je tukaj se zelo v bookstoreu. Zelo se zelo, da je neko strančne. Tukaj je tukaj zelo, da je neko komputografične. Tukaj je tukaj realno. Tukaj je tukaj zelo. Tukaj je tukaj zelo. Tukaj je tukaj zelo. Tukaj je tukaj zelo. in je začel, da je zelo vsočen, da je vsočen. In potem je, ok, je bilo vse, in potem v svoj glasba, da je vsočen. Kaj je vsočen? Zelo se na doga, da je doga. Zato je, da je vsočen, da je vsočen, da je vsočen. Odej, pa je vsočen. In potem, da je vsočen, da je vsočen. More seriously, we are considering turbulence, and turbulence is described by the Navier-Stokes equation. In the limit of zero viscosity, so this is the kinematic viscosity, which is typically a small number in the dimensional number, is a small quantity dimensional number, but in the limit of zero viscosity we have the Euler equation, which is described in the motion of the medial fluid, and the Euler equation is invariant for time reversal. So if t goes in minus t, u in minus u, you have the all the term here doesn't change sign, so you have exactly the same equation. So it's invariant for time reversal, because p essentially is proportional to u squared. And also the Euler equation conserved the energy, the kinetic total kinetic energy, which is nothing but the average of the velocity square is preserved by this equation. Okay, now if you consider a real fluid, which is described by the Navier-Stokes equation, you have an additional term, which is a dissipative term, which is described by the viscosity that pleasure of q, and this breaks time reversibility, because if you change, if you make time reversal symmetry, this quantity change sign, this guy change sign, so you break time reversibility. And also the energy is not conserved anymore, because viscosity is dissipative, and you have the kinetic energy is dissipated at the rate which is proportional to nu times zeta zeta is what is called the anestrophe, but the average of the velocity square, remember the velocity is the carl of the velocity, in any case, energy is dissipated, and the rate of energy dissipation is called typically epsilon is the key quantity in fully developed turbulence. So there is a fundamental law empirical law in turbulence, which is called the essentially the dissipative dissipative anomaly, and this state that the limit of Navier-Stokes equation for vanishing viscosity is singular in the sense that if you let the viscosity going to zero, you do not recover the Euler equation in the sense that the energy is not conserved. Okay, so the limit of the dissipative term for nu going to zero is equal to the energy dissipation rate, which is a constant. This is an empirical law observed in experimental simulation. These are a summary of several simulation, numerical simulation, which plot the energy dissipation rate may dimensionless is not important as a function of the Reynolds number. So the Reynolds number is essentially the inverse of viscosity, so viscosity going to zero is equivalent to a larger Reynolds number. And you see that it starts to decrease, but then you reach a constant plateau. So it means there is a singular, what mathematician call a singular limit in the sense that the limit in nu going to zero is not equivalent to nu equal to zero. There is dissipative anomaly, and the physical reason is very simple at the end. The point is that as the viscosity becomes smaller, the velocity develops stronger gradients, and this quantity here essentially compensates vanishing. Yes, let me finish the vanishing viscosity, and so you reach a finite constant limit, yes. I agree though, this is Mahesh. I don't buy that statement of Srinuva. Can you hear me now? We don't hear you. That's odd. Can you hear me? Can you hear me? Maybe we can. OK, sure. No problem. OK, maybe later. So, I was saying that the reason for this anomaly is that velocity field develops stronger gradients that compensate the vanishing viscosity, so the limit is finite. So, the basis of this empirical fact, there is the chronograph developed theory of turbulence in the 40s. There is a very brief history of the theory of turbulence. It starts with G.I. Taylor, that was the first probably to realize that turbulence has to be considered as a random field. And that, so we need the statistical description, but single point statistics is not sufficient. So, we need to take into account a least correlator between two points velocity correlator for structural fashion, we will see. OK, and then the important contribution was given by Kolmogorov ten years later, or less, where he developed the similarity theory of turbulence, and he derived the fourth inflow that we described in the following for the turbulent flow, and has a fundamental prediction for the energy spectrum, which has this fighter slope, which is what is called now the Kolmogorov spectrum. So, the idea of Kolmogorov is that the Navier-Stokes equation is not only invariant for traversal, but also is invariant for scaling transformation. So, it means that if you rescale the space by a factor lambda, lambda is a positive number, so you rescale the space. And you rescale also the velocity by another factor, which can be, is another number, we call lambda to the h, by another number. And so you have to rescale also time, consistently, so time is rescale