So I'm going to be talking about Feynman rules for wave turbulence. Find minerals of course refers to the method of solving quantum field theory is perturbed notably. And the idea will be that I'll use techniques of quantum field theory to hopefully make progress on classical wave turbulence. So I'll explain all of these things. So this is, this is based on this paper from last month. This is my first time reading a paper that and posting it somewhere other than th so I'm very happy to be giving the stock. So let me start with some motivation. And so when we normally study statistical physics, it's, the entire study is built around the study of the thermal state. And there's good reason for doing this. We start with some initial conditions. We have a closed system, we wait a while. And for most initial conditions, for most systems, after a while. If you ask reasonable questions, the state is indistinguishable from the thermal state. And for that reason we study the thermal state. And we're able to do thermodynamics because we know what state we're talking about. And it doesn't matter if we we don't have to know the initial condition. So that's the thermistor. Also, please feel free to ask questions at anytime to the extent that you're able. Yeah, I'll be your repeater. So just let's pick our planet. So that's the thermostat. On the other hand, there's a large range of concepts in which one needs to study systems that are far from equilibrium. So if you don't wait, if, if you have a closed system and you wait a long time, then of course it becomes a thermal state. But suppose you want to study it before you've reached equilibrium. And there are a number of contexts where this happens. Down here I've shown a picture of a heavy ion collision. So two protons collide. There's some highly non-equilibrium process, and eventually you get a quark-gluon plasma, which behaves like a fluid. But one likes, but like to describe this whole intermediate stage in which the system is very far from equilibrium. Another, so this is fairly recent. This, this is something that's been happening over the past decade. Another contexts that's also fairly recent is called Adam experiments. Where again, one has, the experimentalist have great control over the system. And they can prepare systems that are far from equilibrium and then watch them eventually reach equilibrium. But it takes a very long time. And so one can again studies that are far from equilibrium. So those are two motivations, but it's, it's a very reasonable question to ask how we should describe systems that are far from equilibrium. And this is clearly challenging because the thermal state is now irrelevant. The thermostat is irrelevant, but there's no clear universal state to replace it. So it seems like we have to do this on a case-by-case basis. So you're given some initial conditions, you then solve what happens. But then if you change the initial conditions, whatever you see might be very different. Because you don't have this argument that you approach the thermal state at late times while you do, but you're not interested in late types. Good. So a key point I think, is to focus on a subsystem. So instead of trying to describe the whole system, let's focus on some range of modes. We think we're not sure we have some non-linear field. And there are some, we go to momentum space and there's some modes. And we pick some range between some small wavelength and some larger wavelength. Can we just tried to describe what's happening within that range with boats? And because we have a non-linear system, modes of different wave number are coupled together because the equations of motion are non-linear. So there's going to be a flux, some kind of complicated flux of modes entering our subsystem and exiting our subsystem. And as an approximation, we can replace our subsystem with just a simple open system in which there's this perpetual flux of modes passing through, which is maintained by some external forcing and dissipation. Perhaps at the edges of, of the range we're interested in, which sends modes in and gets them fleet. Um, so this, this was my motivation for studying open systems, of course. Systems with forcing and dissipation or by themselves physically relevant. So one, my motivation is not really necessary. But in some communities, most of the interest is just inserting closed systems. And in that context. I would like to argue that forcing and dissipation irrelevant as well as an approximation to your system at early times when you have a far from equilibrium state in the turbulence literature. So there, there are two things that are discussed. There's what's called freely decaying turbulence and force turbulence. So enforced turbulence, you maintain a constant forcing and there's dissipation to have a steady state of flux passing through and freely decaying turbulence. You just start with initial conditions in which, for instance, you have some low wavelength modes excited. And then as you evolve in time, there will be off. There will be a cascade two. So you start with high wavelength modes excited, and then as you evolve in time, there's a cascade to lower wavelength modes. And for some intermediate range of times it occurs, the system looks sort of similar to as if you had foreseeing and dissipation. So there's just a flux of moles passing through the turbulence literature. It's, it's taken us somewhat evident that in the relevant regime of validity, freely decaying triplets enforced turbulence or the same thing. In terms of equations. In quantum field theory literature. It appears that this phenomenon was called, it was discovered in the early 2000s and called pre thermalization. This effect that if you start with some far from, spend somewhat special or far from equilibrium initial conditions within a string class, there's a range of times in which you have what appears to be a turbulent cascade. So that's the thermalization part. And the relevant point here is that one has a degree of universality in which it doesn't matter which initial conditions precisely you have. Good. So I think I've motivated why we want to study open systems. And once the system is open, there's no longer a requirement that we have thermalization at late times. If we had a closed system that we know that at late times we have to get the thermal state. But if the system is open, there is no such requirement. And so we can now have rich lake time behavior. Blah, blah. Yeah. We, we kind, in this third, fourth turbulence, we like to go to the thing we call steady turbulent switches. Some time in which transients die out. That sounds like you're late time. We have to wait for transient to die out to get me to this debit balance between power and power. And that's right, that's right. So if you, if you have a closed system and you just start with some far from equilibrium initial conditions, then the turbulent regime is after you've omitted for a while for transient to die out but not so late that you've reached the stage where you're close to thermalize and so at some intermediate time. Okay. I think that's the same thing you're saying. So the goal is going to be to look for to take an open system and to look