Do Boxes of Stars Obey Thermodynamics? / Boltzmann, Gibbs and Equilibrium

0:00 Thermodynamics, Friday 2nd of November. Okay, our next talk is Craig Callender, who's going to ask the question, Do Boxes of Stars Obey the Brotherhood of Enix? Or the new title. Okay, well, let me apologize for the talk. I'm going to try to summarize in 20 minutes or so, about three years worth of research. It's actually, since I don't really understand this material, it's going to be a stretch to get this one in the next. What I want to do, just in general, is I want to say that when you look at gravity, it's going to basically upset and ruin almost everything that's ever been said in the foundations of statistical mechanics. So it's a very small, modest goal. and that you have to look at gravity. So you have to be upset. Okay, so the question is, you know, does thermodynamics apply to self-gravitating systems? So, well, let me, the question's important, inescapable, and open. You might think, well, I mean, to motivate it, just, you know, thank you. Both of the systems you usually deal with are all terrestrial. They're all in uniform gravitational systems, all boxes of gas, plastics, and all these sorts of things are all in a uniform gravitational field on Earth, usually. And so you might think, well, you might think, well, you know, thermodynamics has been extended everywhere. Some people think it's practically a priori, so it's a no-brainer that it worked for systems where gravity is turned on. On the negative side, you know, there's not the kind of shielding with gravity that there is with an electromagnetic force, you know, so you have positive and negative charges balancing each other out when you get to a certain scale, you have, you know, Dubai shielding

2:30 and plasmas and stuff like that. And you don't have any kind of shielding like that in the gravitational case. And the more you think about it, you might think, I mean, and then you look at the history of the subject and you see all these, you know, foundational debates about whether the The temperature varies with the height between Boltzmann and Lachmitt, continuing to this day in physics journals and stuff, and so you might... So the relationship between thermodynamics and gravity has always been kind of an uneasy So it might make you suspicious and worried that when you come to gravity, you're going to have problems. In fact, that's what a lot of people say, is Rowlandson, the chemist, giving a kind a statement that you see everywhere. It says, Thermodynamics is essentially a human science. I don't know what that means. It started with steam engines and went out to describe many systems whose size is the order of a meter. Its laws are a highly condensed and abstract summary of our experience of how such systems behave. We have, therefore, no right to expect them to apply to other quite different systems, whether extremely large or extremely small. They clearly are inimplicable to the solar system or to galaxies. there is no equilibrium, the energy is no longer force not to the amount of material, and so there are no extensive bunctions. Clearly classical thermodynamics is not a useful branch of science and cosmology. We have extrapolated too far from its human-sized origins. I think when you reflect on the problems that I'm about to hit you with, it's not a completely unreasonable stance to take. On the other hand, there would be news to the cosmologists who used thermodynamics that it's not a useful branch of science. So there's lots of work in cosmology using statistical mechanics and immunodynamics viewing the evolution structure as a kind of relaxation process. So I think it's a kind of interesting issue because you can be joined both ways. Let me just think about it a little bit. I think it's important for three reasons. I'm going to just really talk, I'm going to focus, this talk is going to focus on this one. I realized that Monday when I did the division, and I saw that I'd have 20 minutes, that these guys would have to go. But you might just be interested in the question, because you might be interested in the scope of thermodynamics. People are interested in whether Brownian motion obeys thermodynamics, and things like that, small things, or what about big things. You might also be interested, because it impacts the foundations of statistical mechanics in various ways, which I'll discuss just briefly as a kind of cancelizing teaser,

