Prigogine and the Brussels/Austin Group Approach to Statistical Mechanics

0:00 Right. It's that to get the recording. Now, it's now recording. It's on the second track. So that is now recording. That's the sound level. And you can check the volume. It's on 2090. It actually goes up to 30. And that, you don't actually need to keep it as high as 30. and record perfectly well in a room like this on perhaps anything from about 18, 20 upwards. Maybe you want to leave it on. Obviously, use up the batteries that much more quickly. Is there a loudspeaker? No, no, no. You can only play it back through earphones. You can only play it back through earphones, which I've got. Now, the one... But on many of the new ones, there is a loudspeaker. So, in fact, the one I'll get, probably, will have a loudspeaker. But that records extremely well. I mean, the quality is superb. Even on mono, the quality is superb. and he talks. The advantage of having it on mono is that you can record twice as long, and then... Testing, testing. One, two, three. Thank you. What I want to do is basically try to give you an overview and some understanding of what it is for you and your co-workers are out there. That turns out to be quite difficult. But what I want to try to do is give you a good feel for that

2:30 and then raise what I think to be a couple of the really key questions that are left unanswered by the reporters that I think need to be addressed. And maybe I have time sometimes to address them or maybe they'll actually address them. I'm not sure which Let me start by trying to give some background and some motivation. This is a quote from a paper by Misra that gives kind of the key question that seems to be involved in those questions. What should we look at from the late 60s up to today? How to what extent the irreversible phenomena observed in the macroscopic domain can be reconcile with the reversible dynamical laws of classical or quantum mechanics is the fundamental question of statistical mechanics. So it's a variation on the problem of having in the trajectory picture a reversible set of dynamics, but in the statistical picture an irreversible set of dynamics and how do we somehow marry these two things together. Let me just briefly say something about conventional approach to statistical mechanics. And essentially there, if you think about classical mechanics, you work with trajectories, particle descriptions. We're interested in point representations of states on some appropriate face space. And almost all of the theories or laws that we derive from that sort of approach turn out to be time-reversible. But what we want, are interested in problems, say, that the air molecules in this room that are intractable on this approach, we switch to some kind of core-screening scheme, say, for example. When we do that, we find that our laws, from the core-screening perspective, are in general time irreversible. And of course, it's the irreversible behavior of these systems that we observe in nature, and thus this kind of question that Misra formulates here that seems to be a constant for the Brussels-Austin group. Well, in contrast to that typical approach, this is, in a nutshell, what the Brussels-Austin guys are trying to do. So instead of starting with a point description, we start with a distribution description. So the distribution, say, rho, over some appropriate state

5:00 space, can be imagined to be something like a probability for finding a set of molecules any other elements that you're interested in, with a set of coordinates, Q1, Q2, and so on, a set of momenta, P1, P2, and so on, at some time T, on whatever the relevant energy surface is. Now, if you start with this kind of description, it probably is not surprising to you that the dynamics are going to turn out to be time irreversible. And essentially what the Brussels Austin School wants to argue is that statistical mechanics should be modeled in this particular way and such models are irreducibly probabilistic and for them they want to take an additional step and claim that these probabilities are on the logical elements of the world. In other words, they should play a role in our fundamental explanations of physical phenomena and that we cannot reduce such things down to some kind of trajectory description. Okay, well, this approach really only applies to what they call unstable systems. That's this third category out here, which they call extreme trajectory instability. And in essence, what that means is that if you consider two neighboring points, and you look at the trajectories diverging from these two neighboring points, the divergence will be exponential. So this is something that can be fulfilled, say, by a case system, for example. And it turns out that at least in the older approaches that were developed from the early 60s through the mid-80s, case systems are the only system that that turns out their approaches try to be applicable for. Their newer approach that I want to talk some about today looks like it's more flexible and can handle a broader range of systems. these kinds of unstable systems that are of interest to them and these turn out to be the kinds of systems that seem to be typical of many statistical mechanics problems that are of interest as well. Okay let me say something about some of the core motivations at least as I as I understand their view coming from looking at a lot of their literature. The first one is one that many people have probably heard of it's a dissatisfaction with coarse graining approaches and oftentimes pretty gene as well as some others will argue for example that

7:30 coarse graining is subjective and that is what pretty gene at least means by that is that we only see irreversible behavior because of how we look at the system so instead of looking at the trajectory picture we choose to look at this coarse-grained average type picture, and it's only at that level that we see the irreversibility, and hence that's a kind of subjective view, because it's a choice that's dependent upon us. That's a kind of argument that Pritchie makes a lot. When he spoke here, he may have even mentioned that. I don't know. He seems to mention that an awful lot. But I don't take that to be the really important objection. It's just the one that seems to get a lot of play. I think the more important objection that they have is that Approaches like coarse graining ignore important dynamics of the system. So one example of that would be that coarse grained distributions don't make any distinction between points that belong to stable versus unstable manifolds. If you think back about this definition I gave you earlier about unstable systems, what you would like is for something about your distribution picture to pick out which of those manifolds are stable, which ones are unstable. course graining typically doesn't do that, so there's something missing in that description that's important to the physical phenomena that we're interested in. And that, I think, is really the more important complaint that the Brussels-Austin School has about the course graining picture. So they're dissatisfied with that. The second motivation has to do with their attitude towards the second law of thermodynamics. Now, typically, the second law of thermodynamics is formulated in an equilibrium context or some context near to equilibrium, right? And that's, we see that in our standards and system mechanics textbooks. What the Brussels-Austin guys are interested in is some sort of, I'll call it appropriate generalization of the second law that would be applicable to non-equilibrium systems as well. Now you might want to ask, do they have one? And the answer to that, the quick answer to that is no. They don't actually have one written down. There are some early attempts. You can find in a paper of Misera from 1978, for example, an attempt at something like this, but it turns out to be only valid at something near the thermodynamic equilibrium itself. But ultimately, they want to

10:00 find something like, say, Newton's law. They want to find a second law of thermodynamics that is as fundamental for statistical mechanics, as Newton's law is for classical mechanics. Okay, a third kind of motivation has to do with the sort of irreversibility that they're really interested in. And here I need to just briefly tell you a kind of distinction that they make, and some other people make this distinction as well, between what I call extrinsic irreversibility on the one hand and intrinsic irreversibility on the other. Extrinsic irreversibility is the kind of typical irreversibility that we think of when we think about open systems. So in this case, the environment for the system can be viewed as a necessary and sufficient condition to bring about the irreversibility. So if we didn't have the system open to the environment, for example, then the dynamics for the system itself would be completely reversible. But they're interested in a different kind of irreversibility, called intrinsic irreversibility. And in this kind of irreversibility, although one can think of it as not being able to get away from an environment, for example, perhaps the environment's necessary for something like heat dissipation, or maybe it's simply necessary for you to have a collection of interacting particles in the first place, the environment itself is not sufficient for the irreversible behavior. It's just the dynamics of the system itself leads to the irreversibility. And that's what they are very interested in, and that's what they want to try to model. And the last analogy is a kind of loose analogy with quantum mechanics. And that's in the sense that, I mean, there are some understandings, at least of quantum mechanics, where we want to say that the probabilistic description is complete. There isn't anything else to do on these interpretations. They give the probability. That's the complete set of information we have for specifying the system. And Pritagine wants to say something like the following. Why not, I mean, if that's good enough for some versions of quantum mechanics, why isn't a probabilistic description as being the most complete description we can have be good enough for statistical mechanics as well? Of course, one would want to say something like, well, it's because we believe that we can reduce a statistical mechanics case down to the dynamics of the individual trajectories,

12:30 and it's just because it's so complicated that we don't do that. and Pritagine wants to reject that kind of reductionist move, right? So this is a kind of loose analogy. He doesn't do a whole lot with it. It's just basically saying that if it's good enough for quantum mechanics to have a probabilistic description, that would be good enough for statistical mechanics as well. Okay, now these motivations, it turns out, remain fixed, pretty much, at least in Pritagine's kind of scheme. You can find most of them actually discussed in his 1962 book on statistical mechanics. And they seem to translate forward through the program year after year after year. Whereas the actual implementation of carrying out these motivations changes roughly with a half-life, I'd say, about every five years or something like that. So what I want to do now, after having surveyed at least kind of the motivations, is tell you a little bit about two of their earlier attempts, and just sort of sketch them for you and the problems with them very briefly, and then get to the newer stuff. Okay. First one I want to talk about is the similarity transformation approach. This is probably the one that is the most well-known, it's discussed a lot in British Jean's 1980 book, From Being to Becoming, Becoming the Being, which are where the time goes. And here's the basic idea. We wanted to find some kind of similarity transformation, a non-unitary similarity transformation, that's important for the mathematics to work out correctly, that basically allows us to to take these unstable systems and take our description in terms of trajectories and transform it into a description in terms of these probability distributions. So that's the idea. And then the hope was that by looking at this transformation it would illuminate the relationship between the time reversible dynamics in the trajectory description and this relationship of the time irreversible dynamics of the probability description. Okay, well, there are actually lots of problems, I only will mention two with this sort of approach. One is that it turns out that it's mathematically intractable. As a matter of fact, it still is the case that there's only been one model for which such a land was being constructed,

