Renormalisation

0:00 What is the difference between the disciplines and the way they are displayed? The methods of renormalization, the methods of renormalization, we will discuss a little about the use of this term and its origin and its justification, we have already talked about it twice during this seminar, the first time with Pierre Cartier who talked to us about the variance of scale as symmetry, I will, of course, address this point, and a second time with the exhibition of Thierry Paul, who presented a part of the history of the methods of renormalization, namely their use, mainly in quantum electrodynamics, as a very technical tool. To sum up divergent series, to manipulate and manage to extract finite quantities of indefinite series, and to understand major, intrinsic divergences. So my proposal today is not at all to make you neither a history nor even a discussion of the different methods of renormalization, which would be interesting but already very technical and would take us quite far, but it is to try to explain to you These methods of extremely technical and sophisticated mathematics and physics tools, or even very abstract ones,

2:30 I would like to try to show you how these tools have ultimately led to a rather important change, I would even say very profound, of our view on the notion of model. So, how did we eventually move from a technical tool to a major conceptual change in theoretical physics? So, that's why the title contains this passage, how to exceed our subjective points of view. Finally, this is where there is really a part of physics, it is how to bring a real phenomenon into a model, into a series of equations. It is of course a subjective approach since there are many arbitrary or more or less arbitrary choices, there are many hypotheses, there is a lot of reduction that is done at each stage of the construction of the model or even more generally of the theory. And these methods of renormalization, through the comparison between models at different scales that they allow, These have allowed us to go beyond a part of the arbitrary of a model, the arbitrary linked to the observation scale. So a model is built at a given observation scale, it is of course dependent on this scale, but by comparing models at different scales, we can go beyond this limitation and arrive at something that crosses the scales, The fact that, to describe something, we have to choose the look we wear on it. We have to choose the magnification of the microscope, the zoom of the camera, and simply by changing the zoom or magnification of the microscope, and by comparing what we obtain at different magnifications, we can obtain additional information. We could call this a threshold by using the words of the common language, the words that I will use must be taken in the usual sense, but this is the point that I will try to develop. So I will start before trying to briefly present to you what the methods of renormalization are, on an example, I'm looking for white chalk, otherwise it will be in color, never mind.

5:00 I will start with an example that must be quite familiar now that fractal structures have almost become a public domain. This example is that of the length of a fractal curve. You all know an example of a fractal curve, the curve of a body. So this curve that we can observe with this resolution, if we look at it more closely, we see other details appear, and if we look at it even closer, as you have understood, we see more and more details. So this object can be iterated to infinity and obtain what we call an ideal fractal structure, a mathematical structure, the iteration is regular, determined, so it is an extremely well known and studied structure. The question we are going to ask ourselves is, considering this object, what is its length? So what is the length of this curve? What is fractal? Well, this length is something extremely poorly defined. Why? Well, you see that if I take this as a unit, I will call it A, and that I look at the first step, the first drawing that I made you. I have a length at the L scale of A which is 4A. See, I'm going to see more details, so my length is going to be bigger. So here, I'm going to end up with 4 times pieces that include 4 times a over 3.

7:30 And if I look again at the smaller scale, a over 9, I'm going to multiply again in this way, so we have a length that will depend on the way... These are examples of the resolution with which I observe it, in other words, with the arpentage step. Here I took the example of a curve. But it is also true for objects of natural structure. For example, the classical example introduced by Mandelbrot is the length of a coast. If you measure the length of a coast on a major state map or on an atlas, you will not find the same. Dimensions are the same, if you measure on the ground, according to the unit of arpentage that you are going to use, you will not find the same length. The length for a circumvolved curve in this way will be as large as the scale used to observe it is small. So we end up with a notion that is completely unsuitable. The notion of length is no longer suitable. And how are we going to replace it? Well, we are going to replace it with something that allows us to compare length to different levels. So what becomes significant and what provides a quantitative characteristic to this object, to this fractal curve, The link between lengths at different scales is expressed as a relation between the length observed when measured with a step equal to k times a and the length measured with a step equal to a and there is a factor of proportionality here which makes an exponent intervene. This is what we call fractal dimension. In this case, we have that df, we calculate it immediately, is log of 4 over log of 3. So something bigger than 1 reflecting the circumvolved character of this structure.