5:00 And then I'll talk about how it impacts the normal explanation of the direction of time. I'm going to say it just basically proves false, the past hypothesis, and I'm going to try to come to a rescue a little bit. But just a kind of teaser, I mean, there is going to be, we'll see, there's no thermodynamic limit for self-gravitating systems. So self-gravitating systems, I mean, well, I'll get to what I mean in a second, But, you know, all long-range forces will become the dominant force. So think if you have particles of gas in the box interacting with short-range forces. Now let those particles be stars interacting primarily by gravity. So you'll have, with these, you're going to see there's no thermodynamic limit. And this is kind of a question I have. What is this going to mean? I mean, when you can't prove the different ensembles are equivalent anymore, what does this mean for these various programs? What does it mean for the Gibbs program? What does it mean? It's certainly not going to help the Lanford program, I don't think. So anyway, if anyone has anything to say about that, I'd be happy to hear. But let's go to the direction of time. Everyone knows the usual sort of story. You know, if this is now, the entropy is low, but if you have a time-symmetric micro-physics, and you have the usual Ultram story, that's going to say entropy is likely to rise for the future, but also for the past, but actually, our actual past is down there, so we have a low entropy state. Everyone's familiar with that. So, people posit the path hypothesis and so this is the phase base of the universe and so you find in Boltzmann, Einstein, Schrodinger, Wald, Price, Albert, so all the most eminent scientists and philosophers all say you deposit a limit to be passed so that's the universe begins there or somewhere around there so now there's an objection it says unusual back and forth that you see in all this literature right so here's the back and forth you see that people say but good god uh you look at cosmology you see the initial state is one of approximately

7:30 uniform temperature and approximately uniformly distributed they say isn't that a high entropy it's low entropy and then people then say so if people make this kind of claim then people say don't worry don't worry uh gravity's attractive we're taking into account gravity clumping is the new anti-clumping homogeneity so what you used to think of this as going from low entropy to high entropy but really you know now that you have your new new fashion clumping being anti-clumping this way now of course you can specialness and momentum sectors of the face face could swap specialness in the position so if you let this is going to be at least we'll just think of those things this guy's at rest or something so everyone says so there's always this kind of objection and then a quick dismissal of the objection remember grab these attractive anyway okay so the hypothesis story fly at all you have to include gravity right does anyone disagree with this if you at least agree with that then you see that there's run of the work to be done and i'm happy already um so uh yeah so but uh you know this common reply presupposes something there's not you know well-defined uh statistical thermodynamics incorporating gravity So, recently, you know, John Ehrman, on his paper, Not Even False, claims that the path hypothesis is false due to reasons coming from classical general relativity, and he says the Hawking Page measure that you get when you have a probability of inflation, you put on these models, it's not what doesn't suit well, but Boltzmann, Hatch, and all that stuff. I myself have a video with zip confidence in the Hawking Page probability measure, so it doesn't bother me that much. I mean, it bothers me, but this bothers me more, because here we're going to just regular old Newtonian good old classical gravity, and if that ruins things, then it's going to be even worse than if some fancy probability measure ends up ruining things. One of the other options, I guess you could replace the past, I mean, if you don't want to include gravity, I guess you can replace the past hypothesis with the not-so-past hypothesis and just think of a hypothesis which places low it says entropy is low whenever the whenever gravitational contributions aren't relevant so I guess you know put one on the sun

10:00 when the sun started and then another one on some other sun and I don't know that would be with many many many many get kind of branch branchy okay so let's look at the problems so what we have is a classical in body the problem gravitation theory so just how many things interacting according to gravity we can give up the other forces and then there's two questions you can think about the So in principle, if you have this n-body problem, can you slap a thermodynamic description upon the features there when these things qualify as macrostates and stuff? So there's that question, and then there's a question about whether it's actually physically instantiated in our universe, which would then get relevant to the past hypothesis. In the self-varietating thermodynamics literature, you know, there's a lot of excitement, especially about globular clusters, because they are, you know, they're not, there's a lot of stars in them. When the stars get too close, then pressure forces, like within a megaparsec, then pressure forces start to operate. but there's a lot of stars that are out so bigger that acting they're away from each other greater than a distance of a mega parsec but there's a whole lot of a whole lot of them and they are close enough so there's interesting gravitational effects nonetheless and it's not kind of this organized motion like in a you know spiral galaxy or something so people uh in this literature are really focusing on you know these one of the problems well soon is i mean the first one i mean rolandson mentions this and then the first thing you think about extensivity of these when you add gravity into the story. So extensive quantities are things like U, S, V that depend upon the amount of material or size of the system. Intensive quantities like T and P are ones that do not. Extensive means that homogenous with degree one which means that if you have