15:00 and that's for the Baker's transformation. And that's after a lot of work has been done on this program. And that's related actually to the second problem, which is that Hilbert's space turns out to be an inadequate mathematical space in which to try to carry out their program. Because the types of distributions they're interested in, the types of semi-group operators they're interested in, can't be adequately or rigorously defined. one needs something broader and in 1981 in an article by Arno Bohm in 1986 by another article by Bramwitz and they mentioned this kind of problem but it's not until about 1991 or 1992 that they began to turn to looking to something broader than the Gilbert space and that's the approach that I want to spend time talking about today. The other one that's maybe familiar to some is what's called sub-dynamics. And sub-dynamics when you first, at least from the way I'm going to describe it now, sounds too easy. The idea is you find some projection operator that would project the thermodynamic behavior and the anti-thermodynamic or non-thermodynamic behavior into two separate spaces. So it simply breaks apart those types of dynamics, and then the idea is to simply enumerate the conditions under which the non-thermodynamic subspace makes no contribution to the evolution. Now, that sounds too easy, but perhaps it's not actually too easy in that sense, but the real problem that I want to simply mention here is that sub-dynamics, as first conceived by the British Indian School was a kind of independent program, an independent approach to trying to get at these questions about irreversible behavior. But it turns out that the sub-dynamics program is dependent upon which particular formulation of their statistical mechanics they're giving you. For example, in the lambda transformation approach, as they modify that particular approach, it turns out that their sub-dynamics approach has to also undergo modifications. Then again when they switch to rigged Hilbert space, they then begin to readjust their sub-dynamics again. So it doesn't appear to be an independent way of getting at the question. It appears to be somehow dependent upon their kind of larger view of what's the appropriate

17:30 way to characterize this irreversibility. And actually the sub-dynamics approach gets very little discussion in their literature from the late 80s on. Could I just ask if you could give the definition for a rig to this place? I will be giving that in a little bit, yes. A very sketchy one, but we can talk more about that. Okay, well, in the new approach, the jargon that they use is this term here, large Poincare systems. Okay, so we need to understand what they mean by large Poincare systems in order to understand what it is that they're trying to do. And the key to understanding the large Poincare systems is the phenomenon of resonance. Okay, so here's a standard interaction Hamiltonian with a non-interacting part and then some small parameter lambda times the interaction term. Okay, now Poincare did a lot of work on these sorts of questions, and one of the questions are questions about these kinds of Hamiltonians, he asked was in what sense can we find a transformation that can take this Hamiltonian and transform it into one where in essence this interaction term is turned off and we just have a set of equations that describe the particles independent of their interactions that are just composed of this term alone. And what he did was looked at the standard perturbation analysis and the standard following where you have these denominators whenever these denominators approach zero that marks out where resonance phenomena is now what Poincaré discovered was that if you have too many of these resonances then it blocks this kind of transformation strategy that he had in mind and if that's the case then you can't have at least in Poincaré's sense a non you can't have an integrable Hamiltonian, you have to have something that you would call non-integrable in the sense that I can't find a transformation that can allow me to look at this Hamiltonian now as a non-interacting Hamiltonian. Okay, well, that's not too bad because we actually, due to the KAM theorem, can find

20:00 out conditions under which we have quasi-periodic orbits and the conditions under which those indicate are those constitute the majority of the motions realized in the face space. And we have techniques for handling things like the periodic and quasi-periodic orbits. But the kind of case that Pritigine and his co-workers are interested in is what they call a Poincaré catastrophe, named after Poincaré of course. Okay, so this is where some of the definitions for large Poincaré systems come in. One of them is that the system is large. In other words, we're going to work in the limit where L cubed, if you imagine a box, goes to infinity. The number of particles can stay finite or infinite. If they're infinite in the sense that the number of particles, the ratio of the number of particles to L-cubed stays finite than we're in the equilibrium limits. So, that's one condition is that we have to have a large system, in a sense of size. Another one is that the number of degrees of freedom in these systems tends to infinity, okay? A simple example of something like that would be a finite number of charges magnetic field has an infinite number of degrees of freedom with it and there are several other sorts of examples you can give so these are not the idea of looking for these kinds of systems isn't too far-fetched and what happens when we have an infinite number of degrees of freedom is the eigenfrequencies characterizing the model become continuous free functions of the wavelength And the denominators in the perturbation expansion approach zero arbitrarily. We have to switch from sums to integrals. And under these conditions, the KM theorem doesn't guarantee us anything about quasi-periodic orbits. We actually get a lot of trajectories that look like Brownian motion instead. And that turns out to play an important role in their understanding of what's happening in these kinds of systems. Okay, so basically, the conditions are that you want a large system, L cube going to infinity.

22:30 Something that they characterize as being persistent interactions, which means there are no asymptotic in and out states, the particles are never free of interactions with each other. that gives you, in some cases, an infinity of resonances. Particle number N is somehow too large for us to deal with it in terms of our classical mechanics description. So if one's interested, say, in doing one billiard ball on a billiard table, then you wouldn't want to go to this elaborate scheme. But if you're interested in gas molecules in the room, for example, then the wood wasn't used to . And these systems are not in the Planck Erase since. What Christian company claim that they've done is they've found an approach to, quote unquote, integrate such systems. And by that, what they mean is that they can actually find a complete set of eigenvalues and eigenvalues At this point, people typically want to say something like, well, okay, what's their fundamental equation? Well, there isn't actually a fundamental equation, because the business of going about and finding eigenvalues and eigenvectors has to be tied to the particular model that you're looking at. And that will lead us in a couple of minutes to this question about recovered space. First, let me try to give some physical motivations for their need, what they claim is the need for distributions as fundamental ontological elements for descriptions. So, okay, the first is this point about persistent interactions. So, okay, consider something simple like molecules of gas in this room, they're all bouncing around collisions with each other and there's never a moment when any of them are not having some kind of interaction with each other. Okay, so this gives rise to a continuous set of resonances in the large plant-free system and resonances basically allow for energy to be transferred in non-local or one might say non-linear ways. This becomes much clearer if one, for example, looks at the these sorts of systems, and then you see if you do a mode analysis that all the modes

25:00 seem to have connections with each other. So you can have transfers from the highest order modes to the lowest order modes. And that leads to a kind of dissipation of the energy that turns out to be very important to their program. So what happens then basically persistent interactions because you have all these channels for energy transfer available to you through the resonances causes the particles to lose energy and their ordered patterns and basically that's a kind of diffusion process and the system's energy ends up being damped out and eventually get to the equilibrium state and that's not all and this basically goes under the kind of description that they call a correlation dynamics, some early discussions of which also appear in the book. But there's also some other developments by Claude George in 72 and 73, and then further refinements. Although the correlation dynamics really plays more of a key role in their discussions in their recent work than it does in their earlier work, interestingly enough. So aside from some of these kinds of diffusion processes that allow for energy, basically, to be dumped from higher orders down to lower orders, you also have a growth of correlations. Okay? So you can imagine just a simple situation, right, where you start with a set of particles in a gas, right? Typical kind of cartoon physics situation. I'm interacting and we turn on the system and then each particle collides with another one and you get kind of first order correlation and the second order correlation is another particle collides with another one and so on. In terms of something like say spatial correlations, right? Spatial correlations arise in a relatively simple way due to the fact that things like the relative position vector, R2 minus R1 between any two particles, doesn't interfere with equal likelihood. And it turns out that as these spatial correlations grow, they begin to influence the distribution

27:30 and the dynamics of the particles. and after a while you can build up kind of long-range correlations and that happens basically because particles in one area of the gas as they undergo collisions and going correlations eventually another part of the gas and they carry with them in essence the memory right of those correlations with them but that also allows for these kinds of non-equilibrium modes of energy transfer to pass throughout the entire gas. Okay, so you have a set of non-local effects that take place. You also have collective effects. And there are many examples of collective effects, which basically are typically due to long-range forces, like electromagnetism, gravity, and spatial correlations, if one's thinking simply of a gas of uncharged particles. A dense gas, interactions are frequent. Spatial correlations build up very rapidly. They couple the particles with many, you know, any individual particle with many other particles on the gas in terms of their interactions. And these kinds of effects can be quite important. they can lead to collective behavior that can help particles come into some kind of current structures we see this for example in things like turbulence shock waves device shielding space charge particularly in these last two for example but actually all of them become quite important say in plasma physics for example there's one reason why it's so difficult to confine plasmus to make them do what we want to be able to do to produce a fusion reactor. So these effects are actually important physical effects. And collective effects, though, only really appear in our analyses when we're investigating aggregate behavior. They don't appear at the level of individual trajectories when we're doing a description in terms of individual point trajectories on some state space. So it's some kind of statistical effect, some kind of aggregate effect. Christian likes to use the term, it's a non-Newtonian effect,

30:00 to try to point out the difference between his approach and trying to do a kind of standard-gram trajectory description. And this, they would argue, implies that a distribution picture is the appropriate picture because there's a kind of ontological status to the distributions. actually having physical effects that at some point swamp what we would take to be the effects through the individual particles and trajectories and their collisions. But that's not all. There's also, it's like these cheap TV commercials, right? Don't buy in because there's more. There's a thing called correlation flow as well out to be an important physical effect. Okay, so as the particles interact right, correlations develop. The correlations grow from lower order to higher order. Long-range correlations develop as the particles separate from one another and distribute themselves throughout the gas, carrying the memory right of their prior interactions with them. So over short time scales, the growing order of correlations tends to dominate in the physics, but on longer time scales, it's the collective effects that become more important in their analysis. And what happens is, or at least one way to look at this, is that you begin, for example, with very short-range binary type correlations, right? One particle with another particle gives you a binary, when it interacts, gives you a binary correlation. And these binary correlations remain finite, but not zero. But due to energy dissipation, the ternary types of correlations, or third-order correlations, they build up, but then they dissipate throughout the gas. Same thing for fourth-order correlations. correlations, they build up, dissipate throughout the gas, fifth order, and so on, such that at a particular point in time, you end up with these variously ordered non-equilibrium correlations that have distributed themselves throughout the gas in a kind of way where you have a very broad distribution of basically multiply incoherent correlations, all transferring