10:00 So, there is really a lot of things in this relationship, since you see that we are going to connect two indefinite quantities, possibly even quantities that can diverge, that can be infinite. What becomes interesting is now an exponent, so we went from a geometry of amplitudes to a geometry of exponents. In the same way, we will go from a physics of amplitudes to a physics of exponents in the case of critical phenomena. So it's really a remarkable change. And instead of focusing on something, on a measure at a given scale, What becomes significant, what becomes quantitatively revealing of the structure, is the link between the scales. So we completely changed our point of view, we almost went from a transverse point of view. What becomes interesting is to compare the different scales with each other. And this comparison involves the factor of scale change multiplied by a certain exponent that will really characterize the nature of the curve. So, abandoning this Euclidean quantity, which is the measurement of a right, the measurement of a segment, we move on to a measurement of an exponent, which we call here fractal dimension. So, in the end, everything is here, all the message that I will try to transmit to you today is here, and I will show you how it can be generalized and how the methods of renormalization have finally made this change, from a physics of amplitudes to a physics of exponents, and how by concentrating on... On the link between the different scales at which we can observe a phenomenon, they allow us to grasp something a little more introspective than what we observe at a given scale. So, let's get into the heart of the subject.

12:30 So, what is a critical phenomenon? And here again, the example of the fractal structure will be very practical. A critical phenomenon is a physical phenomenon that is, according to the usual physical phenomenon, exactly what a fractal curve is to a line that can be drawn on the table. There is also a non-trivial link between the different levels of the phenomenon. Let's give a few examples of a critical phenomenon. The most famous example is the transition from gas to liquid at a certain temperature and at a certain pressure. We no longer distinguish the liquid from the gas and we see inhomogeneities appear at all scales. It is always the same thing, it is at all scales in the sense of a physicist. That is to say, we must not really look at the scale of atoms, nor completely at the macroscopic scale, but what it means at all scales, it is in a range of scales much larger. This particular transition, which is called the second order transition, The presence of inhomogeneity at all scales in the fluid, especially at very large scales in relation to molecular scales at scales of the order of the length of the visible wave, They can distract light and lead to an optical phenomenon that is directly observable. From the point of view of thermodynamic magnitudes, this is due to divergences. A divergence in calorific capacity, for example. So, something very strange happens for a given temperature and pressure. Another example is the transition between a ferromagnetic behavior and a paramagnetic behavior observed in iron.

15:00 Here is the usual phase diagram. I place the temperature and here the spontaneous magnetization, M0, that is to say the magnetization of local magnetic moments. The magnetization of the sample in the absence of any magnetic field, i.e. spontaneous magnetization, the one that a magnet can manifest, like those that can be found in all magnetized objects, in the absence of an exciting magnetic field, i.e. spontaneous magnetization. There is no spontaneous augmentation at high temperatures, and what happens is an extremely brutal transition. Here the curve is really a vertical tangent. We have the symmetrical curve that exists by symmetry of the orientations of local magnetic moments. And therefore an extremely brutal transition, presenting here also a difference in the magnetic susceptibility, for example. If we look more in detail at the statistical properties of the material, the function of correlation in a fluid at the critical transition, the function of correlation of spin in a magnetic material like this, we see a divergence in the length of the correlation, which means that the material is A local excitation of the material will, from close to close, be able to propagate to the whole of the material. So, extremely bizarre properties. It's a bit like percolation. The percolation transition is a critical transition. Since the question is asked, I will also present the percolation networks. This is an example that is perhaps even easier to expose from the point of view of both critical character and renormalization.