for some lake time state, which, which is not the thermal state. And the next simplest possible late time state will be a stationary states, the one that's time equilibrium, but just not a thermal state. And whatever the state is, we can control it through the microscopic parameters, the Hamiltonian, the dispersion relation, and the precise form of the interaction, as well as the external forcing and dissipation. So we take some non-linear system, we are forcing, we add dissipation, and we're looking for what kind of late time state will get stationary time state, which will not be the thermal state. Because we've violated the rules of having a closed system, because we have external forcing and dissipation. And this is essentially waive turbulence. So the thing that was shown in the sixties is that even if we have a weakly interacting nonlinear system, There are cases in which one can find such a stationary non-equilibrium state. And this is the Kolmogorov-Smirnov prostate. And this is the thing that's referred to wake turbulence or weak turbulence. And over the past 50 years in which we have tablets has been known, there's an increasingly large range of contexts in which, which it's been shown to occur. I've shown two here which are recent. So the first is gravity belief turbulence. So I took this from falcons website where I guess he has a lab in which they study waves on the surface. So if the amplitude of the waves are smaller than this, so the non-linearity has to be small. So if the amplitude is smaller than this is described by weakly non-linear equation and one has wave turbulence. Another example is elastic plate turbulence. So they have an elastic plate. And the field we're interested in is the height of the plate and the plate is perturbed and then it oscillates. So this is the height function. This two exhibits wave turbulence. So meaning the height function, one looks at it in Fourier space and there's a cascade. There's a cascade going from high Kate to locate this common chords and prostate which is not a turbulent state. So I want to note that the turbulence was originally and is more commonly discussed in the context of hydrodynamics of Navier-Stokes equation. That is not wake turbulence in this triplets, in hydrodynamics triplets, there's no recoupling expansion. And understanding hydrodynamic turbulences. And the associated non-perturbative phenomenon is very difficult. That's not what I'm going to be trying to do. I'm just studying wave turbulence in which the non-linearity is small. I guess one could ask what the definition of turbulence is. And the definition that I have in mind, which I think I took from a paper by Falcon, which is that it's a state of a non-linear system in which energy is distributed over many degrees of freedom in a fashion which strongly deviates from a equilibrium exhibiting chaos in both space and time. And this is satisfied by wave templates. Any questions, comments? Okay. So to be somewhat concrete as to what I have for non-linear filtering, we have some field, which is, for instance, the height of, of the elastic plate which is being disturbed. We can call that field phi. And it's, it's convenient to switch from Fourier modes phi K2 complex variables, AK, AK dagger, which is just its complex conjugate get. As shown here. This is reminiscent of quantum mechanics of aka dagger being creation and deletion operators. But we're just doing classical mechanics. It's just a canonical transformation which, which is commonly done. And so a Hamiltonian that we can study with quarter contraction is the free term omega p, a dagger a, and some quartic interaction term with a coupling lambda, which depends on the four moment. So lambda is some function of these four momentous of this is very general. And I'm going to, if we have momentum conservation, then lambda is only non-zero when P1 plus P2 equals P3 Plus before. But I don't want to write an explicit momentum delta function so we can just keep that mind or so. So this is the kind of non-linear system that will, will have in mind. It's just a theory with a quarter contract. So I'm curious, if you have a non-linear term and symmetry arrive at quartic interaction. I understand this is some other motivation for doing this in this context. One connected Cuba contraction too. So, so the, the logic that's given is, so the logic is that's given is you write all, you write all possible terms, cubic, quartic, no quintic, sex thick and so on. Since we're doing week turbulence, the higher-order the the interaction, the less relevant it is. So for working to a few orders, we don't need to go beyond quarter four. Maybe we should keep the other ones. Okay. So SHA-2 week turbulence that shows we can stop cortical and there'll be some parameters in this horrible like viscosity or something. There is no viscosity. So you'll expand the parameters, right? Am I right now I'm going to call Lambda. Remember? Yeah, right now we jump. Right now we just have a closed system in which there is a field with some dispersion relation. You omega p, the energy of omega p is the energy of mod p. And there's some quartic interaction, which is described by this function lambda, which tells you the quartic interaction going to be Loco. Now it doesn't have to be. So this, this lambda is just at this point, any function of the momenta. No momentum conservation and momentum conservation. But if there is, okay. So in fact, in the calculations I'll do it will be very general. So this lamp that can be anything you'd like to actually have Kolmogorov-Smirnov cascade. One is omega and lambda have to live within a certain class of functions. They have to be homogeneous functions in the mathematical sense. So omega needs to be some power of p. And lambda needs to be a homogeneous function, which means that it has some definite scaling if you rescale all the momenta by some constant alpha. So an example of lambda might be the product of the P is raised to some power, the product of the magnitude of the piece. And then you can multiply that by any function of the ratios of the momenta or their ankles. That was the case in which Zafar off was able to show that he can get Kolmogorov cascade. But for the purposes of our calculations, we just want to take this Hamiltonian and compute correlation functions perturbed briefly lambda. I'll, I'll get to this. Correct. So, so this is our nonlinear system with this nonlinear interaction. So the main player in, in wave turbulence is the kinetic equation. I guess it's called the wave kinetic equation. So it governs the occupation number of mode, of mode case and k. So it tells you how occupied bookcase. Since this is classical physics and quantum mechanics and K doesn't have to be an integer. And this is the equation for n k. So n k is a function of time. And so this is, this is again a coupling. And there are sort of two terms here. There's, as you might expect, there is an increase in, so there's aquatic interactions, so you might have an increase and then k If two modes scatter, two modes, P3 and P4 scatter and produce a mode K and a mode E2. And you might have a decrease in the occupation number of mode K. If you're mode K scatters off of some other mode, P2 