32:30 all these various modes that eventually lead out to the damping down to equilibrium. And you have only the equilibrium correlations left over, which are the binary correlations. Now, in one sense, it sounds like that might be promising for being able to nail down the arrow of time and irreversibility. But it turns out, as I'll say a few words a little bit later, that's not quite the whole story. Okay, so it's those kinds of considerations that drive them, I believe, to wanting to say that a distribution description is the appropriate description of these kinds of systems. Because over time, the correlation dynamics and the collective effects become more important to the global evolution of the system than the trajectory dynamics do. So So in that sense, that's the natural description, right? But one can still ask a question about this approach, and that is, are these distributions really only convenient for calculation purposes? Or are they irreducible in the sense that, yeah, okay, we have all these correlation dynamics going on, but can't we find some way to somehow take those correlation dynamics and and reduce them back down to the trajectory description. Because after all, one doesn't get correlations unless you have two individual particles on two distinct trajectories colliding with each other, right, that makes a correlation and building up that way. So it sounds like perhaps it still may be possible to reduce all of this stuff back down to the trajectory dynamics. And that's the kind of worry that one might have. And what Pradejian and his co-workers are trying to do in their argumentation correlation dynamics I just sketched for you is to show that no actually these things are ontologically fundamental and we can't reduce them down to the trajectory picture okay well let's take it for the moment that they're correct and at least in the sense that we need a distribution picture okay that drives for an extended mathematical space why well for one reason distribution

35:00 functions particularly singular distributions we want to think about trying to model individual trajectories are not well defined on Hilbert space okay so think of the derived delta function it's an element that doesn't exist on Hilbert space although we use it all the time but it actually lives in a distribution space the evolution operators on Hilbert space are elements of groups. What we really want are semi-group operators. With the group operators in Hilbert space, the best we can hope for is to do some kind of extrinsic irreversibility, where we have an open system, and then we can derive some results about city groups in that kind of context. But if you're interested in intrinsic irreversibility, that we can't get out of So that seems to suggest that all of the work in the 70s and 80s was being done completely in the wrong mathematical space. Additionally, if one is interested in phenomena like decay or scattering and so on, we also need eigenfunctions that have imaginary eigenvalues to produce the decay phenomena. Okay, well, those also can't be well defined on Hilbert space. They also live in a distribution space, and that leads one to having to think about these more generalized mathematical structures like rig spaces, of which the rig-Tilbert space is typically the favored one. And in the rig-Tilbert space, we can do things like have distributions. We can have semi-groups. They're all well-defined. We can do intrinsic time asymmetric modeling in a very natural way. In particular, we can actually construct a complete set of eigenvectors in rig-Hilbert space for these kinds of large co-operative systems. This cannot be done in Hilbert space because the, well, the operators have no eigenvectors on Hilbert space, the ones that are of interest. first. Okay, well, what is a rigged Hilbert space? Well, let me give you a very brief description of a rigged Hilbert space, basically by kind of sketching an efficient method for how one constructs rigged Hilbert space. So I'm going to start with an abstract linear

37:30 linear product space, linear scalar product space. And what we're going to do is equip it with two topologies. Okay, the first one will be our standard norm topology. That gives us a separable Hilbert space. That's the one that we're all fairly used to. But we're also going to use a topology, I'll call it tau sub phi, that's based on the algebra of operators corresponding to the observables for the system that I'm interested in. So So I'm going to construct a norm, actually it's going to be a countable sequence of norms, using these operators, and that's going to allow me to be able to find yet another space. And then I sort of get two for the price of one because there's a dual topology to tau sub phi, which gives me the corresponding dual space or distribution space, and I just have to recall that Hilbert space is self-dual, so you get nothing extra for that. And the result is a kind of triplet of spaces, sometimes called the Galphan triple, since Galphan and his collaborators were the first ones to discover and construct these kinds of spaces. And it looks like this, where phi here is dense in the Hilbert space with respect to the Hilbert space topology. in phi x, which is the distribution space with respect to its topology. Now, what that means is that it allows me to have actually more freedom to model various types of physical interactions using elements in phi, which are typically very smooth functions, or elements in phi cross, which gives me access to things like distributions or these so-called improper eigenvectors, which have complex eigenvalues. And just a couple of interesting facts. Rigober space provides you actually a rigorous justification for the Dirac notation. And that was discussed in the 60s by articles in articles by Antoine, for example. And Roberts and Arnold Baum also discusses this. And this is where one actually eventually finds a rigorous justification for the Dirac formalism, which, of course, physicists have been using for a long time without that. And also, Ray Tilbert Space provides a kind of natural framework for time symmetry breaking

40:00 and semi-group types of evolutions. So here's a place where you actually find a home for intrinsic irreversibility, and intrinsic irreversibility is what you're really interested in, which is what these guys are interested in. Okay. So let me say some things about semi-group operators and irreversibility. Okay, the Brussels-Austin School has a particular kind of time-ordering rule that they use. And it's basically that excitations have to take place before de-excitations. So basically, you want to demand that excitations, which might be related to preparations, for example, are somehow identified as those things which take place before t equals zero, and then the excitations, which might be possible preludes to registrations, are identified as taking place after t equals zero. So I have a kind of time-ordering rule that allows me to be able to pick out the order in which certain types of processes occur, and then the question is, what can I do from the standpoint of my evolution operators to be able to model those kinds of phenomena? Okay, well, what you want to do is take basically your standard evolution operator that we might define for Hilbert space, which we know is a group operator, okay, typically it's U-adjoint, U-dacker, that's fine. It's going to be something like e to the minus ht, right? And what we want to do is we want to demand that we can extend that operator to this space phi x where the distributions are that we want to be able to describe the evolution for such that they're continuous in the topology tau x. In other words, that's a requirement that my application of the operator doesn't leave me outside my space, right? At this point, something quite interesting happens to this u-dagger, which is a group in Hilbert space. It turns out that it breaks apart and becomes a semi-group in the distribution space. What happens is, because of the continuity requirement, I can only define the extension of u-dagger to phi x over two time ranges that I had to break up, t greater than zero

42:30 and t less than zero. And for the T greater than zero, the Brussels-Austin group identifies its temporal direction as basically carrying states into the future. In other words, it models the approach to equilibrium into the future. Okay, that's good. That's what I want. But for the T less than zero group, they identified that direction as carrying states in the direction of the past. In other words, it seems to describe antithermodynamic behavior. In other words, systems that are reaching equilibrium, evolving equilibrium from the future into our past. Okay, that's what we don't want to see, right? And this kind of breakup of the unitary operator into two separate group operators comes strictly out of this requirement of continuity. We can't even define an inverse for these operators. That's how we know that they now form only a semi-group. Now, what happens at this point when the Brussels-Austin group simply tells you that, well, okay, we don't observe this anti-thermodynamic behavior, right? So, what we'll do is we'll select u dagger extended to phi x for t greater than a equal to zero because we don't see systems going from equilibrium, you know, to equilibrium from the future into the past. We see when this future is erected, right? So, in that sense, this is the physically relevant semi-proof. Okay. That, in a kind of really quick nutshell, is their program. What I want to do with the rest of my time is give a kind of assessment of what I think is crucial questions or points about this. First, in terms of the thermodynamic arrow of time, okay, so, right, we're looking for something like physical mechanisms in these systems that explain or the causal features underline the approach to equilibrium. Okay, well, the thermodynamic arrow seems to be governed by things like diffusion, growth correlations, collective effects, and so on, that all translate back to these Pocahry resonances. And the Rig-Tilbert space eigenvector expansion describes

45:00 these mechanisms and does something, for example, like it picks out the difference between stable and unstable manifolds, for example, whereas coarse graining approaches don't pick out those kinds of things. Okay, and of course the heart of of things is this new dagger extended to phi x for t greater than equal to zero, because that's the one that's carrying our states from the past into the future, leading us to equilibrium in the temporal direction that we want. But one can ask, I think, a very important question, what I call here the key question, is there a principal reason for rejecting a t less than zero semiproof? other than just simply saying, well, we don't observe those phenomena. Because I can imagine a couple of possibilities, right? One is that the formalism is just merely descriptive of irreversible behavior. But it's not really explanatory, right? Because if I appeal to observations, right, to throw away the t less than zero semigroup, then there's no sense in which I can give you an explanation, why I threw that semi-group away. So the observations are still there, but I don't have an explanation for the observations. And ultimately, that's what I want if I'm privy sheet. I'm looking for this kind of golden key. It doesn't look like I have it because it looks like at the moment I've got a description, a powerful description. It does certain illuminating features for me, but it doesn't give me an explanation. The other possibility, though, is that the theory actually predicts anti-thermodynamic behavior. Okay, well that would be a bad thing. I want my theory to predict thermodynamic behavior and explain that, right? But I have this other semi-group and I don't know what to do with it other than to say my observations don't agree with this T less than zero semi-group. And one sort of additional, I don't know, I call it here a difficulty. It's hard for me to decide if it's a difficulty or not. But if you look at similar work by Arno Bohm on scattering theory, which he and his co-workers have been working on for quite a while, they also do this work in Rick Hilbert's space. They also get a kind of splitting over the time domain. They get a separate semigroup for t less than zero and for t greater than zero.