17:30 Practically, the most abstract and simplest model that exists in theoretical physics is to represent a material, two or three-dimensional, for example a porous rock. We discretize the space and fill in the cells. Simplify the system and say that a cell, a spatial cell of my discretization, is either full or empty. So for example, you take a rough rock, you mentally cut it into small squares, and if a small square is rather full, we will say that it is completely full, black, white on the table, if on the other hand it is mostly empty, we will leave it empty. A model that is simply made up of empty or full boxes. And this model is really very, very simple insofar as we are going to fill these boxes independently of each other. So it is a model without any interaction. The cells do not see each other. When we build this model, this percolation network, the only parameter is the probability that one box, independently of the others, Thank you for your attention. So, a model with a single parameter, a model with a geometric substrate, so we will have the dimension of space and the geometry of the network that will intervene. Here, I took a square network, there are other models that have been made, triangular or hexagonal networks. But in any case, something extremely simple, since we fill these boxes with a certain probability, everywhere the same. and independently, when we go from one box to another. What we observe is, if we ask ourselves when will I have a path that allows me to go from one side to the other through a connected path, through a touching cell.

20:00 In this case, we say that the system percolates. If we don't have this path, if the environment behaves in a waterproof way when we cross it, we say that we are below the percolation threshold. The term is quite familiar, by the way, it is the coffee percolator. In order to make coffee, we talk about percolation when the coffee powder is not dense enough to let the water flow through. If we squeeze too much, the coffee does not pass. If we squeeze too much, the water passes too much, and there is just a threshold where the water starts to flow. So this threshold is called the percolation threshold, and what is remarkable is that we again have an extremely brutal transition If I draw here the probability of observing a path that crosses, that goes from one side to the other, well, this probability is null, so nothing crosses, no path crosses, no path crosses, and all of a sudden, brutally, a path appears. And again, this transition, by the way, you can notice the analogy that is not at all due to chance, which is a deep analogy and justified between the two phenomena, we find ourselves with a transition of the second time, of a vertical tangent, a divergence of the length of correlation. The control parameter, the temperature analog here, which is the probability of filling the boxes. An extremely interesting model, as there is no interaction, but there are still statistical correlations. The length of correlation here, we will define a function of correlation, which is the probability if a box in 0 is filled, that a box in R is also filled. And this function will behave as exponential minus r over xi, depending on p, and this characteristic size of the statistical correlations present in the system diverges to this quantity. If I trace it behind to give its appearance, we have something that will diverge here.

22:30 So, xi of p diverges. So, we have a percolation threshold which, in the limit of an infinite network, is perfectly fixed, determined. And another characteristic that is easy to explain here is the appearance of scale laws. When we are at, for example, this length of correlation, it will diverge as P minus Pc power minus b. So we end up with exactly what I was telling you here, something quite analogous. When we are far from the transition, what makes sense is this length. This length is significant, it is finite, it will give a characteristic size to the phenomenon. When we are at the percolation threshold, this length starts to diverge. It will depend extremely sensitively on both finite-size effects, if we are in a finite network, or on very small fluctuations that may appear on this control parameter, so itself, as it is, is much too sensitive, much too fluctuant to be able to... This quantity loses all physical meaning. On the other hand, what gains a meaning is this exponent, i.e. the way in which this length of correlation diverges. So we see again, in critical transitions, for critical phenomena, a transition from a physics of amplitudes to a physics of exponents. All the physics of the phenomenon, the deep statistical characteristics and the deep mechanisms of the phenomenon will be contained in the exponents and what we are going to try to calculate are these critical exponents. To understand what happens in this percolation system as well as in the magnetic system, the magnetic model for this transition is the model of Ising, that is to say, we take a network, for example a square network to stay on something that is easy to draw, and at each point we place a spin, that is to say the equivalent of a small compass, which in a way...