produces modes P3 and P4. So those are the two kinds of positive and negative terms in this equation. So the equation looks in fact reminiscent of the Boltzmann equation, which we have for kinetic theory of gases versus waves. And indeed, if you redo wave turbulence or if you redo the computation of the wave kinetic equation. But for quantum mechanics, you'll have a similar equation but slightly modified. Occupation numbers will be replaced by occupation number plus one to account for quantum effects. And then for large occupation numbers, you recover the classical kinetic wave equation that I just wrote. Whereas for small occupation numbers, when particles behaves, behave like waves, one gets the classical Boltzmann equation. So to emphasize again, we have a non-linear system. There's some weak interaction between modes. Because there's a weak interaction, we can write down an equation governing, governing the occupation number of mode K. And it looks like this. And as you can see that it's second-order lambda because we drop our returns. And it has the intuitive interpretation of some mode scattering intimate cans, some scattering out. And it's analogous to the Boltzmann equation. Oh, good. So a natural thing to do with this, with this equation is task what the stationary solution is. So we want to know what happens at late times. And one can check that this station, there is a solution of this equation for which the entity is 0. If you take n to be proportional to some constant which identifies the temperature divided by omega k. So you can check that if you plug this into here, you'll get the entity is 0. And this is easy to check because if you plug this into here, you get omega k plus omega minus omega 3 minus omega before, which is this precisely canceled by this frequency conserving delta function. In this occupation number. This is known as the Rayleigh Jeans distribution. And it's simply the, if you take the Bose Einstein distribution one over e to the omega over T minus 1, and take the high temperature limit of that, then you get genes distribution. And the Bose-Einstein distribution is the correct one to use because that's that's, or it's called Planck distribution. That's the distribution for particles and quantum mechanics. And then if you go to the high temperature limit, then it's like classical waves. So this is a very reasonable answer. So or weakly non-linear system. And at late times it thermalizes, which is what we expect. The surprising. So nothing so far spicing. The surprising thing is that there's another solution in addition to this one that is also stationary. And this is the thing that Zafar off discovered. If we take omega the frequency as a function of momentum to be some power of the magnitude of the momentum. And if we take the, the contraction lambda to be a homogeneous function of the mentor. So it's just, so when I write P, That just means the magnitude of the vector p. So it's a product of these pieces power and you just depends on the ratio of momenta. And there's a momentum consuming delta function, then one can check, but this omega, this lambda, if you plug those into here, that there's another solution to this equation that stationary, which is n, goes like e to the minus gamma, where gamma is this power. So there's a second stationary solution for vlogging Sharia. Because we all immediately in Fourier space, we assume no problem had how many spatial dimensions? D, just any, any, any t is the number spatial dimensions. And you can take the deviation, can be anything, they can be anything. Larger than one. And probably I'm not going to work and 12 dimensions for this one I think is fine too. I'm not D. Notice this. If you plug this into, now this works in any number of dimensions. And that number could take d to be negative and this argument will still work. It might be that you're concealing divergences, but I assure you if you plug this, this, and this into this equation, you'll get the entity is 0. Okay? Um, it, it's probably not physically relevant is if D is below some actual number. So which would be stresses. So, so this is this comp girls aircraft solution. The reason it has both Kolmogorov's name and zeros name is Zachary discovered this in the 60s and it looks like a Kolmogorov cascade and hydrodynamic turbulence, except the differences in hydrodynamic turbulence, these things cause I made up, whereas here the derivable. And again, since the kinetic equation is only valid at weak non-linearity, these solutions are called Weekly turbulent States. However, as you can see, this deviates strongly from equilibrium. This distribution looks nothing like the religions distribution. It's a completely different power of momentum. So much of the work on wake turbulence over the past 50 years has focused on establishing the existence and properties of this turbulent state. There are many questions you can ask about both mathematical and physical questions. Mathematical questions is to what extent the above was valid, the above arguments because it found an exact stationary solution of equations that were on the weekly only valid ID we coupling one can ask physical contexts in which you have dispersion relations and coupling of this form. We can do experiments measuring this and so on. So much of the work has focused on establishing the existence and properties of the turbulent state. The question will be interested in, which is a broader question, is how to repeat everything we know in statistical mechanics, but based on the wave turbulent state instead of the thermal state. So concretely, I'd like to understand how to characterize fluctuations about the turbulent state. So in, in, in, in statistical mechanics we take the thermal state and then there's still an entire book of, of studying fluctuations about the thermal state. We get over the fact that there's a thermal stay pretty quickly and then we're interested in what happens when you perturb that state. And so likewise, here we have this turbulent state, this Kolmogorov's Ashraf cascade, which you get either by explicit constant forcing and dissipation of your system or you get if you set some specific initial conditions. And now I'd like to understand the properties of the state. And studies. There have been very few studies of this, of this in this direction. So that's ago to just redo the statistical mechanics but based on the turbulent state rather than thermal state. And to do this, we need to be able to compute correlation functions in this triplet state beyond leading order. And so we need to be able to compute quantities beyond just the kinetic equation at leading order. So we'd like to understand correlation functions, the field, the different times 2 functions for point functions, the things that we need to characterize fluctuations. And in the process, we'll also see how to correct the kinetic equation into systematically compute corrections to it. So, so, so to contrast with lots of current work, I'm not, I'm not questioning the existence of, of wave turbulence or tried to more rigorously prove anything from the experiments. I believe web turbulence is there and the spectrum is basically what case he tells us. I just wanted to