12:30 a you know the state of the sit the so if you have some function of the system say the entropy you will you know this equation will hold so for instance consider a box of gas in equilibrium the partition in the middle so that a equals 2 then s t 2u comma 2n represents the joint system and the equation says that this is the same as two times the individual entropies of the partition component systems so you can just take the entropy of this the entropy of this and that will give you the same as the total entropy so you get uh additivity there's actually a lot of uh go into literature there's a lot of inequivalent deaf mathematical representations of what extensivity it is but this is one it won't matter of course now let's just quickly put aside the sense in which no system is ever extensive so and you know so-called rigorous statistical mechanics if you think of rule rule massacring his name but if you think of him then you know there's always going to be surface contributions to the partition function as long as the system is finite and so no system the finite system is ever truly extensive let's just put that aside that's what I call taking thermodynamics too seriously in a past paper these are not going to be extensive in any in any sense not just the kind of very strict sense but for finite and they're going to be not extensive too all right now why is there the problem for the most basic assumptions to this mechanics is that the total energy of any thermodynamic system is approximately equal to the sum of the energies of the system subsystems so if you look at the Landau-Lipschitz or SK Ma or all these different really divergent textbooks and statistical mechanics they all begin with this kind of assumption and so you know what's the what's the rationale well the rationale that the interaction energy is going to be proportional to the surfaces of the subsystems whereas the subsystem energy themselves are proportional to the volumes so whenever n gets large the volumes are going to swamp the surfaces and there won't be you can you can make that approximation another way to think about it but yeah so the way you should think about this so if you have short range forces right it's because you have short range forces that you can do that sort of thing right you can think that you can make that you can look at say

15:00 this wants this because the interaction between these two guys is going to be on surfaces of those boxes if it's not short range like that then you have a problem in fact you see for gravity you know if you consider a sphere radius r filled with a uniform distribution of particles now add a particle to your origin and then consider the potential energy felt by this particle to the rest of the sphere you can see that the contribution to the integral is going to come from far away toward long-range forces now suppose those things are non-additive or what happens well all hell breaks loose right so now that as an exercise you know go home grab any book off your shelf that says as the word thermo or statistical maybe there might be some probability books without no thermal in statistical but let's just say a book with thermo in it and you know go through and circle every single derivation you can you can find where additivity presupposed there'll be almost nothing left there'll be something left but not that much left so for instance you know all these various assumptions you know when a system is in equilibrium it's large so subsystems are in equilibrium that's not going to be the case that begins to have an inequality where you can uh get rid of two intensive you can get rid of uh one of these three intensive uh um variables um that's not going to be that's not going to hold think of uh your derivation of the in statistically can't get a derivation of the canonical ensemble from the micro-canonical ensemble that presupposes uh tons of things uh i mean you know right if you think of you know the SK log W on Boltzmann's gravestone right what you know the statistical independence of the W's of the subsystems is because it's a logarithm and directly related to the additivity or we put it in this form and you can prove the s is additive if and only if row which is the probability distribution is factorizable so I should say that they'll still exist if it's not additive right this s is still exists but they're not going to satisfy any of the usual thermodynamic relationships and so if you want somebody to find an entropy via

17:30 and they're like a cut it's be the thing that plays the role of whatever it is you know in these in thermodynamics with the similar net satisfying these functional relationships of thermodynamic parameters you know then that's not going to be the entropy anymore and then the number limit yeah so this will know thermodynamic limit I think I'm not gonna have time for this it's not gonna be it's a it's not surprising because you know that you know the thermodynamic limit the surface of X going to zero and here they're not going to go to zero so not going to have this thermodynamic limit. Everything scales badly. That wasn't the joke, you know. Well, not the joke. It's just the irony that the thermodynamic limit doesn't apply to water systems. All right, so that's problem one. Now the additivity, it just gets worse and worse. Well, actually, I think that's the worst problem. So it gets better and better. So there are all these divergences so you know the local singularity of the Newtonian pair interaction potential the two-point particles approach one another arbitrarily closely they can release an infinite amount of negative potential energy partition function then some over these states and diverge kind of infrared divergence take the integral over you think of the density of states you need this you know the entropy is going to be a log of the density of states, but this is going over all of the range of spatial integrations of infinite, and that's going to diverge. So these are the problems mentioned in the literature. Very common claim. You saw it in Rowlandson. You just see it everywhere. Everyone says, you know, when gravity is involved, you know, there's no equilibrium. Actually, when you look at this, it's not so convincing, and so I'll talk about that in a minute, but there are all sorts of reasons that I mentioned. so they're all these kind of various catastrophes and various models but also just the idea that the you know all so you know the equilibrium points is every single piece of matter in the universe at one point people don't like here's one I can't talk about I think we have the red paper on this stuff this is driving me crazy you have negative heat capacities it's so accepted that there are negative heat capacities in this literature that models that don't have negative heat capacities are deemed unrealistic so but anyway this is