25:00 We will be sensitive to two interactions, two effects, or rather two influences. On the one hand, an interaction which will be of the form . Here we have a spin in I, a spin in J, and we have this energy of interaction, which is the most negative when the two spins have the same meaning, S of plus or minus 1, it is an extremely simple model, there too the values are discrete, so we have compasses that are either up or down, and an interaction that tends to align them, so the energy is minimal if two neighboring compasses This interaction is exercised only between neighbors, so we have a system that is still very simple, since the couplings, the direct interactions are limited between neighbors. In other words, this spin only sees these four neighbors. This is a rather unfavorable case, as three of its neighbors are in the wrong direction, so it will try to turn its neighbors to optimize, to diminish its energy. So, an interaction energy that aligns the spears and counterbalances this interaction energy, this interaction force that tends to order the system. Well, there is thermal agitation, that is to say that each spin will spontaneously fluctuate and spontaneously be able to rotate. And here we understand the mechanism of the transition, that is, we have a competition between this tendency to order, this tendency to align all the spins, and a competition between this tendency and thermal agitation, which tends, on the contrary, to disorganize. At very high temperatures, thermal agitation takes over and disorganizes the system. We don't have privileged magnetization, we have as many spins upwards as downwards on average, and therefore a null spontaneous average magnetization. We have to apply a field to see something, to finally see that there are spins in the material. On the other hand, at a null temperature, Spontaneous and maximum alimentation. Thermal agitation is almost never exercised, there is practically no spontaneous probability of spin rotation, and we have maximum alimentation.

27:30 This compromise between two influences, two opposing mechanisms and the existence of a transition, since we are going to go from a state with a spontaneous augmentation to a state without one, on the other hand, the brutal nature of the transition and the divergence of statistical correlations, the presence of laws of scale, it does not understand itself in such an intuitive way. So we end up with a whole class, I have mentioned two of them, a whole set of phenomena qualified as critical phenomena. The general signatures of these critical phenomena are the divergence of the range of correlations, Correlation is the divergence of time, i.e. if we apply a small stimulus at a place in the system, a small excitation, so we put a field just a little bit somewhere, here we are going to change a little bit the geometry of the system. A localized excitement of the system will spread from close to close to the whole and therefore will put a very long time to relax. So properties of slow relaxation, properties of abnormal response to external solicitations. It is also not by chance that a fractal curve was mentioned in the introduction. The geometry of the structures that appear at the critical point presents fractal characteristics. So here, if we look at the geometry of the mass of... If the black boxes are marked exactly at the percolation threshold, this mass is a fractal structure, quite orthodox, with a well-determined dimension that can be calculated according to the network geometry. So we also have fractal structures as visual signatures at critical transitions and, more generally, in all critical phenomena.

30:00 They all translate or reflect an unusual link, a much stronger link, What happens in a phenomenon, let's say, in a normal phenomenon, that is to say far from the critical point, either here or there, or, for example, a percolation network when the probability is very low, is that we have a characteristic range for microscopic phenomena, and that as soon as we have exceeded this characteristic range, we can average everything that happens at the bottom of the scale. Typically, if I take this network, Here I am in a non-critical situation where the characteristic size is 2, with a probability... If the length of the correlation is equal to 2, it means that if I look at my network with a characteristic size greater than 2, I can average everything that happens at the lower level. In most cases, in all non-critical phenomena, we have a very simple method to analyze the system. In this case, we talk about the separation of the scales, we have a set of macroscopic observations and we go from one to the other simply by averaging. All that happens at the macroscopic level to make it appear only in a few parameters and this is really the approach that is made in modeling to describe a non-critical phenomenon we will take a macroscopic model with effective parameters and these effective parameters we can calculate them simply by means of what happens at the stage of microscopic phenomena. For a critical phenomenon, we can no longer do that, the slightest microscopic detail can, from close to close, be repercuted up to the scale of observation and this is what has led people, on the one hand, to design the methods of renormalization to treat analytically...