compute other quantities which are fluctuations about the case the state. So the plan is to take some classical non-linear filter, like the one I wrote down, which had an arbitrary dispersion relation, arbitrary coordinate contraction. Then I'll add dissipation and external forcing. And I'll take the external forcing to be drawn from a Gaussian distribution. So this is a standard thing to do. We don't take some definite forcing. We average the forcing of a Gaussian distribution, which is what gives rise to correlation functions. And then we'll give a prescription for computing correlation functions. And the basic tool that I'll use is the fact that a classical field theory with stochastic forcing, it's actually a quantum field theory. I'll show this fact. And since it's a quantum field theory, I can compute preoperatively using quantum field theory techniques of Feynman diagrams. And so this will give a systematic way of deriving the kinetic equations. Yes, everything is okay. Kim guy, though, this will get a systematic way of deriving the kinetic equation which I showed and going beyond leading order in the coupling. The standard derivation of the kinetic equation is, is somewhat ad hoc. Some, it's what people based on the random phase approximation which they struggled with going beyond leading order. And that's precisely what we need to do. Because we want to understand fluctuations about the wave turbulent state. So it's not satisfactory to just produce the KDC state. We want to go beyond that. So a note on averaging. So one could have asked why we're discussing correlation functions, that expectation values, since we're doing classical physics. This, I think many of you is intuitive. So since trivially cascades are present and interacting chaotic many-body systems. To perform calculations and measurements, one has to do some kind of averaging. And the, the kind of averaging one does is, is to a large extent irrelevant. So assuming statistical spatial homogeneity, one can average over initial conditions or alternatively perform a time average. Or in the context of force turbulence, which is what we'll be doing. One can average over the forcing function. So I'll imagine we have specific initial conditions and we're measuring on a specific instance, that instance of time. But the way in which we force the system that forcing function. So the stirring at long wavelengths that we're going to draw from a Gaussian distribution. For the leading order kinetic equation, which is what I showed. They, the equation that everyone uses and the wave triplets literature. It's largely irrelevant which of these averages you use. Because one simply assumes that higher point correlation functions factorise into the two-point functions. And this is referred to as the random phase approximation. Like I said, this approximation, it's difficult to go beyond leading order. So for this reason, I want to do with the case of force turbulence with a forcing function which we average over because then I know explicitly what we're doing. Good. So you mentioned some of the, so there's significant current work on triplets. So there's a large community that finds new physical contexts exhibiting classical wake turbulence, both theoretical, numerical and experimental studies of wave turbulence in quantum mechanics and in particular in the non-linear Schrodinger equation. There have been studies of wake turbulence in quantum field theories and as I said, the concept of pre thermalization. And there's a large body of mathematical studies of wave turbulence. And also models such as the empty model, which is in one dimension in which one can wave triplets but beyond leading order in the coupling which they solved numerically. So most of this is largely orthogonal to this work. Work that, that is relevant to this work. There was a paper by Shabbat and Falcon which, which in part inspired us to work on this, in which they stress the importance of studying quantities beyond the mode occupation number. So the thing everyone's studies is the mode occupation number. And seeing that there's a Kansas State. And they said it's time to go beyond that. In particular, they were interested in finding out how information theoretic quantities. So the, the, the thing they were interested in is finding the invariant measure for the turbulent state. And they argued that, that there will be divergences when computing higher point correlation functions. So quantity is outside the mode occupation number. They felt the wave turbulence will break down. This seems somewhat odd to us because in quantum field theory, the computation of higher point correlation functions is, has the same ingredients as computations of loop corrections to things like the mode occupation number. So it seemed implausible that perturbation theory breaks down for water, not the other. And so it seemed timely to do systematic study of how to compute higher point correlation functions and go beyond the reward and the coupling and wake turbulence. So there, so that's the reason they were, there was a somewhat remarkable pieces by gravity in the nineties in which he used in which he was slightly mysterious techniques to find the next, the leader board a correction to the kinetic equation. So we'll reproduce a result, but in a straightforward way. The, the main technical tool we're going to use the connection bit, which I mentioned that stochastic classical field theory. So it's same as Klinefelter is, is, is well known. It's been known for a long time. And it's appeared in the turbulence literature. In particular Migdal and Falcon, which used it in the nineties. But the cases in which they were using it was to study something harder. They weren't studying week turbulence. There were studying things like burgers turbulence. So they re-expressed a stochastic classical field theory. So a case in which there's turbulence in terms of a quantum field theory. But they were not a weak coupling. They weren't strong coupling like an hydrodynamic triplets. So they would try to evaluate the path integral and look for instant towns and so forth, which is a difficult problem. The thing we're going to do is use the same connection between the gas, the classical field theories and quite a few theories whenever the contexts in which perturbation theory is valid. And so we can actually systematically compute correlation functions. So it seems to me the wave turbulence has been known for a long time and the connection between stochastic classical field theories in quantum field theory has been known for a long time. The thing that appears to have been missing is applying one to the other. So that's what we'll do. Questions, comments, carry on. Let's see, I'm a crush. Any questions? Okay, so the outline is, is the following. So the first thing I want to do is show that a classical filter with Gaussian random forcing is equivalent to quantum field theory. The argument for this is actually only three lines. And in particular, we'll see that the Lagrangian for this quantum field theory is the square of the force three equations of motion. I'll describe this person. Once we have a quantum field theory, then we know what to do. We just use this