20:00 complicated to go into now I really just don't understand what's going on in that literature okay so let's let me see if I can give now let's see if I can fight back for the sake against the threat of gravity and see what can say well there's Here we have a path, a choice we can make. On the one hand, you can go and say, convinced by all these problems, you then say, that's it, I give up, I'm going to invent a new theory, non-extensive statistical mechanics. So you see the Thalus statistics, you see this a lot. It's very controversial, it's been heavily criticized in the literature. I myself don't like it that much but I don't have anything to say about it what I want to do is try to go the more conservative route we're going to stick with regular statistical mechanics regular thermodynamics and see what we can do with gravity okay so here is where the body is a ton of literature so the scores and scores of papers applying statistical mechanical techniques to self gravitating systems and if you're interested you know this was huge physics report by this man in 1990 the Bible the stats law book is very nice as well okay so let's look at the divergences well you can make the divergences go away if you cheat so if you but not so bad so you can you can regularize and replace the potential with a softening potential. Choose this well and the problems go away. I don't think that's such a bad sheet, because you know there's quantum mechanics, right? So you know something has to happen when the things get really close together, so it doesn't seem so bad. Infrared divergence. Various things are done, but one thing that's done is, you know, put the system in a box. put the system in a box in the range of spatial integrations is going to be finite not clear that problem is special to gravity anyway right because of the ideal gas in an infinite universe you have the same kind of divergence happens so you didn't think that that caused the end of the dynamics in the end of the world so probably shouldn't think that this caused the end of the world here either and you know if the system is relatively isolated you know its

22:30 evaporation rate is going to be important is proportionally small so it doesn't I could sleep at night back in these two things. Now, what about equilibrium? Well, here are the two options. Here's the mathematician, Kiesling. He has a long, long paper about this. He says, it's obvious that both lines of thought are closely oriented to classical thermodynamics, which was developed for systems of laboratory size. However, it's tempting to argue that the above difficulties with equilibrium arise because of a biased standpoint regarding what must be the properties of a thermodynamic equilibrium state. Let us therefore try to take a less biased standpoint and consider a generalized meaning of equilibrium thermodynamics simply as a tool that describes the average fate of physical systems possessing overwhelming large numbers of degrees of freedom without making too many restrictive assumptions about the properties. So he thinks that the main objection is really a kind of intuitive objection. He then has a whole way of... So he thinks it's okay to have an equilibrium point being just that point comes on it there is then a problem with defining a partition function and stuff but he has only he has a whole paper dealing with this probably one way going would be to go read the kiesling paper but most people in the field actually go and say things like this so they'll stick with metastable states or stationary states and they'll say you know in the real world we do not need stable equal we do not need the stable equilibrium states to describe yourself gravitating state local entropy maxima metastable states are in general more physical than global entropy maxima for the timescales use in astrophysics the time required for a system to place them in a metastable gaseous state to collapse increases exponentially with the number in particles so that's the way they'll usually go they'll say replace true equilibrium with you know, a metastable state. Non-extensivity so that one question for philosophy would be what do we make of that kind of move away from equilibrium as the thing to these metastable states. Non-extensivity is probably the most important problem, but even here you know, the question, once you get away from rigorous statistical mechanics things on the infinite limit being strictly extensive you know what you're really looking