32:30 This kind of phenomena, and above all, is what forces us to change the way we look at these systems and not to look at what is happening on a given scale since what is happening will depend on our choice, there is not one intrinsic characteristic scale, any scale is good or as bad as one or the other, we will say. And this change of gaze consists of looking at how The critical phenomenon will appear on different scales and what will be significant is the link between the different perceptions that we will have of the same phenomenon on different scales and even more than the perceptions, it is the link between the different models that we are able to make of this same phenomenon on different scales. So the model that we are able to make at the scale of 1 spin, the model that we are able to make at the scale of 4 spin, of 16, etc. And so on, until the macroscopic scale. And it is this link between the models that will be significant. This link that you are talking about, is it made only for the critical state or is it also made for states, let's say, near or far from the critical state? It is made in the vicinity of the critical state with... The idea is that if we start from here and look further and further, the system will appear less and less critical. If we are far enough, we will be able to regularize the system in this way. If we are at the critical point, we can really see the variation of the chains and in this way we can access the exponents. So, both ways of doing things have been used. At the beginning of the development of the method of renormalization, the ambition was quite modest. It was precisely by taking the spins by a set of 4, then... We used larger and larger blocks to determine the effective parameters, i.e. the local effective field and the effective interactions. We simply hoped to get back to a less critical model for which disruptive methods, for example disruptive treatment, were applied. By doing this, we discovered that we could be much more ambitious.

35:00 and quantitatively determine the exponents, which is precisely what is at the heart of the renormalization method. So, how will this group of renormalization work? Well, on the example of the disease model, it consists of precisely, quantitatively, These are the different Hamiltonian models that we can obtain, either at the scale of a spin, or at the scale of a network, and then we can look at what happens if I cut out the system into blocks of four spins, And I assign each block to an elementary unit. In this case, everything that happens inside will contribute to a local field. Everything that happens between a block and the neighbor will contribute to an effective coupling constant. In this way, we can build an effective Hamiltonian for this spin block system. We can even do better. We can make a change in the spatial scale to bring a block back to the size of a single spin in order to be able to really compare exactly as we can do a zoom that allows us to superimpose the visions that we have of the same system at different scales so in this way we will have an Hamiltonian for For example, a network, this time of size 2. And then we can continue, we can make an Hamiltonian for a network of size 2 squared, and then the Hamiltonian for blocks of size 2 power k, etc. And in this way, we can explain the transformation that goes from one Hamiltonian to another, and this transformation is what we call... The transformation of renormalization, the group, the structure of the group here is trivial, it is a semi-group structure that simply comes from the iteration of R, that is to say, the structure of the group is R iterated M times, that we apply to R iterated M times, it is the same thing as applying it to M plus.

37:30 So the group of renormalization, the structure of the group is finally quite trivial, it can also be expressed in... RK, if K is the scale factor, RK prime equals R of K, where K is the scale factor. So either we can vary the scale factor, possibly continuously, or we fix it once and for all. Here I have it pre-equal to 2, and in this case we have a discrete group structure. Here we can have a continuous semi-group structure. So that's why we're talking about a group of renormalization. In this case, technically it can be interesting because if we exploit the structure of continuous groups, we can only look at the generator of the group and technically it is more convenient to analyze the generator of a group than the group as a whole, but it is a little marginal in relation to the general range of these methods. So I have built a transformation in this way. And so that's when there was really an advance that was not really anticipated, since at the beginning the goal was to reduce the apparent criticality of the system. Because you see that by doing this operation of taking the system in blocks and then by making the change of scale, well, for each operation, I divide by two the length of the correlation. So we will say that this system here is much less, its length of correlation is divided by 2 power 4, so it is much less critical than the initial system. So if it is much less critical, I will be much more comfortable to study it. So at the beginning, the ambition was ultimately quite modest, but once this operator was built, this renormalization operator, we realized the immense scope of this approach. For a very simple reason, it is that we really have again this change of gaze, this transition from a vision at a given scale to a transverse vision, crossing the scales because instead of taking as a starting point a certain Hamiltonian and then deducing all the configurations, all the properties of the model associated with this Hamiltonian, Instead of working in a configuration space with a given model, a given Hamiltonian, we work in a Hamiltonian space. So we completely changed the level, we went from a configuration space to a model space.