47:30 But in his case, both semigroups are oriented in the same time direction. forward directed. However, as a note here, Bohm has assumed a more restrictive time ordering rule. And basically it's one that's tied very closely to laboratory operations. So that suggests that it's not immediately obvious at all how one could generalize that rule without losing the kind of restrictions that give you these two semi groups going in the same time direction and that's a kind of open problem at the moment that I don't think has been answered at least I haven't seen anybody answering that let me say something about the so-called quote-unquote collapse of trajectories that's the current phrase that these guys are using an unfortunate choice so I'll try to explain First, as a kind of warning, the Brussels-Austin group actually never meant to argue that there were no such thing as trajectories in unstable systems, or that there's a total absence of such. Now, I know from reading many of their papers, it's somewhat easy to conclude that that's exactly what they were saying. But that has a lot to do with the fact as to how poorly they write about these sort of things. So even Balasquiu, for example, was confused about this point. And Pritigine told me one time, as Zavi said, I can understand why people think I say there are no such thing as trajectories. Well, if your native language is English, like me, and you're looking at people whose native language is not English, trying to describe what they're trying to describe, you can begin to understand why there's a confusion on that point. They don't quite get it right. Although, in the most recent literature by these guys, now takes pains to say, well, some people have accused me of saying that there are no such thing as trajectories, but that's not what I'm saying. What they have in mind is that there's a particular type of trajectory which is inconsistent with these types of large-point array systems, these unstable systems. And that's the kind that we get from our typical solutions to Hamilton's equations that have nice features like being infinitely differentiable, for example, these kinds of smooth trajectories. And I have a kind of quick example that I won't explain too much about other than to

50:00 say that if one looks, for example, if you try to consider a case of an exact trajectory for a system, and you want to look at something like the momentum for that and analyze it in one of these large quant-grace systems, it turns out that in this particular limit, The resonances in the denominator give non-vanishing contributions to the dynamics, even for short-range interactions. So you're going to build up these kinds of long-range correlations and other sorts of correlation dynamics and so on. And if one looks, say, for example, in the thermodynamic limit, this kind of momentum that I've built up out of the individual trajectory description ends up diverging, okay? So that's one form of argument they try to make to say that these kinds of smooth trajectories are not physically admissible. In other words, they're somehow inconsistent, say, with the thermodynamic limit in large quantum rate systems because we get these divergences out of the trajectories. What's happening? Well, they want to argue that physically what's happening is the dynamical effects of these continuous resonances basically converts our smooth trajectories into Brownian motion. And that happens rather rapidly, and that's actually what they mean by collapse of trajectory. Not that somehow trajectories disappear from the system, but a particular type of trajectory breaks down at a certain point and becomes a Brownian motion type trajectory. Okay, so that's really what they mean, but I challenge you to find them to say that clearly anywhere in the literature. Okay, a couple of other points, in cases where there are relatively few particles, one can show that their approach recovers standard results in classical mechanics. So if you wanted to do a billiard ball on a billiard table with their distribution approach, you could do that and you can actually get the results correct. But that's a lot of work and one doesn't want to do that, right? In a case where you have a large number of particles, but you don't fulfill the conditions for a large Poincare system, then they can recover other kinds of standard results from statistical mechanics. They can get Fokker-Planck equations, they can get Master equations, they can get Boltzmann's

52:30 equations, and so on. That's fine and dandy. However, in the case of LPS systems, these large Poincare systems, and cases like thermostatic dynamic limit, which is a special case of the large bond-grade systems, they have both analytic and numeric results to show that if I try to do the approach using just trajectories and building up course grading, and I try to do, in other words, kind of attempt at reducing the distributions down to the trajectory picture compared to their approach, then we get different results. And one way to see this is that it's very interesting if you take any subset of the total number of particles in for the system, no matter how large, as long as it is still a subset of the total system, then these dissipative effects disappear, so you lose some of the key dynamics that seems to be important in driving the system to equilibrium. Okay, so that seems to be important. That looks like at least their formalism is pointing out, or at least pointing us to think about some other mechanisms that we typically don't think about, perhaps, when we're doing our statistical mechanics. So at least from that standpoint, it's useful. The last sort of critical comment I want to make has to do with their notion of probability. Okay, so if distributions are now going to be the fundamental explanatory elements, right, we still want to know what's the nature of this kind of distribution. What's the nature of the probability? Right. So you'll find them in their literature interpreting this to mean that probability, right, is fundamental to these kinds of systems, fundamental to our picture of dynamics now. merely epistemic as in the case of coarse-graining and so on. But the question is what kind of probability is it? Well, it's not a kind of probability due to indeterminism, like in a kind of collapse picture, say, von Neumann or GRW or something like that. That, of course, would be a case where the indeterminism is ontological due to some kind of collapse phenomenon, right? But that's not what they're talking about. On the other hand, it's not like typical Hamiltonian chaos, where we know the underlying

55:00 dynamics is deterministic, but the outcome is unpredictable, right, and we can describe it with a kind of Karl-Mogorov major and so on. That, we would typically say, is only a kind of epistemological construal for the problem of the system. And Pradegine ends up saying something like the following. What is emerging is an, quote, intermediate description, that, like, somewhere between a deterministic world with an arbitrary world of pure chance. That's as clear as they get, and I simply note that it's insufficient to characterize this probability as Kolmogorov, because that doesn't tell us whether we're ultimately talking about deterministic processes or we're talking about some kind of indeterministic process. And one of Prudegine's ideas is that probability is a kind of fundamental or ontological picture of our classical world, just like it So it better not be reducible to some kind of underlying deterministic process, or it better be something that somehow the deterministic process generates in a way that it takes on a kind of independent ontological status. But I need to be able to describe that in some way to clarify and understand what those kinds of words mean, and that they have not done yet. So, my final slide is sort of the upshot. Well, they need a physically principled distinction between thermodynamic and anti-thermodynamic behavior. Basically, we need some principled way to argue for throwing away the t less than 0 semigroups. I also think that they need some sort of clarification about what this probability is supposed to If it really is an ontological out of the classical world, we need to be able to clarify it in a way that we can understand why is it different from the Hamiltonian chaos case or the quantum mechanical collapse case. So I think those are two outstanding questions that need to be answered in this approach if it's going to be able to deliver ultimately the promise of an explanation for irreversible behavior in the era of time. Now, on the other hand, the formalism does seem to be quite powerful in illuminating various mechanisms that we possibly haven't thought about as much as we should. And it also has a kind of feature that I haven't said much about here, but that it does provide a kind of natural way for being able to include both classical and quantum mechanics within

57:30 the same formalism. That is, provided I can properly cash out this claim about probability such that I and make it applicable to most of the classical and quantum worlds. If I can do that, then I can come up with a kind of unified description for both realms. That would be very nice. That would be very nice. But one needs to answer this kind of question before one can make a kind of strong claim about being able to unify classical, statistical, and quantum dynamics in one form. Thank you. Well, if you could make a clear rundown of the very complex topics, a little kind of a little silice coming out of it. But I suppose if you sort of read primitively some of these acolytes, they tend to, they have a formal stuff which you describe, but they also have a kind of sort of philosophical message running through them. I mean, the very type of figurative, the popular point of being in the coming, I mean, it seems to, I think, in some sense, everything is sort of a process. I mean, you think of the kind of symmetrical thing, something, like painting, swinging backwards, or whatever, an example of being, where that's the self-dynamic processes, the original processes, are something that's becoming, But in the slightly more extravagant moments, figuratively seems to think that the world is really a world of becoming. I mean, sort of quotes. People like Hague or Perks. He likes white hair. I mean, these are the same before. I mean, none of the class were just totally ignored in the analytic condition, because they don't really understand what they're all about. They think that's just kind of intuitive, kind of making stuff. And clearly, it seems to me to be, if you want to talk about philosophy more, is to try and say, well, look, let's have all this data mathematics in how we have the name of it, but let's use it to really make sense of what he looked like as the great intuitions of our time, which he has associated with his own works. and so forth, but he was one of the favourite people of the film. Now, there are two comments I could make. First of all, there's a sense of it, I think the British in his brilliant philosopher when he came to talk to the signal car, it was a far bit disappointing experience,

1:00:00 because he just went on, sort of, mainly about the sub-dynamics of some years ago, of early start of the programme. And if you try to sort of thin it down on the issue of the notion of the probability of people, for example, you then sort of start waving his hand, you know, and then start talking wildly, as I thought, about people like Birx, for example. So my impression was, you have really studied that philosophy. It's like the whitehead, even to me. I mean, whatever you may think of whitehead, it's hard to know exactly what you're talking about. But there is a sort of propensity about whitehead, right? I mean, in any way it's not from what about it. It might have slogans to be picked up, processed, and that kind of thing. It's not a real philosophy of physics. And that's what I find disappointing, because lots of physics that I would understand is to take the hard science and the mathematics and all that sort of thing, and they're kind of related to a real understanding of certain philosophical conditions. It might be due to the white element, but you don't do so much of that in the LSE, but I mean, there could be a legitimate thing to do to a religious process process, but you can kind of match that on the traditional. So, it seems to me that Brigadier talks in a way, it might be accentuating this process theory, but he also talks about this complementarity to the description. And this, again, is a sort of curious thing because the mentality, I guess, of both descriptors are legitimate, that you can't use it simultaneously, right? Well, if you really dig into a regime, it doesn't seem to mean that's not the handiest attitude to the trajectory of description, which is sort of despising this kind of something that comes out of Greek philosophy that's going back on many different ways of being. And so I don't think he talks about this cost of mentality, and particularly in his time operator, an entropy operator, which doesn't even use the believable operation. And he sees that as sort of corresponding to the Heisenberg uncertainty, although now