25:00 for is if you have the internal energy and you think there's the correlational internal energy between two systems a and B other cases where this is the case where things are approximately extensive can you find so can you find out there in the cosmos self-gravitating systems where that holds answer yes so I'm running out of time so I'm gonna go quicker apologize so what I want to do there's a lot of this literature and all these different approaches that are very different and so really kind of defies a succinct summary as I found with preparing these PowerPoint slides and so what I want to do is kind of just give you the impression a taste I'm sorry it's so it's gonna be so informal here but just give you a kind of feel for the picture that's emerging from the self gravitational literature the literature doesn't stop well the thing is here focus on the time scales and the space scales okay the two-body relaxation time for such a system is huge it's yeah so if you think of this crossing time the amount of time it takes a star to go across the system in the end times that think of the evaporation time is going to be huge the single particle distribution function describes mostly collision with evolution close encounters are infrequent all these you can find time scales and space scales where the energy the mass the chemical composition n are effectively fixed for long long periods and so you can define that this effective this hypersurface in the phase space which is effectively where those the usual thermodynamic parameters you usually use are effectively fixed even though that you know the real if you were to take that go the real equilibrium you could be going down to that single point and none of those things one wouldn't be fixed but you could still find these regions which are really big which may be so big that you know they encompass the whole observable universe which are good we can do thermodynamics again so there's this paper by Cibriani and Bettini they argue that the fast making so the big bang then it's a fast mixing takes place you get into this effective

27:30 regime and they think that that fast mixing justifies a uniform probability measure we all know that's trouble but let's just give it to them because we don't know why we justify it with the you know a glass of water on the table never mind a globular cluster so uh give them that and then they have this sort of secularly you have also these they'll introduce these other terms they can get a Boltzmann entropy for this system which is changing where the Boltzmann entropy will also be a function that's not only the usual thermodynamic things but also things like the variable ratio ratio which will be kinetic energy of the system over the potential energy of the system and you can get that so that thing evolves slowly so you're going from one quality equilibrium state to another and so it looks pretty good and so like I said is they can use mean field dynamics you can do also some other things and again you got find always in these regions that are huge that where the thermodynamic properties are effectively fixed for large regions they don't know how do we think of this as a foundational people but I mean it should now explicitly talk this way but it certainly should make you think of you know not a classical non-equilibrium thermodynamics there you're doing just this sort of thing right you're you're you're looking at non-equilibrium in homogeneous systems with some parameters varying in space and time and then the game is to find systems small enough to be nearly homogeneous but large enough to apply statistical techniques is what they're doing and you then try to see if you can export the principle of local equilibrium which it says you know that the gift state will hold in different other regions so you find these small fit and then you try to export it and so there's a kind of thing that's going on here these effective regimes are effect essentially one of these small volumes but a small volume just happens to be huge but if you think of small in terms of extent with respect to the range of the interaction then you know the universe is a small system gravity is taking into account I'm sorry about how quick that was but let me now come to the conclusion let's go back to the past hypothesis what's the conclusion well the past hypothesis usually says false sorry so you like it it's gone this problem it's not equal if it's not an

30:00 equilibrium state then there's no I should be defined in classical equilibrium thermodynamics so it's all the whole universe entropy may diverge if there's no guarantee that it's approximately expensive or anything so there's no guarantee there's no reason I think that it's satisfied even for usually thermodynamic relationships no putting this down I thought you know why we haven't think the path hypothesis state was an uniform was an equilibrium anyway as the universe just bang big bang why the heck was that thing in equilibrium do you think that way we included gravity was wasn't going to change did you think that when you removed a constraint of the universe what constraint is there to remove right so why do we ever think it was an equilibrium um but rather than be so negative let me you know and being positive uh so what i see emerging from this is you know so what i should say is the universe as a whole was never an equilibrium never had an equilibrium entropy it does however I have plenty of metastable and stationary states, and, you know, non-equilibrium thermodynamic entropy can be defined here. So what I'm saying is just grab the sort of, all these entropies that are being produced by the people working in the self-gravitating systems literature, grab one of those, those things then work for these effective regimes. We can then imagine a new past hypothesis, which says the non-equilibrium balsam entropy, that in an ideal paper, I would have grabbed one, I would have fixed on one of these in particular just as a kind of example, but I certainly could, and will in the paper, and say that that's low at the beginning of the largest effective regime we're in. A lot of work needed to be done here, so not so much as you might think, because a lot of it's already been done by the people in this literature, but there will be a real question about non-uniqueness, I think, which would be a little kind of worry. um and it's kind of a mystery as to why any of these techniques work usually in non-equilibrium thermodynamics uh in a rarefied gas things don't work uh because there's not enough collisions here there's really not that many collisions but why the heck it works is a mystery or the mystery before anyway so um and that's for me the big question then is is how large is the largest effective regime so this literature shows that you can include gravity and get a kind of entropy