as lambda to the one minus h, because it is a scale over velocity. OK, and you put the scaling transformation in the equation, the viscous term fix the value h equal one minus one, sorry, and you get exactly the same equation. And this, when known, is called the similarity transformation, and people use the similarity transformation to make experiment in scale of turbulent flow. For example, you want to study the flow behind a car, you can put a model of a car scale by a factor lambda, the velocity rescale, and on your model in the lab, you have exactly the same flow that you have in reality, because the equation is the same, it's just rescale. So this is the classical idea, it's not the idea of Kolmogorov, it is one. The idea of Kolmogorov is that when viscosity becomes small, so at higher or not number, in some sense we can expect the scaling transformation is not fixed anymore, because this guy doesn't play any role. And so the idea of Kolmogorov is that we have a scaling variance in the equation valid for any scaling exponent, so the scaling exponent now is three. And we have to fix the value of h in some way, so how to fix h. And the way to fix h was derived by Kolmogorov with the 40th law. And the 40th law, this is a little bit technical, if you know, you already know what I'm saying, if you don't know, don't worry, I mean, just the result, the final result is important. The 40th law is very important, because it's probably one of the few, for sure, is one of the few exact and untrivial result that you can derive from the Navier-Stokes equation. And starting from the equation of motion, now there is an additional term, which is forcing, which is needed in order to have a stationary condition. You write two point correlators for the equation you take the average, and you introduce the observable, which are very important, the velocity increments delta l of u represent the velocity differences between two points at a distance l. So in stationary condition, using this auto-remogeneity compressibility, you end with this expression, okay, where sp represent the moment of order p, the average of the velocity difference to the power p, which are called the structural function of p. And so here, three different, sorry, three velocity increments, so it's three, s3, this is s2, with a viscosity is s2, and this is called for the forcing term, okay. I rewrite this equation in this way, and we are stationary conditions for the time derivative disappear. Gido, sorry, can I ask the question from the chat? Sure, sure. So this is from Mahesh Bandi. Okay, hi Mahesh. I don't buy the Falcovich and Srinivazan statement. I've stared at turbulence videos for hours over years, and I can't tell forward from backward. The statement of time irreversibility can only hold on average, isn't it? Time reversal symmetry is almost, but never completely restored. We have strong backscatter, so only an average could possibly make sense. Then where is my understanding wrong? Okay, maybe, let me go on with the discussion of irreversibility, and maybe we try to answer the question. Okay, sure. Okay, thank you. Okay, now the idea of Kolmogorov is that if you have a scale separation between the scale at which the forcing is pumping energy in the system, so this term, this is the scale, which is the scale at which viscosity removes energy from the system. You have an intermediate range of scale, which is called an initial range, in which the first term you can show saturate to a constant, which is the energy input, which in stationary condition is equal to the energy dissipation, so I use the same epsilon for the two. And in initial range, so this term converges to epsilon, to a constant. This term is negligible, because a large scale is negligible, a scale larger than the dissipation scale is negligible, and so this equation becomes simply what is called the fourth flow. That the third order structure of fashion is equal to a constant, which is minus 4 over 5 epsilon, which is the key quantity, the flux of energy times l. Okay, so it means that there is a flux of energy from large scale to small scale, and this is again a signature, this is already a signature of irreversibility, because of this sign means that you have a flux from large to small scale. For example, in two dimensional turbulence, where the flux is from small scale to large scale, the fourth flow is similar, you have a different coefficient, but the sign is plus, because the flux is reverse. Okay, so already the fourth flow indicates the breakdown of irreversibility in initial range. The fourth flow is a verified in simulation, in experiment, is experimental data pretty old by the tabeling group. So what I plot here is the third order structure of function compensated with epsilon l, so the minus, so the prediction is 0.8, 4 over 5, and you see that a large Reynolds number, you have an intermediate plateau of scale, which corresponds to the inertia range in which you reach this plateau. Okay, so this is a verification of the fourth flow. So now we have a way to fix the scaling exponent. Remember, we have a global scaling variance with an exponent h, which has to be determined. And now, thanks to the fourth flow, we can fix the exponent, because if the third order structure of function has a scaling