40:00 And what will become interesting, what will become significant, what will capture the very nature of the critical phenomenon, are the properties of this transformation. which connects the different models to the different scales in a space of models. So instead of studying a given model and all the four figures it generates, we have completely changed the level and we will now compare models describing a given phenomenon at different scales. And it is the comparison of these models that will allow us to catch something more intrinsic than the amplitudes, which will allow us to catch the critical exposures and the laws of scale. A sketch of this operator's action in the space of Hamiltonians, I will make a path extremely reduced and simplified, but just to make you understand. What will become central in this approach are the fixed points of this transformation. Why are these fixed points a particular role? Well, because these fixed points, writing that if we change the scale, we get the same thing, these are models that are, by their characterization, by their definition, they are exactly the same in varying the scale. There are two kinds of scale invariance, just like there are two types of fractal structures. There are trivial fractal structures. A cube is a particular fractal structure. It is of full dimension. So, in the end, it is quite banal. Here, it is the same. We will have systems of trivial scale invariance. It will be the stable fixed points. And we will have critical fixed points, which are called hyperbolic points, that is to say that there are directions in space, so here I am in a model space, in an Hamiltonian space, The points that I will draw here, all the lines that I will draw as trajectories of a dynamic system, are in fact trajectories under the action of this operator. What happens, what varies along a trajectory, is the factor of scale.

42:30 And so these are points where the trajectories will have this shape, with a hyperbolic fixed point here. We find a first property that I was talking about, namely that this operator of renormalization, if we go beyond the critical point, it will decrease the criticality. That's what we saw. And it leads to a stable fixed point. But what is interesting is its action at the vicinity of this fixed point. And we can show that the properties of this operator at the vicinity of a critical fixed point of this type We have access to the exponents, so it's a little bit technical, we have to linearize the operator and look at the action of the linearized operator and we see that the proper values ​​of this linearized operator are directly linked to the critical exponents, so we have direct access to the critical exponents. So this is the extremely powerful technical aspect, extremely fruitful, of the methods of renormalization, but there is even more, I would say, in any case, for today's proposal, is that, see this drawing, it gives a structure in the space of models, it releases a kind of organization in the space of models, in the space of the Hamiltonians. When you say linearized, do you mean that you approach it directly or do you make a development when we know that the rest is negligible? We really make a development in the rest. The idea is to have growth rates. In this direction, in the neighborhood, you just have to catch this direction. Is it enough in a first approach? In a first approach, it is enough. Of course, if we really wanted, if we could, we would try to control all non-linear terms. To have the numerical value, linearization is enough. It's very rare, there are very few cases, such as percolation. To have critical exposures, these exposures are linked to their own value.

45:00 On the other hand, of course, to really do the analysis in particular, to determine the position of this curve that appears as a frontier, it would be better to go to the... to control the non-linear terms, but it is very rare in situations where... But at the same time, it is a hyperbolic point of criticism, it is really generically hyperbolic. In any case, the system is equivalently differentiable to its linearization. Yes, locally, but for example, we would like to have the equation of this curve. And to know, for example, a point, is that ... No, no, no, that's for sure, but from the point of view of behavior, the proper values ​​are ... Even locally, it's not a dynamic system, it's in another dimension. Yes, that I don't like. If it's really hyperbolic, it doesn't bother me. Is the CTP equivalent? Yes, yes, yes, of course. So, when we arrive at this pattern that we call a flow of renormalization, with my fixed critical point here, which represents the ideal model, In the end, we can subdivide the model space into behavior classes, called universality classes. Even worse, we can obtain different fixed points and each one will translate a given critical behavior. So we will structure the space of all the models envisageable around the different fixed points. Each fixed point is the representative of a class of critical behavior called the class of universality. In this whole class, we will be able to divide those that will tend towards different stable fixed points. I will try to make a 3D drawing. Here we have those who will be more sensitive to this, so this allows us to distinguish the different critical behaviors with for each of them an ideal representative, a typical representative, ideally self-similar, we will therefore be able to