32:30 and a kind of past hypothesis. So we don't have to go to the natso-past hypothesis that I mentioned at the beginning, but it might not be, you know, how big is it? Is this the gal at the Milky Way, or is it the whole thing, or what? That's it. We do not have time for questions. Our next speaker, David Lavis, on Boltzmann Gibbs You'll have to suppress this technology, yeah. Damn, I'm doing it right now. I'm scared of you, too. Look at that projector. The reason we're doing our own projector is that Intel's charge only $400 a day for the use of her projectors. Oh, that's okay. The cost of the coffee is about $8 a plate. A little lunch place for various ingredients, right? $50 a plate. You don't need that. We need a little bit more. I used to be able to say, well, it was only $8. Well, today we heard you dropped a penny yesterday. That's just a kind of a little glitch in terms of the Prime Minister saying, I lied. I know they did, but you shouldn't have promised them, wouldn't you? Okay, that's the title, that list is not so informative as it looks, some of it are just comments, essentially. Okay, what's the problem? in statistical mechanics there are three levels of description there's the mechanical level dynamic level there's the thermodynamic level in the middle there's essentially two different versions of the middle level a macroscopic version due to Boltzmann the statistical version

35:00 to Gibbs, the trouble is that both Boltzmann and Gibbs have different approaches with respect to equilibrium, different entropies, different approaches to reconciling reversible mechanics and irreversible thermodynamics, and they also have different treatment of fluctuations. Just to briefly summarize, the Boltzmann approach, and I characterize that in terms of the Boltzmannians, on Goldstein, etc., accept the reversibility mechanics and qualify the reversibility of thermodynamics. This is a single-system approach, and that's very important. The Gibbs approach, or, as it were, the extension of the Gibbs approach, which I can characterize by the work of Prigine and the Brussels Austin School, modifies mechanics to get irreversibility. I mean, there are other people who say the same thing, Michael Matty, for example. On the other hand, this is the most spelled out version of this particular approach, and it's an ensemble approach. There's also Jane, which I'm not going to talk about. Okay, so what's the aim of this one? the usual approach of working physicists I think to this problem if they think about it at all is essentially a sort of schizophrenia or sort of hand waving they would use Gibbs for the work they do in equilibrium but when they're asked what's actually going on in this gas they tend to resort to a sort of falsehood view of things Our aim in this work is to clean up a sort of mixed approach so that those of us who believe that Boltzmann essentially got it right, and when I talk about Boltzmann I'm talking about Boltzmann post-1877 essentially, not the kinetic bit. Those of us who think that Boltzmann essentially got it right can go on earning our living using the kids' bridge. I think I'm talking myself rather than anybody in the audience in this respect, because I didn't something to use a philosophy. Anyway, so what do we need to decide? We need to decide what we mean by equilibrium, we need to decide what

37:30 we mean by entropy, and we need to decide what our object of study is, what we're talking Let me give my answer to three at this point to get it out of the way. We're talking, as far as I'm concerned, about a single system. We're not talking about an ensemble. Any reference to ensembles, and I think you can more or less teach statistical mechanics without using the word ensemble, Here's essentially, as far as I'm concerned, a way of pictorially representing relative frequencies. So, the rest of what I want to say will be concerned with a discussion of A and B. Okay, this bit, it's going to be over that from what Roman said, so it's time to go through it, no, it's a decrease. We've got the two end levels, as it were, of our system. We've got a microscopic level, an autonomous dynamics, a measure, we've got essentially which normally in alternative dynamics means that you change the sign of the velocity and momentum, and then you have a measure on the set, on the space, on the base space, or at least on the singularity measure on the subsets of the same space, which is absolutely continuous respect to the base measure, and it's preserved by the flow. So, there's no equilibrium there, of course, because equilibrium occurs in dynamic systems when they're dissipated. So, any attempt to relate in some way or other equilibrium in mechanics or equilibrium in systems mechanics is a non-starter. They're no attractive. At the phenomenological level, then, in thermonomics equilibrium is such that the system is either is or is not in equilibrium, it's a binary property. The system in equilibrium never evolves away from equilibrium, and a system not in equilibrium