1:02:30 in the world of this classical description. So my first question is, how do you see these so-called uncertainty principles ? Yeah, yeah. But you seem to think very important. So whether you agree with me, that's brilliant. You believe they can't . I think, yeah, so we can do this two ways. you can talk about TL not equal to zero, or ML, not equal to zero. Okay, so what this means is that M is a kind of micro-entropy operator. And this is developed by misreads of others in, like, 1978 through 82. And then it gets related to how one can build a kind of time operator, which tells you, like, the age of the system and so on. One's thinking of seeding everything within Kupemann's formalism, for example. Then you can very easily show that you don't get commutation between these kinds of operators and the legal operator. And then the claim from that is that, well, all other important operators have to be simply the multiplications of factors over phase space. And those would commute with L. But part of the claim is that all of the other important operators have to be of that form. And that may not actually be true. Nobody has done any kind of work to show or give a theorem, for example, that says that all my other operators are of interest. Well, you see, I mean, that was the group I mean, it's operated by a kind of things like the . And they're operated to a sort of technical sense, that they're not just the original human space, the distribution of function. Now, what it seems to want to do is to be fairly trying to define a Lyapunov value for the function that could change on the product. Yeah. That's how I may derive something like this. Well, that's right. If you stick with the Kuhlmann space, the space of energy, being the operator,

1:05:00 the famous theorem, I think, due to Pankai, for example, I think the existence of Pankai recurrence is that you can't define a Lyapunov value as a function defined over the space space. That's individuals of Kuhlmann space. So if you want to get a liapura variable, you have to go to something which is not defined as a function, but it defines itself as its operator, which ends, the thing which doesn't commute without. So you're sort of taking yourself out of the space of functions in order to create this liapura variable, which you never want to equate with the entropy of the entropy, right, so forth. Right. But once he came, it seems to be sort of, it's more a kind of descriptive exercise. You know, he says, well, in terms of dynamics, we do have, therefore, not valuable. Maybe the entropy of those all this. You know, of course, it's going to be increased positively in time. We know from the part-hari recurrents of the factors of mellows, and what we can't have such a function defined just over the space. So we've now got to look at a mathematical description which will enable us to produce the atomic value. And so it seems there to say, well, without the description of that, it's going to be anything. Well, the anything is not going to be anything. Right. So all that side talk about the analogy with Heisenberg, and certainly the principle of what we can, seems to be totally misguided. I mean, they have a long philosophical discussion about what Heisenberg would actually do. anything like what these connections mean. And then once you start to define Then, of course, it's not the old problem. It always seems to resuscitate itself in all these discussions, particularly everybody else in the .. You always have the anti-therapodonetic solution. This is what you clearly commented on. And then you throw those away, because they don't describe what you have to observe in the voluntary. But that's not a kind of principled way of getting rid of it. a cheat to regard that with any sort of foundation of statistical behavior. So it's going to be the end value of the table. It still doesn't really solve the problem of the arrow of time. Well, I was thinking about behavior. That's correct. That's correct. I mean, they don't, this description, for example, does only that describe.

1:07:30 Gives you a kind of formalism. It does give you a kind of advantage in that you can write down a kind of time operator. That's actually quite interesting that one can write down a time operator. It's not clear how deep this time operator really is because they can give me a kind of age for the system, right? It's not like I can write down some expression over here that has some factor of h bar or something like that that's telling me how precise I can make my time measurement versus whatever this is supposed to be described. It doesn't work that way at all. Well, it's quite unlike the Heisenberg, right? Yeah. It was . Right. And that's why when Misra talks about this, he talks only in terms of complementarity in the sense that the description due to this and the description due to this can't somehow be used at the same time. But that requires certain assumptions about the operator that there are no proofs for, aside from the fact It looks like you have somehow illicitly mixed spaces. As part of my complaint that they needed to be in a recuperate space all along, they're beginning to develop some ideas about what a time operator might be like in that perspective. And we'll see how that turns out. But I doubt if they can get the right kind of complementarity relation, or even any sort of uncertainty relation that makes any kind of sense. My final comment is that although they claim that you are tragically dead, you don't, but I think you can nicely say you're tragically down there at the back, and then they're all thrown in the type of tragically, not differential. And I think that does seem right, because they start the whole thing with just classical called Hamiltonian Dynamics in relation to processes to domination of the world. Look how it's the sort of problem that the point that it gave. It all looks like standard across chemistry dynamics, where the beginning of the book looked like that. Almost every article that it looked like sort of standard undergraduate courses in Hamiltonian mechanics in molecular consummation. and then from that he sort of builds up, I mean having got the distribution sense in that space he notices that you get the trajectories, you mean singular distribution, they've got to enlarge the space we want to work in the direction of the recruitment type of form, and then he goes to the room for the space

1:10:00 and so on. At the back of all that seems to be the thought that yes, there are these singular distributions What are these secret distributions? They are old friends. We do love the tragic truths. That's another way of describing it, that it's vastly more complicated than doing it if you use this extended proof of the . In the old kind of . So the tragedy is really hanging around the background all the time. And the idea that you've somehow got rid of the tragedy. Well, I mean, maybe it's a sort of pulling Yeah. I don't know about the tragedy. It is a tragedy-based thing, although the description of the tragedy is sort of using a vast sledgehammer to do what I can do like for simply a simple system, which is amazing. Or you could do it. You could do it. You could do it. it would be a ridiculous kind of way to proceed. But I'm sure other people have thought, and of course the only question what probability could be . Well, I have a question that's kind of related to this. You mentioned particle interactions building up and building up and so on. Maybe a way of approaching whether the probability can be objective or subjective is to look at how they treat these sort of correlations. What's the zero point? How do you pick a certain point at which the correlations start building up? For real world systems, that's pretty hard to do, yeah. I mean, if you can successfully model no matter when you start, then these are subjective problems. Right. I mean, the sort of examples that they do are the kind of things that we start out with our textbooks. We build a chamber with gas molecules. They don't interact yet. And you sort of click a switch and turn on. And that leads to two kinds of questions. One is your kind of question of, well, okay, so I can build a formalism that describes that nicely. Now, if I want to translate that into something like the real world where I, it seems that I don't ever start out in such a situation, then how applicable is it? That's something that's totally unaddressed by them, as far as I can tell.

1:12:30 So there's no answer that I know of within the Perugian approach for your kind of question. of sort of promise that well if I have this formalism and I can demonstrate it on these kind of toy problems, then I can also use it on more realistic problems. And they have, they sort of have a bag of more realistic problems, particularly to talk about scattering in terms of Friedrich models and so on, which are fairly realistic models that we use for the discussion of scattering phenomena. But if I want to talk about and of course yeah I mean I can build up a set of gas interactions you know molecules for example in terms of my analysis of individual scattering events using the Friedrich model quite but that still doesn't answer for me the question of what do I do with a system that's already got some kind of structure in it and then I see that structure dissipate away where I have a a kind of equilibrium distribution, right? Nor, this is the other question that possibly is somewhat related to your question. It's not clear, because I mentioned partway through the talk that with these kinds of correlations building up and then dissipating away, that it looks like they might give some kind of promise for being able to sort of anchor the flow of entropy, right? Because it looks like it's a kind of one-way process. but it turns out that there's no kind of no-go theorem to say or impossibility theorem to say that that can't somehow reverse itself but what you can do is describe in essence kind of growth of an entropy barrier that does have an exponential form to it so very quickly it looks like the kind of energy that's required to overcome that turn around the requirements are very steep but kind of impossibility argument that you can give. It's more of like a kind of standard argument we always give that, you know, I mean, the probability that all the air molecules in this room would rush out that window is like 10 to the minus 65 or something like that, right? Okay, well, I mean, it's the same sort of argument that it seems is the best you you can do in this case and so that doesn't look like that's sufficient to be able to nail down from a physical perspective this arrow of time it looks like it can give you it supplies some more of the answer or perhaps not all of

1:15:00 the answers and it's possible that these two questions are actually related to one another because what one wants to be able to do is to be able to just go to any kind of system where, okay, so I consider these kind of, you know, Rayleigh systems where I have an inner chamber, an outer chamber, and water in between in the ninth. I get differential rotation. I get all kinds of really interesting phenomena, right? Well, there's no case in which no matter what I do, how long I I wait after I put the water in a chamber that the water molecules are non-interactive, right? So I have to start with some sort of interacting situation. I apply some kind of constraint to build these kind of non-equilibrium structures, and then I watch them dissipate, something like that. I need some way to be able to pick out, A, how I can simply start up with a system that's already got a bunch of correlations in it, even if the correlations may have died down. So I need some kind of measure, for example. add this to my list of requirements, the unanswered question. I need some kind of measure that can tell me when I've got a kind of equilibrium situation before I disturb it again to find out how the system might decay the equilibrium. And I also need some way to come up with a kind of impossibility argument, which is related to this question of do I have a principle way of getting rid of the anti-thermodynamic semi-group. If I have a way to do that, I think that's the flip side of that question. And I can then say how it is that these physical mechanisms, it's impossible for them to somehow overcome the energy barrier because there is no corresponding real-world process out there for the list of reasons I give you that can overcome this. Now, one might think, oh, well, what about spin echo experiments, for example? Those seem to look on the surface like they give antipoamodynamic behavior, possibly. But the Brussels-Austin argument for why that's not anti-therbo-dynamic is the following, right? So what I do is I have some, you know, nonlinear field that I have on right from T0 up to some time T1, right? that I have, it's not only right so that all the magnetic moments all spin at different