47:30 We will be able to determine the different critical behaviors, different class universalities, but we will also be able to determine what makes a type of critical behavior different from the other, what makes a domain governed by this first fixed point different from a domain governed by another, what we call crossovers. There is also the term, this physics is full of terms devoted between scaling, crossover and relevant direction, well, relevant direction is essential direction. You see that there will be disturbances that will not affect the critical behavior, that will ultimately preserve the fact that the starting model is in the domain of influence of this fixed point. On the other hand, there are other terms, other perturbations of the initial model that will make it change its domain of influence, just as a very small displacement of the source of a river can make it change its hydrographic basin. So, in this way, we will be able to determine a whole classification of the space of the models according to... So, I should have said it from the start, it is that in all... All that I've just told you, there was... there is still a... A price tag should be more explicit. This price tag is to look at the properties on a large scale. What interests me here are the macroscopic properties, the critical properties, the properties on a large scale. And what triggers the transformation of renormalization is a similitude between the properties on a large scale of the different models. Depending on the way they behave under this transformation. So it is in this sense that I say in the title that these methods of renormalization, which at the beginning were an extraordinary tool, This is a purely technical tool, a rather sophisticated analysis tool, providing a change, in any case providing a change of view on the notion of model since now we are able, thanks to this tool, to work in a space of models, to compare models with each other, to be able to appreciate the range...

50:00 Is it possible to add or remove a small term that seems trivial, where we have a polynomial model and we add a term of degree 17? Does it change everything or does it change nothing? So we are able to structure different model spaces in this way. I spoke to you about Hamiltonian models. The method of reorganization was also developed for dynamic, discrete or continuous systems. In this way, we are able to gather, for example, different families of evolution presenting the same transition towards chaos. This is also a remarkable example of success for the transition towards chaos by doubling of periods. In addition to looking at only macroscopic properties, either in the thermodynamic limit or asymptotically in the long term, These methods will work when we have a critical phenomenon, that is, when we have properties of scale invariance. These are methods that are mostly developed to determine critical exposures. However, their ability to evaluate The robustness of the macroscopic properties of a model in relation to the small disturbances of this model, this is really structural stability, we disturb the model itself, this can work independently of a strict scale variance. So, applications in statistical mechanics and equilibrium, in dynamic systems, applications also for many spatio-temporal phenomena, turbulence, transport phenomena, diffusion, for example, We can construct an operator of renormalization for all random steps and thus distinguish the Brownian movement from the abnormal diffusions, the fractal Brownian movements or the Levy waves. So we have a whole structure of the different partial derivative equations describing transport. Are the classes of universality you are talking about well known today or do we still have to look for them?

52:30 I would say that the ones that are known are still there in phenomena that we would not have observed yet. For example, for turbulence, the work is not finished at all. For diffusion, it starts. There are a lot of phenomena where it is not yet completely finished being exploited. Not to mention the ones where it is simply the obstacles that, for the moment, kill, kill the work. No, no, there is still a lot of... But the principle, in any case, of this change of gaze, this passage of... The study in phase spaces, in the space of models, was explicitly said and written by Fischer in the 70s and 80s, so it's not something new, but I think it's finally a kind of... These methods have been given much more than what they have been developed for and have a very, very large scope from this point of view. So a change in the notion of model finally shows what has been known for a long time in physics. That is to say that a model is something quite relative since it depends on the scale at which it is placed. Since the first theoretical studies, the centenary of this article by Einstein of 1905 this year, there has been Einstein's theory, there have been more macroscopic theories, there have been more microscopic theories. We know today how they are articulated, including quantitatively, that is to say that we control all the approximations well, for example for the example of diffusion. We have to have about a dozen models that we know how to articulate to each other, so the fact that a model is limited to be used in a given context, at a given scale, This is something that has been consciously perceived by physicists for a very long time. On the other hand, there is this tool that allows, in some cases, for a certain class of models and for a certain class of questions that we ask ourselves, which allows us to link these different models and finally put in the same bag all those that will correspond to the same type of behavior.