exponent l, so to the one, it means that the scaling exponent is one third. So the velocity increments scale as l to the power one third. It is a similar tier of commodor that predicts the structure of function of order p as a scaling exponent p over 3. The dimensionless coefficient we don't know apart the case for c3. And by Fourier transform, you can predict the energy spectrum as a scaling exponent five thirds, which is the commodor's spectrum. And these are some examples of the commodor's spectrum observed in laboratory experiments with two different fluids. It's a gas, it's a water, different scale. This is a stratospheric wing, so a completely different flow. But in all the case, you see an intermediate scale, you see the commodor's spectrum. Okay, just because you have the same equation behind this phenomenon. Okay, so there is a universality in this sense. Okay, of course, this is not the end of the story. Otherwise, I won't give this thought, because if you look more carefully, data already commodor did it. You can observe there is a breakdown of a similarity. The simplest way to look at that probably is looking at the probability distribution of the velocity increments. So what I plot here are experimental data of the PDF, so probability density function of velocity increments between two points, a different separation. And they are shifted just for clarity. Okay, so the lowest one is for large separation. And this is very close to Gaussian. To Gaussian means that the two points are completely uncorrelated as well by Gaussian equation. But if you go to smaller scale, you see that the PDF change the shape, the shape of the PDF change. And so it means that you have, you don't have a similarity in your flow. And so it means that the single scaling exponent cannot describe the statistics, the statistical profit of two points. And indeed, if you compute, remember the prediction by Kolmogov, is that the p-order stretch of function is going to be over three. But if you compute the stretch of functions, this is again experimental data in the lab. So this is a stretch of function of order two. So sorry, his p is the same. Order two, four, and six. And the blue line represent the Kolmogov prediction. You see that order two borrow less words, but of course there are strong deviations for order four and six. And so if you fit this line with the power law, this stretch of function with the power law, you get the set of scaling exponent, which is represented here. These are the scaling exponents for the velocity increments as a function of p. The dashed line is the Kolmogov prediction, and the red point are the experimental data. OK, so there are strong deviations from the Kolmogov prediction due to the breakdown of similarity. And so far there are no theories that can predict the scaling exponents. If you want, this is the problem of turbulence, fully developed. The biggest problem probably is to determine this exponent starting from the Navier-Stokes equation. As Kolmogov did for zeta three, which is equal to one, but for the other there is no way to derive them from the Navier-Stokes. But there are some models that can describe the scaling exponent. Probably the most famous model is the multi-fractal model, which was proposed by Giorgio Parisi in Uriel Frisch many years ago. And the idea is to replace the global scaling invariance of Kolmogov with a spawn in one third with a local scaling invariance, in which you have a set of exponents, and each exponent is realized in a different set in the space with a given probability. The probability is a scaling function, which for historical reason is expressed in terms of the Fratta dimension. The idea of age represents the Fratta dimension of the set in which you observe the given scaling exponent. And then at this point you can compute the average given the probability, but the age you don't know if you need a model for the age. But given this probability, you can compute the structure function, which are the moment for the velocity increments, because the velocity increments are scaling exponents h. The moment p is p h, with the probability you add with this expression, which in the limit of small r you can estimate with a steepened descent. And so you have an expression for the scaling exponent zeta p in term of the genre's form of age. Of course, this is not a theory, because we don't know what is d of h. But what is very important, what has been very important for the multifrata model in the past is that it is a consistent method to take into account the intermediary to compute different quantities. For example, you can compute d of h by inverting the Legendre's form, so you measure in the lab experiment, you measure the zeta p. You invert this expression, you get d of h, and now that you know d of h, you can make prediction for other quantity. And this is how the multifrata model is used to make prediction. So one of such predictions, which would be useful for the second part, is about the statistics of the acceleration. So now we are considering, we are switching from the Eulerian global flow to local Lagrangian trace. So we are considering now a single particle, particle, which is transported by the turbulent flow, which is a very simple model, for example, for dispersion