1:17:30 rates, become more complicated, right? And then in essence I apply, I reverse the field from time equals T1 to T2 and everything's realigned again. All very nice. And so their Our argument is that what happens is the entropy increases up to this point from d equals 0 to d1, and then there's some kind of outside intervention, right? I mean, I am the one who switches the magnetic field. then introduces a kind of jump, a discontinuous jump in entropy that then approaches the equilibrium value here at t2. So the entropy actually does turn out to be, with the exception of this jump, always increasing function, right? And they would then argue that this is an example of a kind of extrinsic sort of situation because there's been this kind of intervention here. The outstanding question, though, I mean, I want to ask is, well, can the system by itself, not intervention, okay, so I have this increase here, can it somehow do it itself? Right. And that's where I'm looking for some kind of, you know, no-go, you know, that they I think there's a sort of argument that you get in there and there. The second edition that we discussed is . I think it's sort of a counter where you have to reverse the screens, where you take all those entry balancing into a counter, it will start there until it starts to increase again. And it seems to me that there is a kind of real problem. But if you did, actually, spontaneously, this magnetic field has tripped on the end of that, it seems to me that there is a sense in which the echo is . That's a really common reading of the experiment. Right. That's the kind of natural case versus the intervention case. And there, I mean, that's where you want this no-going theorem. I don't know what I do about the natural case. I mean, I agree with the actual . Well, that they don't discuss because they think they've said enough about the intervention case.

1:20:00 In terms of the Spend ECHO experiments. They don't know what to say about it, but that sounds like that. But could I just ask you one other thing? I thought it was very interesting. You said that on the bustles of the practice, that you have this system of n-part, usually a large number, and then you take a subsystem, that subsystem doesn't to exhibit some of the natural behavior. Well, it's on the sort of extra because it's an interventionist view. It's exactly the other way around. It is the subsystems that they actually do export by correlations to . And this is what explains the increase of entropy. So it's the subsystem that actually exhibits the natural behavior. And the whole system, of course, on that sort of interventionist view, that's a continental group. view is that integrity of the approach to equilibrium is a kind of global behavior as opposed to something that's developed locally with the kind of proviso, right, that I have this kind of correlation dynamics going on that somehow is ultimately tied to the trajectories. But at some point, the global effects due to collective effects and so on, this is my language that I talk about. It grows up over a swamp, or it makes the trajectory effects basically ineffective for affecting the global dynamics. But that, I think, is the kind of picture that they're actually driving for. They don't state it that way, that then all these effects become a kind of emergent phenomena in the sense of what happens at the global system. And they're not something that you could look at, say, just this part of the gas here and see it. But then we go into the opposite case, you know, the word correlation, this is the most standard interventions to try. And they would say, well, the build number of correlations is what prevents the entropy from increasing. That's what keeps the five-grade entropy constant, as you can view it all. The entropy is constant, but because of the build number of correlations, if you want to have the entropy increase, you've got to export the correlations into the environment, get rid of the correlations, that's how you increase the entropy. people seem to think that the combinations of cells are what produce the . As it is the word . No, I think that's pretty close. They're not clear, although in some of the early literature

1:22:30 they do have some sort of disparaging comments about the fine-grained equilibrium and they think it's not the right formula. Because it's only applicable to some neighborhood around the non-equilibrium version. They don't have one. But their intuition is that if they had one, that it would probably give you the answer for such questions as this. So one way of interpreting them is that in the absence of having this master formula for non-equilibrium entropy, they have to then focus in on particular physical mechanisms that they see as generating this kind of barrier. And so I think that's the best that they could do, is to try and show why the interventionist approach has some kind of fundamental misunderstanding of the situation, which is one reason why, you know, people are violent controversies, right? Because the trouble with the intervention system, with that, it actually gets done in the big system. And the big system is the whole universe. Well, the whole universe is fixed. The whole universe is increased with wise cosmologists. Yeah, yeah. What's your own feeling between the extrinsic and intrinsic approaches in that? My gut intuition. Well, I mean, . My gut intuition is that one actually wants to think about both types of irreversibility, but not necessarily always in the same context. There might be some systems which are intrinsically irreversible, and one candidate for that seems to be something like chaos decay. There doesn't seem to be any way to make sense of chaos decay since it requires an environment or any other kind of interaction of subsystems. something that falls right out of the dynamics of chaos themselves and that's that's a typical example of that Arnold Bohm and others point to when they say there are phenomena that have this feature let's go look at other things like this so let's take a more complicated example like scattering theory okay what do I do in the scattering experiment well I have a

1:25:00 target. I set up my particle accelerator. I have a certain energy range that I want my beam to be at an environmental target. I form these resonances, these temporary states that then decay into all kinds of products that I measure. Now there again, both the Brussels Austin School and R.O. Bohm want to talk about this decay of the resonance as being an example of intrinsic irreversibility. And that seems quite plausible, because although it is true that I fired the beam into the target, so there's a kind of intervention there, there's nothing else that seems to be required by the environment to bring about the decay, that seems to be something that happens in the internal dynamics of the interaction Hamiltonian, say. And then I detect this decay of particles. Okay, so that story might actually go through quite okay. Arnold Bohm kind of leaves it off there, that's his interest. The Brussels-Austin guys want to go a step further and say, OK, well, if I think about this, maybe I can then build up. If each of these kinds of collision events are kind of resonance events that are intrinsically irreversible, then perhaps I can then attribute this property that they don't say quite this way, but I think this is what they have in mind. Then I can attribute that property to the dynamics of the whole, after all, each component of it is built up out of these sorts of intrinsically irreversible phenomena. So the whole must have also this kind of intrinsic irreversibility. Okay, but at some point, you want to ask a kind of question about, well, when do I cross a boundary in which it looks like the environment's playing a role, but I need to take a look out, and when am I not playing, when have I not crossed that boundary? And for the Brussels-Ostin At the moment, their interest in their new program, in contrast to the old program, they talk a lot about open systems. In the new program, they focus almost exclusively on closed systems because I take it that they want to try and, well, actually a couple of them said that I was right on this. That what I mentioned that they wanted to try to do is to say, if I can show you a clear example of intrinsic irreversibility and closed system, But it makes it easy for me to show how those dynamical processes are just as valid in open systems, although I might have to talk about sort of the competition between the effects of the environment and the intrinsic dynamics itself in the open system.

1:27:30 But let me try to convince you on closed systems that we can get these kinds of really interesting effects. And then we'll worry about this other problem of kind of the competition between two sources of irreversibility later. But first I need a kind of proof of principle right to just to show you that intrinsic irreversibility is important And plays a role in these kind of that's if that's true then Eventually for real-world systems, we're gonna have to think about both types of irreversibility somehow interacting And that's going to be messier. Yeah, but perhaps that is actually the way that perhaps the world's messy. That's okay I don't mind a messy world, but But can I think you have a point you may like at the beginning, Robert, about subjective objections to your cause, isn't it? I mean, that sounds all right. You choose this level of description to make yourself depend on that. But you could put it in a very different terms. Say, well, give me the description. You get whether it's my description or whatever. A description. Relative to that description, the behavior is this. Relative to another description. Human beings would all be bright and dark, and they would all be like dinosaurs or something. But the perfectly objective facts that relative to such descriptions of behavior will be such or such. I mean, that's sort of a, what's not feeling about that sort of response to the accusation of subjectivity of cause-brainy people? I don't think he's on cause-brainy, that's a suggestion. Then I think, no, I think that's an appropriate response. That's why I say, I mean, Pritagine talks about that particular argument I've seen him give, he starts with something about this being subjective. I don't like this because it depends upon us, how we look at the system. And that, I think, is too quick because, one, just think of a kind of a simpler case, something out of Hamiltonian chaos, where I could look at, say, a simple predator-prey model which gave rise, say, for example, to, you know, dimensional map I could follow the trajectory around and around if I'm very careful and describe it in a very deterministic way up to a point that at some point in time initial sensitivity on initial conditions is going to cause

1:30:00 me some problems but that's one way in which I can describe the system and it seems perfectly valid to describe as a deterministic system on the other hand I I can look at a kind of statistical picture of it that looks like it's a probabilistic picture, right? And in some sense, one might want to say both pictures are right for their purposes. And what that illustrates is that the world is the kind of place where, I mean, we make cuts. But the cuts that we make don't give us a kind of fundamental knowledge about what the world is like. They simply are one of the cuts that we make to examine the world. still be available to, say, a particular system So it seems to me to be wrong to want to simply pass this problem off as somehow easily as being a problem of subjectivity on our part. That's why I think now that the really important objection, which they don't talk as much about, is whether or not a particular cut is focusing on the appropriate dynamics. if the coarse-grained approach misses dynamics that's crucial to the picture, then that's a problem for the coarse-grained approach. Maybe we can fix the coarse-grained approach, or maybe not, right? But Pritagine's attitude is, I don't want to worry about trying to fix the coarse-grained approach. I want to try a different approach. Somebody else can do this project, but I'm going to focus on this one. So why it is that he doesn't talk about the second objection I mentioned more forcefully, I don't know, perhaps he's convinced that this subjective kind of criticism is one that can be delivered to broader, more popularized audiences more easily and is now convincing. But a group of philosophers is typically not very convincing. We just sort of hold our breath, try not to laugh out loud, about whether or not subjectivity is really the issue. When it's actually not, it's the question about whether the dynamics has been appropriate looking at the data that used to be real. What do you think about that point already? You were nodding the idea. I don't care about it, but what is the agreement between the other? Well, I agree that this objection is not, I mean, I agree on that dependence on description does not make anything subjective. That's all. We give an index to the behavior of them and so on.