Lagrangian. We work out the Feynman rules and we start computing Feynman diagrams, which gives us correlation functions. So the tree level correlation functions will give the leading order equation and the loop diagrams will give the next leading order terms. So let me start over. So here's our Hamiltonian. So we have a field, we switch to this space of a and a dagger. The canonical transformation to these new fields. We have a quarter contraction. We have equations of motion, which is just a k dot is minus IDH SDK. You take the HDA k, then you get a cubic term here from, from this term. That's the equation I wrote before. And then to this we're going to add forcing, which I call FK if T and dissipation gamma k. So the problem any, yes, this is a big Gail. So the dissipation, they just prediction. So that kind of sounds sensible. Yeah, I'm horsing, you'll just make it Gaussian. Correct. Okay. Got it. So the problem is very straightforward. So we're going to average over the forcing which will be drawn from this Gaussian distribution. So the function will not be some definite concrete function. It will be something that we picked from a Gaussian distribution that looks like this. So different modes or unquote. So there's something called the diagrammatic approach, which was used by Zara and 75. And it's very straightforward. One just takes these equations of motion and solving them particularly in the coupling to find a hot, then after that one averages of the forces. So if you want correlation function because of a, by solving the equations of motion retributive and you perturb it if we get the solution for a and now a correlation function for a, it is something in terms of a correlation function for f, which you know, because the only correlation function for f is a two-point function. So the entire problem is very straightforward. You just perturb it really solve the equations of motion and then do the averaging over half. Good. And so the key thing is that this is straightforward, but it's tedious. So in this paper in 89, an attempt was made to find the kinetic equations beyond the dewatering the coupling. And perhaps find some other quantities, some non-trivial quantities and the expressions that they haven't, this paper basically unusable because they're so complicated. Because as you can imagine, solving this pre-operatively and the coupling generator a huge mess. So again, ten times in, in, in hydrodynamic turbulence, there are many conceptual difficulties related to the fact that there's strong coupling in WE weak wave turbulence, there is none. It's very straightforward because the coupling is weak, so everything can be done perturbed of the coupling. Good. So the thing that we would like to do is to streamline the procedure of taking some definite forcing, solving the equations of motion than averaging of the force. So what we'd like to do is just average over the forcing at the outset, because we're going to do it eventually anyway. And this is where quantum field theory will come in. So we're interested in computing the expectation value of some operator, where again, the a could be like the occupation number, aka dot gray K or something else. And so as I said, the procedure is 1. Solve the equations of motion, computes this quantity of a for each value of f of K, and then averages over the f of k as prescribed by probability distribution P of F. So one is doing this integral. So when computes overnight and then averages over the app. And keep in mind that a is depends on if you change the equations of motion change So you have a different day. So to enforce these, these equations of motion, we're going to explicitly insert a delta function for the equations of motion. So to this quantity, I add an integral. Over the field a and a dagger, because it's a complex field and answer a delta function for the equations of motion are, the equations of motion are just what we wrote before. So, so hard versus purely classical. Everything Muslim former enforced sharing or low Lagrange equations of cost or comic, comics are good example. Because I think everything I have dissipation I'm driving so that you don't have a background chamber. That's right. That's right. So this is just classical, I'm just formalizing what we've been doing. So now we do a few more steps. So these delta functions for the equations of motion, I write in terms of an integral, which is familiar. So this is like delta E. We know we can write this integral over eta e to the I ATE. So we're doing the same thing but the functional version, because these are functions. So let's write that function as an integral. Sorry, so what is this? And k, Where did that, okay, Where did that come from? Or you're integrating over it, okay. Okay. And if you do the integral, you get a delta function, right? Yeah, yeah, and and so it's really not a polite second. Okay. Keep going. But so I into here. And after I did that now I can perform the integral over f because I have f appearing in this probability distribution of that word appears Gaussian is e to the minus x squared. And then I have an f appearing linearly because I have a to E. And within the E, the equation of motion, I have an F. So the integral over f is a Gaussian integral, so I can do it. So I do it and this is what I end up with this effect of Lagrangian. And now I should get some determinant. Does your dog for the permanent determined is equal to one. So I, to simplify notation, I didn't write the gym. But you're right. There is a determined Mm-hm. Mm-hm. Job you should take home to one. It's not obvious, but it is equal to 1, okay? With trust her. And then after that, I do the Gaussian integral over 80 k, this fake variable I introduced, which is also a Gaussian integral. And I finally end up with this very simple expression. So the expectation value of this Olivier, which can be anything, It's just this integral with this effective Lagrangian, which is the square of the equations of motion with F equals 0. So whatever the square comes from, It came from the fact that I'm doing a Gaussian integral. So when I do this, in this L here, I had a to E and I have a Gaussian integral over your company Square in New York, for example, completing the square. And that's it. So this is, this is a very sensible results that we started with equations of motion. F equals 0 with no forcing. We added forcing and then we average the forcing of a Gaussian distribution. And the result is ineffective Lagrangian, which is proportional to the equation of motion with F equals 0 squared. And this, this, The, only, the thing being integrated over here is a. So now it's just a quantum field theory because a has now become, what's the capital F? Capital F, Good. It's in my probability distribution for F. Here. Capital F was the variance of the forcing. Okay? So the whole thing hang some being forcing, being Gaussian, does route makes this looks like the usual stochastic field here. Yes, this is, this is 10. Yeah, okay. And what quantum about that a is the fluctuating variable. So the end result is there's no forcing. We just have a feel a, which is the fluctuating variables. So this is precisely a quantum field theory with Lagrangian That's EMF