in the atmosphere of pollutants, or a dispersion of phytoplank on sail in the ocean. It is transported by the velocity. And it is well known that the particle transported by velocity, which are called Lagrangian particle, they display extreme acceleration, which is extremely intermittent. What does it mean? It means that if you compute the acceleration along the trajectory of the particle, and you plot the PDF, these are experimental data from the laboratory of Bode Schatz many years ago, you see that this is the distribution for different Reynolds number, and it is very far from a Gaussian. The Gaussian is this dashed line. This is compensated with the RMS value. So this would be the Gaussian, and you know that the Gaussian at the sigma, and zero probability to observe something larger than the sigma, while in this case you can observe with a finite probability, fluctuation which are 20 or 30 sigma, so extreme fluctuation in acceleration. And in numerical simulation that we did also some many years ago now, also in the simulation we observe for the acceleration the same similar shape, so we are able to observe acceleration without 60 or even 70 times, 70 sigma, the average acceleration. And it is possible to predict the shape of this acceleration in terms of the multifractal model, again the details are not important, but essentially you express the acceleration as a time derivative of the velocity, so it is the velocity increment of the column of the scale, active scaling turbulence divided by column of time, and you rewrite in terms of the Reynolds number of the flow, so this depends, the shape depends on the Reynolds number is not universal, and then usually the multifractal machinery you end with an expression for the PDF, which is very complicated, but the key point here is that there are no free parameter, the only parameter is d of h, but d of h you know from the experimental data on the structure of fashion. You can put the d of h, you can integrate this numerically for example, and you get this blue line that you probably can see here, which is not a fit, but it is the prediction of the multifractal model that works very, very well up to 60, 70 sigma. So the multifractal model is used as a model to derive to describe a new statistic starting from the velocity structure of fashion. But I would also like to understand what is the origin of this stream statistic that we are serving in Lagrangian trajectories. So this is an old movie from a simulation, so the different point represent different particle transported by turbulent flow, and so you see that the, okay, it's horrible, I'm sorry for that, but they generate the trajectories, which is a kind of spaghetti diagram, but all this trajectory contains all the statistics where we see in this red line, the statistics of the acceleration. Now I want to detect what contributes at the level of particle to the extreme events, so I select among all the trajectory the one that contributes to the extreme tails in the PDF. You end with very few trajectories, but you can see probably that all the trajectory has a common feature that at certain point the particle is trapped very clear here, or here is trapped in a small scale vortices where it starts to rotate very fast. Okay, the rotation time is the order of the curve of the time, and they remain trapped for a long time because both the vortex and the particle are transported by the flow, and this fast rotation gives the stream acceleration that the particle. Okay, so this is the physical origin of this stream acceleration. So now the point that I want to discuss in the following about irreversibility is how we can detect irreversibility along a single trajectory. Okay, now consider that you have a trajectory of the Rangian particle, you can run forward in time or backward, I don't know which is which, forward or backward in time, and we want to detect if it's possible to have a signature of irreversibility at the level of a single particle trajectory. Okay, so this is the second part of my thought. It is possible to detect irreversibility at the level of two particles, this is somehow trivial, because if you consider two particles at a separation r with two different velocities, from the fourth flow you can derive, this is the Rangian equivalent of the fourth flow, you can derive this expression, but the velocity difference square is equal to minus fourth epsilon. Okay, this is just the Rangian version of the fourth flow, and so it has an important consequence, which is very interesting, that now if you consider how the particles separating time starting from the initial separation r for short time, you can make a Taylor expansion essentially, so r square of t is the initial separation, then the term of order t for symmetry disappears, so the first term you have is t square, which is trivial, it is just ballistic, and then at the order t cube you have this term, which is not in that, but this one, if you take the time derivative, you have an acceleration times the velocity, so delta v, delta a. And so you have that for short time, the separation between two particles has a constant term, a quadratic term, and a cubic term, which is minus two epsilon t cube. And so this means that you have a signature of the symmetry because t