1:32:30 that indicates the type of coarse graining that we're choosing. And I don't see why this is problematic. The fundamental problem is to justify the probability distribution using the dynamics. This is the fundamental problem if you choose to put coarse graining on this once you've sold this fundamental problem. Then add to the description, I don't see, I just don't see what's the problem. This is why I was not interested. I just don't see why people think it's a problem. This seems a simple problem. I saw it very often. I do feel like it depends on your ability. Right. Because I don't think bridging is the only guy This kind of, I mean, he's just, he just uses it a lot. He'll say the same thing about collapsed versions of quantum mechanics, because he eventually wants a kind of different approach to quantum mechanics, but that takes us back to this, trying to qualify the probability to be of the right type to do the job. But he has a similar view about that, where he thinks that that just happens subjectively well, because it has to do with how we cut the system and look at it. It's a more complicated question of quantum mechanics, for sure. But the subjectivity is not really, again, I don't think the subjectivity is the core issue. I think that was really sharp with the interpretation of the probability, because if you think that there's a sort of privileged, fundamental word where everything's deterministic, well, then you're going to interpret the probability of that. Exactly. Right. On the other hand, if you think that these different descriptions are all kind of co-legitimate, to pull up the coarse-grained ones, and not believe that he's willing to be great. A kind of distribution. A distribution kind of approach. Then he might well claim that they're wanting. Well, I don't know exactly what he means by that. Well, I think the other big level of probability can't be something out of that distinction of wanting and ever seeing it. And he's really a discussion of limited description. Well, I'm raising them. what I am as a great opponent of the propensity type

1:35:00 news, probably building your name's objective, and making it some stupid thoughts. I don't know whether he talks in the language of propensity and that kind of, that kind of. No, he's not even, he doesn't even say that much. I mean, well, you know, this quote, this is, that's sort of it. you know, that basically, in this lesson, I'm going to read this, but the new laws of nature deal with the possibility of events but don't reduce these events to decidable, predictable consequences. All right, I mean, this is all suggestive language. But in the absence of being able to say something about what the notion of probability is, and I presume he's heard differences between, you know, intensity interpretations, subjective interpretations but nobody in the group talks that I know of in those kinds of terms or tries to apply those kinds of categories. And actually, to my mind, it's not clear that at least our traditional categories or probability help to make sense of what they think they have an idea. You want something intermediate between things that were, you know, in essence, he's looking for something in a realm where we haven't really searched a whole lot. Or the people who have searched for it, didn't find anything, and they gave up and went off and did something else, right? So that quotation now, it's like. This comes, by the way, from this book called The End of Certainty. Yeah. But I mean, it looks like a publication of politics has become a hope for this in the future. And it's just kind of a hope for the tangency to bring it out. Well, I know you very like tangency, but I think you've done a few more questions. But I mean, it doesn't sort of affect. This is not going to feel too involved. So how would you want to cash out that notion of open possibilities on your account? Well, I'm a little more sympathetic to the subjective objection that you were just discussing for the reason that I would want to deal with this as a matter of a useful way of describing things, but not something that you really want to attribute to the system itself. But at the same time acknowledging what you said, which is that there are a system as such that it can be usefully cut this way, or it can be

1:37:30 usefully described and predicted to a certain extent by using this tool or this exchange. But it's not, I'm not wholly unsathletic with the Christians' uncomfortableness. That's a bit of subjectivity. I mean, it is compared to the sort of the lower Popperian propensity open future, right? Although part of that depends upon, perhaps, let me try this out. I think that there is a kind of assumption built into that, that there is, at some base, level one, given once and for ontology. It has a particular kind of nature to it, block universe or something like that. But if one thinks about, say, Putnam's idea about internal realism, where there doesn't seem to be, well, if there is some kind of base ontology, it's more of a kind of holistic thing, and by trying to look at it, we, I mean, we can't do the holistic picture. where we always end up looking at some level. And in some sense, I can make, and some people have tried to develop this idea a little bit more clearly about, well, OK, we have these various levels at which we can look at things, and there are relations that we can sort of give from one level to the next level, whereas one level turned out to be a kind of ontic base for the next level, which appears epistemic, which they can be transformed into a kind of ontological level, and so on. But there's no sense of objecthood which makes sense of the notion of all the objects there are once and wrong. Right, right. Which is precisely what Prigogine seems to assume there has to be with his subjective, his stress on the subjectivist objection to the cut, which, as you say, Michael has said, clearly, still, if the human race died out tomorrow still be objective fact that right I mean the dynamics would still happen even if we're here to name it is perhaps all these arguments are about what people expect science to provide and what is the notion of explanation and description that they use for instance if we say that whether or not or or the the way in which entropy changes. Suppose that coarse-graining is an acceptable tool,

1:40:00 then they expect, they will reject coarse-graining because they think that physics should be able to describe a world in which human beings are not present. In other words, that there as a world that is not described by us in any way. I'm not sure this is possible or feasible. This is a general problem in epistemology that we have to account for before we decide on the issue of cross-draining. And so, this is the requirement to have a theory in which description does not appear. He's asking too much from science. I just think this is not what science is doing in general. It's really like the problem of natural kinds, right? It is a version of the natural kinds of problems. You know, people have a way of carving . It's not a very large tiger as well, but a large tiger, right? Different ways of describing the world. And it's one more legitimate than another. I mean, I can't find it when it gets out. I don't think that's the message, but when I get to it, I think it seems too rigid in one description because this power business to build up in the 17th century is the great way I describe it. Yeah, I mean, I think that's part of the reason why at a certain point he puts in this term non-Newtonian Yes, exactly. So you can try and draw a strong distinction between, OK, And it's clear to your world. He talks about what is emerging, as in this is something new. The new laws of nature. So that's a rejection of certain Newtonian formulations of the laws of nature. So the new laws of nature deal with possibility. They don't deal with certainty, which is what looks like Newtonian laws. So I mean, linguistically, I think he actually does a lot to try to draw this kind of distinction out and say, well, don't be wedded to this picture that somehow it's delivering not only an effective tool for analyzing the world, but it's also delivering to the ontology for the world.

1:42:30 Although, on the other hand, it's not clear to me that he's nuanced enough to say that, well, my view may not all, you know, you should look at my view as also not giving you the ontology of the world. I think he sort of thinks he is getting on to the ontology of the world. It's a probabilistic kind of world and not a kind of block universe exact trajectory. Well, then you think of the correct way to go with the actor, so there's no privileged description. Give me a description, then I'll tell you what the world is like according to that description. What else seems to me, of course, that I quite agree with this, seems to me to be trying to privilege a new description over the old, he despises the Newtonian description. Right. And that means to go back to a point what you made in your first set of comments. But to my mind, there's a kind of attitude shift that happens in the 90s versus what you see written in the 70s and early 80s, where he was trying to argue for a kind of equivalence between the trajectory and probably the descriptions. Now, I think the equivalence thesis doesn't make sense, because even in the older program, they were talking about various sorts of things that, OK, well, this land of transformation picture to a probability picture without loss of information, that's the claim. So perhaps, okay, perhaps you don't lose any information, but it looks like in the distribution picture you gain information that isn't in the other picture. Okay, if that's true, then in what sense can you say that these things are equivalent? Because you ought to be able to get as much information on one picture as you can on the other. And there are other arguments that you, where you can show from, I think, from within their own, by their own lights that it was a mistake to try to say that these pictures are somehow complementary or equivalent. And then you notice in their discussions in the rigged over space approach that the talk of equivalence thesis completely disappears. And now they're interested in trying to show you and convince you that, well, if I try and do this kind of standard classical mechanics approach and then build everything up and compare that I don't even get the right results, and I can't describe through the mechanisms that I can between the two. And it seems very clear that they're making the point that, no, these are not equivalent descriptions. Ours is better on at least some measures, maybe all measures, but it's better on at least some measures,

1:45:00 such that the equivalence thesis disappears, and then that could be read as a kind of argument for, well, they only have a better description, but perhaps it's pointing us to a better understanding Well, that's in some other thing, complementarity. Yes, these complementarity arguments, yeah, these complementarity arguments aren't made in the new approach, in the regular approach. They simply drop out of the discussions. I think their approach to, their tendency to prefer one restriction over others appears also when they distinguish the intrinsic and extrinsic reversibility. Because to do this, to make such a distinction, if you disregard the K on decay and things like that, if you really try to understand what's happening intrinsically and extrinsically same dynamics, for instance, the Newtonian mechanics or something like that, then the distinction is blurred because you first have to describe what is the system with which you're working. And a system is not a natural kind in the sense, at least not a large system, and perhaps one particle is, but a large system does not have clear boundaries. And if you take the interaction between a system and the external world, this interaction, perhaps, you have to, if you start with a system that interacts, then the external part with which it interacts becomes part of the system to a certain extent. And so you have to make a distinction, and this is description dependent. Also, the choice of the system is not always a natural one. For instance, if you take the particles in this room, you can find, say, three particles that spontaneously start moving in the same direction. And so their entropy, if they are a system, spontaneously decreases. But why don't we see them as a system? This is a matter of usefulness or choice.