equals 0 square. So this is a quantum field theory, gives out imaginary constant EI, and if it's out h bar, That's right. That's different from your instructors to cultural chair there. So if you would the normal Lorentzian feel theory, there would be an I here. But the reason I'm not calling it a Euclidean feel theories because time here is little rents and so the T here is the physical actual time. So it's a quantum field theory, but the action is missing an eye, which you might have expected, which is actually a positive thing because if in Laurentian, in the rents and guilt there is the action never just has an eye because then it's not well-defined. So then you have to, in, the interesting thing is that in standard quantum field theories that we study, you secretly are always adding dissipation to make the path integral converge. Now dissipation people called the i epsilon prescription, but it's essentially just dissipation. And here we don't have to. Do any of that artificial stuff because we truly have participation. So unlike normal quantile through here, everything will be finite and physical because we're doing a physical problem. So we have a quantum field theory and our Lagrangian is the equations of motion with 0 forcing squared. And so here I wrote it out explicitly defined this 20 d, which is just aka dot plus I omega k plus gamma k, and it's AK. So it's just this term squared. So you can recognize this, this was just the equations of motion we had in the beginning. So these were our equations of motion for this with the gamma cancer. So we have a quantum field theory in which the Lagrangian is the equations of motion squared. So I emphasize again, we had a, we had a classical field theory in which we added Gaussian random forcing. And that's completely equivalent to a quantum field theory in which a is now a quantum field with this Lagrangian. Good, Um, and now that we've reached this stage, now we know what to do. Now we can perturb it in lambda, so we can expand this out. And this, the square of this term is, is the free part. That's what gives the propagator. The cross term here is proportional to lambda, and it's just a quartic interaction which I've drawn here. And this term squared is the sex, the contraction because the squared we'll have six A's. And I've drawn that here. So it's just a quantum field theory with a propagator, a quarter contraction and the contraction. And that's, that's essentially it. Now we, now we just start computing correlation functions. And the way we know how to do in quantum field theory. In I'm, in previous, in previous line these by Mughal, in fact we should so on. They had this expression but there are the Lagrangian they took was here for the equation they were doing, for instance, burgers turbulence. So the equations of motion they took was what you have for Burgers equation. And so you have a path integral which you can evaluate that then they wanted to evaluate it. So maybe I'll try. The square of the equations are much less. What's commonly done is people don't like the square, so they introduce an extra variable. Often people leave it in this form in which you don't have the square root, the equations of motion, but then you have two fields. Yeah. That's what's commonly done. But for us it is, It's better to just have the square of the questions about it. So in these studies with burgers, triplets and so on, people then try to evaluate this half is for growth in cases in which you don't have weak coupling and then they look for instant tons and so on. But in our case we have weak coupling, we can just do what we know how to do. So, so I drew the tree level diagrams here or the one loop diagrams. If one is computing the 4 function, they look just like one loop diagrams and quantum field theory. We want to scattering scattering particles. But these one loop diagrams arise from the quartic interaction term we have here, this quartic interaction term in the sex, the contraction term. And you get these 12 diagrams. And so we, we compute the diagrams, we summed them and then we get before point function at up to 12. Of course what you're going to any order and the coupling that you like. So this is one loop, so this is second-order in the coupling. As you can see, that each, each vertex comes with the, with the coupling. So in particular, we can compute the kinetic equation. Because the kinetic equation is, It's an equation for the occupation number and k, which is just the two-point function. So I've written down the kinetic equation, so let me zoom in. So the first line of the kinetic equation. So again, the kinetic equation tells you the occupation number of node k. So the occupation number is still a real number, not an integer. It's not contrast. That's right. It's all classical, so that's the number quantized. So do you have a lot of ends, but the other ends or not? The occupation number on the right-hand side, you have events or the occupation number. I've just, so this first line of the right hand side has the same kinetic equations before. So N1 denotes and P1, so as the occupation number of mod p one and I'm summing over these *** camp. So it's like, let me go back. So it's like here. So on the left-hand side you have NK. But the evolution of n k depends on the occupation numbers of these other modes. And P2, P4. Because the occupation number of Moltke can increase if two modes scatter into McCain, it can decrease if mod k scatters against some other mode. And that clearly depends on the occupation number of these other modes into which a scattering. So this is difficult to solve. But the thing people don't wait, turbulence are interested in is the stationary solution. And then one is just setting it equal to 0. The right threatening you have totally free coupling constants, lambda, but Bureau make them all the same or something, something happened that makes it possible to do this calculation. Let's define this stationary solution. I just need lambda to be dysfunctional form. Ok? Construct this. This is a solution of that. This is a, this is the case. The spectrum is one for which the NDT is 0. And this is delta P1, P2. That's some kind of momentum conservation. And this is our shorthand for momentum conservation. P1 plus P2 equals P3, P4. So some Sophie quo or P1 plus P2 equals P3, P4. Okay, gotcha. So the first line of the, so that was the kinetic equation that everyone knows. So the thing we've computed as the kinetic equations next order, which most people do not know. So the first line is the kinetic equation. Everyone loans which has this interpretation of high, you lose or gain an occupational modes, mode k from scattering into hardest mode k. The second line or these correction terms, which is now third order in the coupling. It has a bunch of terms here, but all of these terms just corresponds to this, this processes here. So you can gain mode K. If, so, you lose mole K If K scatters off some other mode, and this turns into two other modes, which then turn it to other modes. So these diagrams reflect the physical process that's happening. Alternatively, you can gain mode k If k is and is not one. Then some two modes, they scattered to other modes which are 56, which