40:00 evolves into equilibrium. I guess one might need the brown infig minus one law for that, but I don't want to go into it many times later. Now, the middle level. In the middle level, we've got the Boltzmann approach, which, as Roman said, involves setting up a system of macrostates, they have certain properties, they're invariant under permutations of the microsystems, the size of the macrostate for the system and the system with the versatility the operator apply on the same size, in many cases they're action-state, macro-state, and the Boltzmann entropy is defined like that, and then, according to the established account, equilibrium is then the state where the system is in its largest macro-state. In the, for the Gibbs approach, then of course, we have a probability density function, the In every M, when the probability density functions are the explicit function of time, the entropy is the Gibbs entropy, and we know about all the problems that's sending that to non-equilibrium in terms of the fact that the entropy doesn't change under the operation we live. Now, it's very useful to do simulations, and so let me just briefly mention a couple the two systems, and I shall return back to the few simulations. This is a rather special sort of perfect cast. All the molecules are moving in horizontal lines separate from each other, and therefore, of course, the system, because it's simply essentially a combination of a lot of one-dimensional systems, it's symbolic, and this is just the simulation idea of the evolution of the molecules when they stuttered in one end of the box. The other example I like, used quite a lot, is the Baker's gas

42:30 wheel, or the Baker transformation, I guess. I put a picture of it there. You simply stretch your box and move it over, and that's the inverse. If you start all the particles of there, it evolves in the same sort of way. And for that, we can work out the, for those gases, we can work out the polydeneficies. Let me talk about the problems with Boltzmann approach. We talk about the problems associated with the Boltzmann approach, and then try to make some suggestions for how we So, a lot of discussion of Boltzmann entropy tends to concentrate on one particular thing, which is the increase in entropy. It seems to me that that is a too local way of looking at it. One should actually look at a more global situation. One should look at what I like to call thermodynamic-like behaviour. thermodynamic-like behaviour for the Boltzmann entity would be something like that the entity stays most of the time with its maximum value with lots of small fluctuations and the occasional large fluctuation. Now there are two problems with this. One is what is equilibrium in this context. And the other is that we would like that behavior to be typical, typical being the word that John Eberwitz and John Digmoner use all the time. Let me deal with the first of those, what is equilibrium in this context? Well these are simulations for the Baker's gas, and the maximum, the entropy rises, and you can see that, and this is the number of particles, that's 2,048, because it's a Baker's gas, it's in power to 2, and that's how it works best, and you can see even for a large number, and the n

45:00 corresponds to the fact that the number of cells of 4 to the M, when I make the macrosate, so you can see that in fact it doesn't actually go to the LIDIS, go to the LIDIS macrosate. The LIDIS macrosate, the value of the LIDIS macrosate is scaled here, and it is equal to 1, and you can see it sits below that. And that is true for any finite system, particularly your airing example of that is if you do something really trivial and you look at a gas of eight particles and you can see that most of the time it stays on that line there. Why? It's because, of course, the biggest part if you're measuring the entropy in terms of the largest amount of space Of course, there are macrostates which are below the largest one, which are degenerate. So when you add those up together, of course, that's a bigger part of space. So you always get that kind of situation. Put that aside for a moment. I'll come back to where I'm going to deal with that in a minute. Let's look at the other part of the problem, which is in terms of typical behaviour. Now, this is the standard error-in-first picture. We have a macrosate, and there are, if you look at page 33 of the Encyclopedia article, the 1912 Encyclopedia article, there are four possibilities in terms of the way that the system behaves in relation to its macrosate. It either starts at a smaller one and goes to a smaller one, or it starts at a larger one and goes to a larger one, or the probability is mu. And I'll divide it up into mu plus plus mu minus minus mu plus minus. And of course, reversibility means those two bits are the same size. And the hash bit is the bit that goes to a larger matrix. Now, there are two things here. like that to hold in order for, in order for, to get the, well, to get the pictures I've