in minus t is this number, this term change sign, for example, if you compute the separation backward in times, minus the separation forward in times, so this two term cancel because they are even in t, and so you have at the end a positive number, four epsilon t cube, and so this means that there is an asymmetrical relative dispersion between two particles forward and backward in time. In particular, backward in time dispersion is faster than forward in time. So if you have two particles in the turbulent flow, the time they separate, how they separate backward in time is faster than when they separate forward in time. So this is a signature of reversibility, but at the level of two particles. As I said before, we want to see if it is possible to detect the reversibility at the level of a single particle. Okay, so I go back to the first statement of Falkoviće Srinivasa. So here are a movie of turbulence. It's a numerical simulation. Each particle is a Lagrangian particle, and the camera is placed at one of these Lagrangian particles in movement with the flow, with the flow. And as you can imagine, one of the movie is forward in time and the other is backward in time. So we should be able to detect which is which, but of course it's difficult to do it. Okay, you cannot. I don't even know which is which. I don't remember. Okay. So we need some statistical analysis for this. So the simple idea is to look at the velocity difference along the trajectory, but this by definition cannot detect time reversibility because if you look at the velocity along the particle, and compute vt minus v0, for time reversal t goes in minus t, but v goes in minus t, v, so you get exactly the same. So this is, by definition, is invariant. But this is not true, for example, if you consider, for example, the v square, because v square doesn't change sign under time reversal. And so the energy increment, so the energy is velocity square, now you remember, the energy increment in principle are not time reversibility. And so is a natural candidate to detect irreversibility along the trajectory. And indeed, if you compute the energy increments along a single Lagrangian trajectory, you see that the statistics is not symmetric, so this is the probability density function of the energy increments for different lag tau. Okay. You don't see the symmetry from this plot. Okay, they're almost symmetric, but there is a little symmetry because the average is zero because they are stationary conditions. So the average is zero, so you can compute, for example, the third moment, w to the power three, and this is not zero. It's negative, okay, and increase, a short time increase, but this is for some simple reason, but for intermediate time, which correspond to the natural range in the physical space, you have a plateau, and the plateau changes with the Reynolds standard. So, this is the origin of the energy along the particle. And what is the origin of this asymmetry? Again, it's very simple to understand once you see that, I mean, the beginning was not so easy, but for example, let me consider this is a particle, a trajectory of a particle, this is a real particle in the experiment in Cornell, and this is the history of the energy along this particle, or it is the important quantity. We have several times in the data set, this typical pattern, in which the energy in the particle start to grow, it takes some time to grow, okay, about here is about four, a little four kilometers of time, and then increase much faster. And this break the symmetry between t and minus t. The particle takes more time to accelerate than to decelerate, what we call the flight crash event, okay? And the flight crash event is the signal to irreversibility at the level of the energy particle, and I want to mention also that there are several other systems, like for example in traffic flow, but also in stock market, for example, people know that in stock market you lose money much faster than the time it takes you to gain money. There is this typical estimate for completely different reason that is kind of universal low. Okay, so in order to have more look more in detail, let me now consider the limit of very small time differences, so I take the derivative of the energy, which is essentially the power, derivative of energy, so whether the statistic of the power along the particle is not symmetric, so this is the pdf, the probability density function of the power of the particle, and what I plot with the continuous line is the positive side of the pdf and the negative is plotted as a dashed line, you see they are very similar but they are not the same. The negative is larger than the positive, so it means that the third moment is negative, and indeed this is the third moment as it goes with the Reynolds number, so we have that the probability, sorry, the statistics of the power is skewed and the skewness goes with the Reynolds number. And we have this third moment goes more or less like Reynolds to the power 2 and the second moment like Reynolds to the power 4 third, but this number is not a simple dimension and dimensional number, so if they are exactly the same, this skewness spoon and the skewness is constant, okay, it's not the skewness, but the third moment is the skewness. So these numbers are