then scatter into your mode can some other boat. And that's what's captured by these diagrams and this equation. And this is now like condensed matter dishes really Hamiltonian. Time has direction. And I have to add up all possible time order things. What do you mean by in condensed matter in these diagrams? You have to have holes. You have to have no, no, no, no, no. We posted on Canvas marijuana to increments are narrow. These are just normal Feynman diagrams. And in quantum field theory there are no, none of this whole business. It's just, yeah, just signify momentum, direction was that branch. The arrows it because we have a complex field. So the arrows, the arrow signifies is that It's because we have complex fields. So it's like if you have electron and positron. So that's me, yeah, okay. But to use this, you can just use this equation. So if one takes this kinetic equation and from this kinetic equation, one can compute corrections to the Casey solution. Um, because the KC solution had that a stationary solution is, and if k is k to some power which differ strongly from the power you have in thermal equilibrium. So if you, you, you can use this equation to turbulence the correct that and find what the more correct power is from the equator. And presumably if one is doing an experiment and measuring the case, the spectrum, the power you will measure will be closer to what you get from this equation then from just keeping the first line. And the occupation numbers are time-dependent? Yes. They're all time dependent. But the key is because of Casey, at late times, there exists a stationary solution. Cilia will evaluate it at a later time in stationary. Yeah. Yeah. Yeah. Yeah, that's right. That's right. Well, actually, for now I won't be doing anything else because my talk is almost done. But the key point is that we can, so we can compute two things which are not formalism are the same. We can either compute corrections to the Casey state because again, case, the aircraft drive that at leading order in the coupling. So we can compute fractions that and we can compute fluctuations about the case the state, which our correlation functions like a four-point function of the field at different times. Which characterize fluctuations about this name. For which we have equations two. And those should in principle be measurable. Because if, if you take this, this picture here, if you take this experiment, they do, they set up either here or here at this wave turbulent state. If instead of just measuring the occupation number of mode K. So they set this up with the elastic plate, they hit the plate, or they're constantly forcing the plate and then it's bouncing around. And then they measure the occupation number of mode K. If they in addition measure some 4 function. So the correlation between four different modes. Um, then they should get an answer which should match the answer we derive. Have to force Xing Xiang Yao Shen Wei. Yeah, they're doing that anyway in their numerical simulations slash experiment, an experiment second UGA. I think so. I think this is a mental picture. Well either that or he just put the picture on his website. But so to summarize, we took a classical field theory with an arbitrary but small quarter contraction. One can do this with a cubic interaction or any other interaction with dissipation Gaussian random forcing. And we get a prescription for computing correlation functions pre-operatively and the coupling. And then we applied this to compute 2.4 functions, the next leading order in the coupling. And again to emphasize in the limit of vanishing forcing and venison dissipation. One might have thought that the properties of the system should be the same as that of a closed system whose late time status, the thermal state, and the discovery of wave turbulences that this is not so. If you take interactions and dispersion relations of a particular form than the late time state is actually far from equilibrium state, which is the case. The state which has this occupational number from 0 to k, which is very different from what you get an equilibrium. And our formulas in this case give corrections to the Casey state and fluctuations about the case the state. So in my mind, there are two broad goals which, which motivate the study of correlation functions in the turbulent state. On the one hand, the turbulent state is a stationary state, which is not a thermal state. And it's relevant even in contexts of closed systems if you have far from equilibrium initial conditions. And so it seems reasonable to develop linear response theory for it. Do the analogous thing that we do in statistical mechanics for the thermal state, just do it for this triplet state. So understand how it responds to the fluctuations and find the turbulent state analog of transport coefficients and the fluctuation dissipation theorem and things like this independently. Which was also part of my motivation. In recent years, there's been enormous interests within hydrogen, that's matter in many-body chaos. And perhaps during wave turbulence in light of these developments might be productive. Though I don't know how specifically how in terms of future work, one can repeat everything and go to higher orders in the couplings, higher point correlation functions and compute information theoretic measures like entanglement entropy. If for a fraction of the turbulence scales community, an interesting question is what the invariant measure is of a of k. So this is the probability distribution of FK I feel at late times. And this follows from the correlation functions at a fixed time. So once we know the correlation function is, we can reconstruct the probability distribution for the field, and that is the indirect measure that people thought up. So the invariant measure is this non-trivial thing because one does not have a closed system. So the probability distribution for a is not the thermal state. I've kept emphasizing. It's something more interesting and good. And as I said, these corrections to the correlation functions should in principle be measurable quantities. And finally, it, it seems useful to generalize this to quantum field theory. Within quantum field theory, in fact, even thermal field theory. So quantum field theory of the thermostat is a fairly recent thing. It was only developed in the eighties. And given current experiments like with heavy ion collisions and other contexts such as heating and there are the universe. There is no interests in five from equilibrium quantum field theory. And the context of wave turbulence seems like one in which one has a controlled far from equilibrium state. So it seems reasonable to develop wonderful report. Okay, Thank you. Okay. So people online, you can just like a tomato, you can just turn yourself on and ask questions. People in a room, just tell them to me or I'll give you your fonts. She has a question or actually let me yeah. I can hear you. Thank you. Yeah. Vladimir, this is really nice. Stop the pleasure it to hearing about your stuff. And then graduating. I was trying to figure out if I've seen this mapping from the forest and with dissipation classical theory to the quantum field theory. And