1:47:30 extreme example but to choose to distinguish the internal and external interactions is also distribution dependent and there is here the choice is natural because we know what we want this is a system and those three particles in this room are not a system for us but this is there is no I couldn't find any physical objective criterion for defining a system as such Yeah, much more pragmatic. Yeah. Roughly. Centric. Yeah. And Pritagine, I mean, near as I can tell, Pritagine doesn't say much about what his view is of science. I lean towards looking at science as kind of human activity that's quite pragmatic, although that doesn't mean that somehow the radical versions of social constructionism are correct correct because it turns out that our best pragmatic accounts sometimes fail miserably when we try to mount on the world. The world seems to have something to say to us over and above our Soviet activity. But Pritching's view, I don't think, is inconsistent with some kind of critical realist approach that goes something along those lines. But he does seem to have a kind of stronger ontological message that he wants to deliver. Well, the world is not the kind of Newtonian world of certainty, clockwork world that we used to think it was. It has a different flavor to it. And he might be willing to admit that, okay, I don't ultimately deliver for you the ontology, but I can tell you what it's not like. Well, I just wanted Well, you may open up a pile of a can of worms, in which case leave it to know to afterwards, but quite apart from the motivation from that Phrygene's approach in statistical mechanics and thermodynamics, has anybody in mainstream philosophy of quantum mechanics looked at, well, for instance, the basis problem in Rigg-Hilbert space, and is there interest in Rigg-Hilbert space or motivations other than the? Well, yeah, that's, there's a lot of activity in about 65,

1:50:00 66 up through the mid-70s where people write several articles on regular space and quantum mechanics. With apparently two goals, one is justification of the drop formals, and that's a very important thing. And the other goal was to try to sort of explain a little bit about what it is and how it can fly to quantum mechanics. Now the interesting thing about that, when you look at Antlon, or Roberts, or Melsheimer, or even early Bohm, they are decidedly, for the most part, operationalistic in their treatments of rituoclase. So if you want to ask questions about, OK, how am I supposed to understand the elements in this kind of privilege, in terms of the elements of phi versus the elements of phi x, the operators, and so on, am I supposed to understand them realistically or not? People are very cagey about what they want to say about that. It's only in, so far as I've seen only in Arnold Bohn's work in the last few years where he's begun to say, well, you know, I don't know, it's like gamma vectors, which look like they're exactly the right kind of mathematical animal for describing the scattering phenomena, the kind of resonances that indicate the resonances that live here in this space by x. Those appear to be things that we can't interpret as being realistic because there's a kind that they map on to experimental phenomena we see in the laboratory every day. Dirac vectors seem to have the same sort of thing, and then, I mean, it's interesting because in like the 70s, Bohm says things like, well, I mean, the main reason to switch this is because it has a lot of mathematically nice conveniences to it. But recently he's saying things like, well, you know, there are other elements in here that await their physical interpretation, which is a kind of hint or suggestion that... That there is a connection, there is an overarching connection with the whole program of looking at non-unitarity. Right, and that the elements may actually suggest new physical phenomena that we haven't described or discovered it. Now, nobody's produced those things,

1:52:30 although there's a sense in which one can read, I think, the British group is trying to say that, well, when I switch to this kind of formalism, where I have this broader mathematical framework available to me, and it sheds light on the correlation dynamics in the way that I tried to sketch here, but that's some insights, a new insight that's been delivered to us. Now, we might not want to say that those are new mechanisms, because we sort of knew some of those things were around anyway, connect up in a way that leads to part of the explanation for irreversibility. But it does at least point out that looking in this kind of framework sort of amplifies our illumination power. The second question after that is whether or not it also gives us some new physics that we can find, you know, go to the laboratory and check it out. And if one finds that and is verified in the laboratory, then that that would be quite interesting. But so far, I haven't seen anybody produce any kind of candidates for new physics yet. And that's because there are, aside from our moments of his co-workers and the Brussels Austin group, and not everybody in the Brussels Austin group is interested in this framework, which many of them are. Outside of them, I'm not aware of anybody who's really actively pursuing much in terms of the super space stuff. I mean, I hope to do something along those lines, but it's a kind of lonely world out there for a few parents in this framework at the moment. Although, I presume that if somebody comes forward with a new proposal for physics and it's discovered in the laboratory, then it might not be so lonely anymore. Well, thank you very much indeed, Robert, for the passage before, and to share your ideas with us. Thank you very much. It's not very tomorrow after all, but it's Wednesday weekend. It'll be from 3 to 5 in the afternoon, so we don't have that. It might last centennial there.

1:55:00 All right. Excellent. What is it? Scientific genius. Oh, yes. I saw that about Einstein on the first day. Sort of the old line. Yeah. That's good. Thank you. Thank you. Thank you. Thank you. Thank you. Oh, no, no, no. Now she's going to be next to the street, but it's really cool. I'm going to be here. I'm going to be here. I'm going to be here. Thank you. I'm sorry, you almost saved somewhere in my house. I'm actually in the middle of a tour and I've got a group of people, I've got to take a look at theatre. In fact, I'm already left, so... Oh, it was an absolutely fascinating talk. Actually, what is the term now? No, actually, I haven't got to... It's just a very quick one.

1:57:30 Thank you. The reason I asked that last question about the Rick Gilbert space was that I wondered I wonder if there has been anybody that has been working on connecting up this whole approach and particularly the motivation for relaxing your character requirements with the investigation of underlying algebraic structures in the phase space. particularly in terms of the underlying of the great structure, which is, of course, the motivation of the other and, you know, some of his. I think there were people like Basil Hyland. The Richard was saying, you know, it's caught on, isn't it? People reckoned that he was respectful to the other. I mean, . Well, I mean, in a sense, The spectral theorem gives you all the tools you need to do your spectral analysis, it's only the eigenvectors. But it turns out that the spectral theorem can only be rigorously proven in the framework of the Richtilbert space. That's one of Galpant's triumphs, but I think the typical physicist's attitude is that tools like Dirac formalism, spectral theorem, and so on are there, can be used, okay, if somebody and eventually I don't really care because as a physicist I want to use those tools to get to my results.

2:00:00 If one really looks carefully, one discovers that physicists, when they do quantum mechanics, don't even work in Hilbert space. What they're working in is probably more like a pre-Hilbert space, which can be a little bit dangerous because you might not have, for things like, well, you might not have the complete sediment of products that you want. You might not, you're not guaranteed, let's put that way, you're not guaranteed to have all the commutation relations that you want. You only have those if you have a broad enough structure, and the red silver space can guarantee those sorts of things. But this is just simply pushed through, no matter what. That's also problematic. But I mean, typically what it seems to me physicists do is they might start off an article by saying, okay Hilbert space L squared functions and so on and then they immediately start writing down formulas that don't make sense in the framework and they write down say expectation values well you can't really take an expectation value in Hilbert space because the vectors in Hilbert space really are equivalence classes of the elements of Hilbert space right so I have to do some extra interpretive work to make sense of what this expectation value means. Okay, so perhaps it's one element out of that equivalence class, and as long as I think I have the right equivalence class, maybe that works. But if I'm doing something like taking the expectation value for the momentum, well I don't have any eigenvectors for the momentum operator in the difference. The problem with the Kaplan theory is always a kind of problem. We're not only about discussed that. Yes, yes, exactly. Although S-matrix theory gave us a set of tools for which we could do a lot, but it's not until the 60s that you see people in the 70s where they begin to formally justify these tools, and again you have to step beyond Hilbert's face in order to do it. Did the rock himself not say that he thought that Hilbert's space was the wrong structure of the dynamics? I don't know. Well, so I have been told. I mean, I know Van Neumann has this, you know, there's this famous letter that Van Neumann wrote to, I think, to Burckhoff.

2:02:30 He says, you know, I no longer believe in Hilbert's space. But I think what he has in mind is he's already shifted gears to an algebraic approach. we now do know today as algebraic quantum mechanics, right. And then there, yeah, and there, I think, well, I mean, I think personally that there's a very interesting question as to what connections there might be between the Ray-Kilbert space approach and algebraic quantum mechanics. Right. But to my knowledge, nobody has actually looked at this question. So Fred Krons and I, Fred Krons is one of my thesis advisors, have plans to try to dig that question after we finish an article that's discussing the interpretive issues about free killer space and fun mechanics I was actually going to ask you if you want to have some of your overheads to think about it more deeply. I can give you my email address. I can't give you my overheads. It's the only set I have. Oh, no, no, no. I would not. I was going to ask you to give me some of your ideas. I was wondering if I could maybe sometime send me a set of copies of them or something. Right. Do you want to? RCB at IGPP.de. Gosh, that's easy. Of course, because you're in Germany, aren't you? In Freiburg. Right, okay. And when are you going back to Pryborg? I should be back sometime Friday evening. Oh, okay, right. Well, I'll send you an email there. That would be great. Thanks again, Michael. Very nice to see you. Well, I am going to see you for your lecture. Oh, yes, yes. I finish with these people on Sunday, so I'll be around next week. I did, and thank you very much indeed. And also, I meant to write to you to bring you up today of course he's not coming over now, he's not coming over this month, he's going to be in

2:05:00 Italy for about six weeks in July and August, he's giving a tour of course in Pellugia, and after that he hopes to get here if possible for a week or so, and he says thank you very much for all the kind suggestions in response to my discussion with you in general. He has, however, he has actually now become an Italian citizen, and he's very, very much set his heart on creating this thing in Italy, if it can possibly be done. That's the line of work. But I'll certainly keep you in touch on that. Oh, yes, and I'm sure that's... No, don't worry, I'm working on it. Anyway, thanks ever so much. I'll see you next week. Thanks very much. Good luck. Thank you. I'm going to slow you down. Thank you.