55:00 This is still something quite remarkable. Which is one of the morals of the history of this group of renormalization. Thank you. I'm a retired physicist. I've never used these things on my course, but on a personal level, let's say, the history of the length of the body, all that, that's always going to interest me, and... on the other hand... I may not have read the good books, but something that was enlightening for me is the sentence you said and that I did not know. To say that one replaces a physics of amplitudes with a physics of exponents opens a horizon for me and makes me understand everything. Or it brings together things that I did not understand under the same hat. I must say that there is no word, I heard it by Luciano Petroniero. It is easy to find even in books the right sentence that clarifies the matter, regardless of the formulas and all. Second thing, in classical physics, I would say... In general, we consider that there are privileged scales, as you said, where we can make small scales or large scales, so we will make one model or another, we will use such and such equations or such formulas to calculate the averages. So precisely, when we have a fractal phenomenon, there are no more scales. There are no more scales, they are gone. And practically, I found this problem. I'll give you a practical example. It's the migration of radionuclides, the waste that we're going to bury at 500 meters. I worked for six years, more than six years, in the Andes, among other things. And then there are two models for permeability. Since radionuclides are carried by water, so permeability. Or you have a medium for them, clay, for example, very little or very little for them at the moment. There, we have good equations, we know how to calculate, on one condition, nevertheless.

57:30 What is important is that we can well determine the physical coefficient. Yes, the example of the porous environments is extremely interesting because it allows to illustrate precisely two types of porous environments. One where we can homogenize it. It's very simple. The porous environment has characteristic size ports. In this case, it is enough to place oneself on a larger scale than this characteristic size and the middle becomes gray. In this case, we can define, for example, the coefficient of effective diffusion. Or the porous environment is fractal, there are holes at all scales, if we change scales, we will always see holes and planes, and in this case, all homogenization methods defining effective diffusion coefficients fail completely. In this case, we are obliged to build models based on the normalization of partial derivative equations describing transport. To describe this type of porous-fractal environment. So here the example is clear since we have the two that we can compare in a really... That's what I wanted to say. Second case, in practice, I'm talking about the engineer who does a modeling on a computer, etc. I'm not talking about thinkers who will think somewhere in the world to make a new model, etc. Certainly that Mr. Demarcy, who is a hydrogeologist, he knows all that. You have your second model, which is the fractured middle, a block of granite, small, large or even in the kilometrical order, it doesn't matter. We admit, in principle, that fractures follow a fractal distribution. What do we do in practice? What did I do in practice in my calculations? Since we did not know what scale to take, we said we will take a porous medium equivalent. We will do an average. And then, if we say, yes, but be careful, it's not very good because ... By the way, the results are much worse than compared to what should be found in the porous mediums. But we feel, in practice, by saying, ah yes, but you understand, and De Marcy himself told me, between four gods, to measure the physical coefficients on the ground to know the good physical coefficients that must be put in the model, all geologists know that it is horribly difficult to have a good precise value.

1:00:00 There are physical coefficients that must be included because we cannot measure a coefficient of a block of 1 km of a cube, of 1 km of a side or of 100 km of a side. No, but the problem is much worse in the case of manufactured mediums. It is that the physical coefficient itself is poorly defined since it will depend on the scale at which it is measured. No, normally we do not feel it. We are going to place it in a medium equivalent for them, or rather that it is large enough. And that's how we make the calculations. So, the question I ask you is, when we have fractured environments, what would be the right answer to, let's say, take something equivalent to the average? Precisely, we cannot take something equivalent. So, what would be possible is to describe the phenomenon by scale laws, by trying to catch the scale laws. I don't know much about literature, but there is a field that is quite close to that of growth, growth phenomena, where we have growths that can lead to a homogeneous material, not homogeneous enough to be described by an equivalent environment, or models of much more complex growth, a model like... Known under the name of Karnar-Paris-Isandl, where scale laws appear and where re-normalization methods have been applied for this model. So this is one of the standard examples. I don't want to hear you say that. If indeed we have fractures at all scales, we no longer have privileged scales allowing us to separate a microscopic level from a macroscopic level that we could look at in an equivalent way. So there are indeed methods of this kind. Of course, each time we have to adapt them to each type of figure, to each particular situation, This is precisely to overcome the difficulty of taking into account, in the same model, in the same study, the microscopic scales and the scales of macroscopic observation that, to take all the scales into consideration, the only way is precisely to look at this link between the scales and therefore to abandon any horizontal vision for a transverse vision inter-scale.

1:02:30 It is important to look simultaneously at several levels at the same time, to look at the links that they present to each other.