trying to figure out if you have seen it before. You've seen out. Yeah. So can can you help me figure out what's the context? It looks a little like a habits televisions and you get that squared. So within condensed matter, this, this is called Martin say girls. They did this in 73. The connection between stochastic classical figures, though I find the paper slightly principle within different communities. This is attributed to wild and 61. In McDonald's, mcdonald's papers in the nineties, they said it well, they also cited Xin Justin's textbook. So different community site, different people, but the derivation is really three lines. You just, you just, it's the three lines I gave. You have an integral over the forcing. You have an adult a function to, to enforce the equations of motion, you write that delta function is a path integral. You do the path integral over everything except your field. And you end up with a point of view. It's very, very simple variation, cool looking. I don't know that's a reference from the seventies. I'm sorry. I should ask you. I didn't to be honest, I didn't understand any of these references, but It's just three lines. Yeah. Yeah. Okay. Yeah. Egalitarian should say, is, it's an archive which you want to read the paper, find references. Where has it gone? Archives. If you happen to have the archive link, put it in the chat so people can, okay, I'll do that. Yeah. I mean, because it looks very nice and I I guess I haven't seen it used that often. Even though it looks straightforward enough, it shouldn't be very generally useful. Yeah. Like I said, the papers the papers people say to her tombstone. And then a I don't know if it's a question that can be answered. When you derive the final expressions at the end. How hard is it to, to figure out some realistic parameters like the interior, some kind of for a given experimental numerical simulations, microscopic value of lambda, that net loss, it's not going up. It's not like if one is doing the plastic plate with the one knows one's Hamiltonian, so one nows length no matter how it's trivial. So do you know actually how it compares to numerical simulations? Is going to the CR quadrupole. Is that good enough? So for the, for the reproduce the KC spectrum, whatever people use the tree level, that's good enough because they reproduce it. I think the, so the corrections, the case you will be small so that, that is not the interesting part. I think the interesting part or correlate the higher point correlation functions, especially at different times which characterize fluctuations about Casey. And I should that's measured because there is no theoretical answer for it. So I don't see why they wouldn't be measuring it. In numeric station. They should have access to it. Yeah, yeah, of course, it seems, it seems easy. It seems easy to numerically or experimentally. Well, since he's a tilde. And to compute and compare, though, it seems like I'm, one needs to. Aside from just the four-point functional at unequal times, one would like to ask a more physical, more physical question. Which is why I had the comment about an analog of the fluctuation dissipation theorem abandoned Casey state. So when one wants to, Yeah, one doesn't just wanted to compute random correlation functions just because you can put, one would like to do what we do in statistical mechanics and have relations between them and understand what they correspond to physically. Okay? Yeah, that makes sense. Okay, Thanks. Yield the floor to any other key. It's me again, but there are lots of these lambdas. So you have a big, ugly formula if lambda is four, mm and page etc. But presumably to connect the two experiments, that is 1 lambda that some phenomenological parameter, that diffusion constant for wave chaos, spreading on the surface from a swimming pool or whatever, you know, whatever, yes, it is growing. So how do these Lambdas connect? That? They'll be phenomenological parameters you would measure in weak wave chaos experiments and it'll be 1 lambda or there'll be 1000. For like if you take this surface gravity waves, then the lambda is some particular function of the momentum. It's not a totally trivial function, but it's like one can derive the function by just writing down the equations of motion for surface gravity waves. And it appears in various parameters that transition is weak or strong or something. I mean that mass, Yep. That's right. So lambda, lambda is some function of the momentous with some number out in front. And that number you, it's not computable number for you, it's shunting. Be measured in particular. Well, it's computable. Just have to tell me what the Hamiltonian is. So for surface, for the surface gravity turbulence, one knows precisely what lambda is because one derives it so. Computable. It's just my form. The formulas I'm showing are general. So any, it just applies to this wide class of Hamiltonians. Thing. You know, you're hungry. But actually your question is good because lambda depends on the momentum. So these equations, if you, if, if you either go to very high momentum, very small moment that even if you tried to set the coupling to be small, it will become large because the coupling is momentum dependent. And so these equations are only valid within a certain range of momentum. So this is another thing people worry about within wavetable is that even though you're trying to make it weak, at some moment it becomes strong. Okay, The other question, sure. What is the something or shorted that chunking a chat. Shari. Thanks. Have a look. Okay. But if you try Rosenhaus are not cover, you will find that that's the last pre-print suggest. Is Casey state stable? That's another thing people in the way of turbulence, the richer way that I think so because it's, it's measured in experiments and it seems fine. Though one can, one can. One can. Actually. So I, so I think the answer is yes because an experiments, it's fine. From a theoretical viewpoint. It's also not totally clear because one is doing perturbation theory. But if you wait long enough, perturbation theory might break down. So it's not totally obvious. In study turbulence, nothing is stable but the ensemble of things you see Rancho where in our set, stable but not individual solutions Class a cultural shift. I guess because we're, so, I think because we're averaging over the first thing, it's stable. The only question is if perturbation theory breaks down after a long time and the approximation is going into week turbulence or become wrong. But I think since it's measured experimentally, that probably doesn't happen at least for reasonable times. And how to see that theoretically is, is a question which one can try to address. We have the corrections to the leading order stuff. So then you can see how the subletting corrections in higher-order lambda, how they compare with leading order lambda. Okay? Anymore questions? If not, thank you. You can hear me clap and thanks a lot. Thank you. And deny if anybody wants to see. You'll have to Google here and there. But it's smart. Hack has a repository of physics stocks. So it will be possible to find the talk. I'll take five days or so for sure. Okay. Thanks a lot. Thank you.