very simple, but they are just a fit of the data, and they are not dimensionless, dimensional number, so it can make a very simple dimensional prediction of a la Kolmogorov and the idea is that the power is the velocity term in acceleration, the acceleration again is the velocity at the Kolmogorov velocity over the Kolmogorov time, and so it's proportional to a Reynolds to one alpha, and so the Kolmogorov prediction will be the third moment as the skewness spoon in 3 over 2 and the second moment of Reynolds to the power 1, and there are the dotted line which do not fit the data, okay. So we can do a field exponent, this is exponent 2 and 4-3, because they are due to the anomalous scaling due to the flight crash event which are extremely intermittent event. So we can try to do, is it possible to predict this exponent using a multi-fractal model? So to do that, we first need to translate the multi-fractal model in the grand-jum framework, it is just been done many years ago in Borgas. I don't want to enter into the details but the idea is that the velocity increment within a single particle at time t is as the same scaling exponent of the velocity increment at the distance r which correspond to add it to another time tau. Okay, and so starting from that, you can make a prediction for the statistics of the power which at the end as you can predict the scaling exponent alpha as a function of the Reynolds number and you have this expression which is a little bit complicated but again, there is no free parameter here because the off-age is given from the linear statistics, okay? And this is remarkable because remember that the multi-fractal model is a kind of generalization of homogorodimensional argument is a refined dimensional argument which has no information about a symmetry or sign in the statistics. So it's not clear that we observe the data. But it did, it works okay, because these are different data these are more recent data that we obtain. So we have simulation of a different Reynolds number and so this is the power this kid sorry, this is the power to the power 3 the third power of the power and the point, the blue and the yellow point and the prediction of the multi-fractal model is 2, is very close to 2 remember that fit was 2 the prediction is 2.1 but the prediction is perfectly compatible with the error bar in the data and for the second moment the prediction is also a little bit smaller than the fit it's not 4.3 but it's 1.2 more or less 1.17 but it's still compatible with the data okay so this is okay, I think I can skip this one because a little bit doesn't add anything but I want to discuss a little bit what are the main contribution to the power in order to do the symmetry of the power and understand the relation between the viscosity so the reversibility of the Navier-Stokes equation and the reversibility of the grand genre. So the power is essentially the velocity time the acceleration and the acceleration you can use the Navier-Stokes equation you can write the acceleration as 3 terms is the pressure gradient term the viscous term and the forcing term okay, so numerical simulation you have all the statistics so you can disentangle from the statistic of the power you can extract the contribution of the 3 different terms and this is the result so the blue line represents the probability density fashion of the power which should be already seen and the green line which is exactly almost exactly the same point is the contribution for the first term the pressure gradient term the black line is the contribution for the viscous term which is strongly skewed because dissipation mostly reduce the power but sometimes it gives some power to the particle locally, in average no but locally can do and finally the red line is the contribution from the forcing so from this value you see most of the contribution to the power comes from the pressure gradient term but what is remarkable is that the pressure gradient term is not skewed, it's almost in matrix or even if the skewness, the recent skewness is the opposite way, it's positive so it's dominant in the statistic of the power but it's not dominant in the statistic of the other moment, in the third moment what is important is that we have found dominant term in the third moment of the power, so the skewness of the power is a complicated term it's a cross correlation between the pressure gradient and the viscous or the forcing term but the point now is that there is not a simple link between the viscosity and the irreversibility so the viscous term which is skewed, is not dominant in the statistic of power so it cannot be responsible that we observe the power so at this point we can ask another question Is real necessary to have a dissipative time irreversible system to generate irreversible trajectories? So we have seen at the beginning that the viscosity is what breaks the time reversal in the Navier-Stokes equation and we have seen that the grand general trajectory are time irreversible the power in the grand general trajectory are time irreversible so one can ask is the Lagrangian reversibility related to the symmetry breaker in Navier-Stokes equation or to the dynamics or the dissipative dynamics of the Navier-Stokes equation so another question is possible to have irreversibility in the grand general sense starting with a time reversible dynamics a very common example of state is thermodynamics thermodynamics if you consider mechanics you have a perfect microscopy dynamics which is time reversible but the microscopic quantity are time irreversible so the idea is to use a time reversible Navier-Stokes equation there are some proposal to change the Navier-Stokes equation what you have to do essentially is to change the viscous term this is not for the Navier-Stokes equation this is for the simple dynamical model which is called the Schell model but as the same quantity for this argument is very similar to the Navier-Stokes equation so the idea is to replace the viscous term which breaks the time symmetry with a different term which is symmetric in a time reversal this can be done the original proposal was done by Gallavotti many years ago you had to use a dynamical viscosity which change with a field and you have to keep some to force the preservation of some quantity so we studied this problem in the Schell model and the result let me skip all this stuff the result is interesting the result is that irreversible Schell model reversible Navier-Stokes if you want you have that the grand gene trajectory the power along the gene trajectory is still time irreversible so there is not a direct link between the reversibility of the equation and the reversibility of the trajectories ok, you can see here this is for the reversible system this is the third order moment of the power they are negative they scale with the Reynolds number as in the normal Navier-Stokes equation but the scaling exponents are different they should lie represent the scaling that we are solving in Navier-Stokes equation here the scaling exponents are different but ok, there is no reason to have a universality we are using different systems ok ok, this is the end this is the end of my talk we have seen that we have seen that the reversibility in the energy fluid patient along the range of trajectory thanks to the flight crash event we have seen the dependency on the Reynolds number and the agreement with the multi-fractal model I did not mention but we also studied this problem in other system for example in two dimensional turbulence you can also study different turbulence systems like MHT and we also studied a little bit preliminary study in Shell model like energy reversibility in reversible Navier-Stokes equation apart of theoretical interest of this problem I want to mention the fact that this can be have some relevance for Lagrangian model so there are several models for example mobile for dispersion of pollutant in the atmosphere on the ocean which are not based on the Navier-Stokes equation because you cannot simulate or the field is too big so you use a kind of stochastic model that describe how the particle moves in your flow and this stochastic model do not take into account so far the irreversibility along the Lagrangian trajectory so using these results can be an improvement in developing Lagrangian model Lagrangian model for dispersion so with that thank you very much for your attention and if you want to have more details about the results you can have a look at this paper ok, arigato thank you very much for the talk and also for the nice review of basic turbulence so let's see if there is any question in the audience let me start from internet in Mahesh I have the follow up from Mahesh Pandivim online so he says thank you Guido, I understand my observation was in line with what you explained later in the talk I wish I had had the patience to wait another question can you offer any thoughts or comments on how this picture changes for 2D turbulence ok, thank you Mahesh so 2D turbulence I don't think I have the slides here we study also 2D turbulence and what is remarkable is that we expect since 2D turbulence I don't know if you know 2D turbulence but the main point is that in 2D turbulence there is a potential turbulence the energy flow from small scale to large scale ok, because of several reasons the conservation of anstrophy is the main reason and so we expect that since here let me go to the ok, this one since the power ok, is proportional to minus epsilon to some power this is proportional to epsilon cube times the constant we expect the 2D to have the opposite asymmetry but this is not the case this is somehow surprising that 2D turbulence we have very similar result at the level of the statistic of the power in spite of the fact that the flux is opposite, from small scale to large scale then the details are different are very different, for example the distribution ok, that we have here in 2D turbulence are different there are different contributions but the main point is that the skillness of the power is negative is envired with the dimensionality which is somehow surprising but I don't know why thank you very much what happens with entropy in this time and turbulent developed flow is it to continue to grow what happen in which sense can you apply for entropy calculation I'm not sure I understood the question you mentioned the entropy entropy which sense the entropy the microscopic ok, this is a microscopic system so I have no access to a microscopic entropy here you can define some kind of entropy in the turbulent flow but also I don't know very well about this sorry for this but microscopic entropy yes, entropy is similar but it is a little completely different quantity yes what happens to this Lagrangian irreversibility in like Euler flows if you have a turbulent Euler flow in Euler flow haha, this is a very good question I don't know so in Euler flow you know that Euler I think you have said the problem is that the Euler the solution of Euler which typically develop singularity but the fact that they can develop singularity for example makes that the Euler equation which formally conserve the energy can also dissipate energy in a weak state you have weak solution of the Euler equation which dissipate energy in a finite so I don't have no idea what happen for Lagrangian the problem is that doing simulation to compute Lagrangian tradition you need to do simulation you cannot do anything else and doing simulation with the Euler equation for sufficiently long time in order to see dissipation I don't know the statistics of the Lagrangian tracer I don't know if you can do that but in fact they use the same the same feature because the dynamics Euler equation there are some work on Euler equation in Navier stocks equation probably you know from the French group and they observe a cascade like in Navier stocks equation even if they do not have any viscosity the flow start to generate turbulence and you have a direct cascade and then at a certain point the energy start to accumulate a very small scale and this is the signature of the formation of a singularity then the singularity is regularized by the fact that you have a finite resolution but in principle you get a singularity but in the intermediate time in which you have a flow of energy I would expect that the phenomenology of the Euler service would be the same as in Navier stocks and is it because the pressure term is the major contributor in that? Exactly, this is one of the problems because it does not play in your work Thank you for the talk In your reversible shell model so even though what I understood is even though the model was designed to be reversible it showed a reversibility so how does the forcing there affect the formulation of the forcing can affect that So the shell model is designed so but the point is that it is irreversible because this quantity is u over t so it doesn't change sign u goes minus u and t minus t this is u squared doesn't change sign and the dynamical viscosity is built in a way that this is proportional to u and if I remember this is proportional to u to the power 3 and then it is multiplied by u so this is symmetric also so it doesn't change sign so formally it is irreversible but the point is that the dynamics is very far from an equilibrium because you have the forcing which is a large scale and then you force the system to dissipate a small scale this is a very dedicated technical point and the way you force the system to dissipate a small scale is not by adjusting the viscosity control of the viscosity but you ask to have a finite a given a stop so this quantity is designed in the way that you preserve the a stop in the system in this way you have a scale separation between the forcing scale and the dissipating scale because the a stop is a small scale quantity we first try with the viscosity where it fixes the total energy in the system and this doesn't work because everything is a large scale you don't have an inertia so you have an irreversible irreversible dynamics at the end because of the separation between the injection of the energy and the dissipation of the energy this is what makes the system dissipate ok, thank you ok, so there is one more question from the internet I have another one from the chat from Amal how does the Lagrangian irreversibility differ from the Eulerian irreversibility ah, yes the difference is that in the Euler irreversibility is based on the structural function of the flow of the structural function which is two point correlators so two point statistics in two different points if you look at the distribution of the velocity in a single point everywhere in the field if you get something which is Gaussian perfectly symmetry you don't see any signature irreversibility you need two point quantities why the Lagrangian statistics is irreversibility along with a single trajectory so a single point statistic a different time this is the difference thank you hello, I was surprised by the histogram of the how it sometimes injects energy into the flow I thought the viscous term was always taking energy from the flow in Fourier space this term just becomes k squared times u squared which is always positive no this is not true in Fourier space is the average k squared u squared in order to make this term u squared you have to make integration by part this can be done only on the average and then one of the two double goes on the other side so you have double u squared and this is definitely positive but this term there is no reason why this must be negative or positive indeed it is positive in the plus and why it is negative so locally the dissipation can absorb energy locally but the average of course is negative and you see here pdf thank you is there any other question? so it's thanks again thank you very much