47:30 got here essentially, that this, these two bits are the same size, so that this bit must be very much bigger, that is to say the bit that goes to, that's come from a larger one and goes to a larger one, and that's larger. So that inequality, which is what the analysis depends on, of course depends both on the dynamics and your choice of macrostates. And it seems to me there's no general analysis for deciding if that's true. You can do it as case-by-case for particular systems. Now even if that is true, that's only half the problem. because, of course, if that was true, then you could say, let's choose a point for the system. It's in a particular macro state. Now, if you use a sort of general kind of replacium, you're probably, then, of course, you're likely to choose it in here so the entropy is likely to increase. But, of course, that's not just what we want. We actually want the entropy to go on increasing. In other words, we want it to be in the corresponding bit of the next one. Of course, it's actually somewhere in that bit, and the dynamics would tell us whether it was in that bit, as it were, down here. You cannot make a new choice. So, I mean, the whole discussion in Neanderthal seems to be completely flawed, as far as I can see. But I don't think it matters, because I think in fact this local way of looking at it is not the way to look at it. It seems to me that the way to look at it is to look at it in a global context. And if you look at it in a global context, what you're looking for is not simply, I want the entropy to increase, you're going from one matrix to space to another. Well, heads for a lot of this too. I don't know where it's gone anyway, never mind. want is to pack the argument up in such a way that we've got something like thermodynamic-like behaviour and we need it to be typical. Okay, so let me first, there were two things, so

50:00 let me go back to what I said about equilibrium and the way I propose to deal with that. Remember that it's not simply sitting in the largest macro-state. For any finite system, it's not entirely true. So what you could actually do, of course, is to make some sort of band and say, well, if it's in that band around the largest macro-state, it's an equilibrium, but that's totally artificial. And then you have some artificial division between fluctuations out of equilibrium and fluctuations within equilibrium. Seems to be the easiest way to cut the Gordian knot is simply to abandon. equilibrium, and the two-value quantity, and replace it by a continuous property of equilibrium likeness, which in the original paper I wrote about it I called commonness, and I'm stuck with that title now, I'm not sure I would do want it, but anyway, it's with me. Anyway, so that's the way I would deal with equilibrium, I'd get rid of it as a binary property. Now, What about the getting typical behaviour? Well, in order to get typical behaviour we want the following to the case. Or thermo-lambda-like behaviour, I should say, on a particular trajectory, isn't it? We want most of lambda in phase space to be in macro-states close to the largest one. And then we want, on a particular trajectory, we want it to spend times, the time it spends in a macro state should be proportional to the size of a macro state, because then it will spend most of its time in the diverse macro state. So the time it, on a particular trajectory, at x, it spends in a macro state mu would be that. Well, of course that's the definition of the system being egotic. And, of course, most systems are not egotic. So how do we deal with that? Well, my suggestion is that we weaken this. I mean, this is more than we need, which is a good thing because we don't have it or only have it very rare. Let's suppose the system has an egotic decomposition,

52:30 which means that the phase space is divided up into components, all of which are invariant under the flow. In that case, of course, within a particular member of this lambda kappa, the time spent because it's ergodic in there, the time spent will be given by that, and then it will be it will be thermodynamic-like in that particular member of the decomposition if these are roughly equal to each other. where does that take us? Well, having got our, I should say that I don't think a godic decomposition is as bad as all that. actually has an ego key composition and in fact that was that's the example the britain one uses all the time to show that the god is not necessary so I think I've sort of smuggled it in at this point to try to show that in fact it is necessary but it was rather weak performance. I think we need two sorts of probabilities we need the probability of the system being in a particular member of the decomposition, which I've put Kappa, and we need a probability of the system being in a particular macro state within the member of the decomposition. Now, if you follow von Plato, and in fact Boltzmann at one point also made a similar suggestion, then you can take that probability to be, since this is component, you can take this simply to be retired, it spends in there, and then of course the total probability is simply going to be given by the usual formulae, and we can also write down, I mean, within each member, we've got a probability density function, which essentially might be a chronological distribution, and then we've got a probability density function over a whole lot. The system will, thermo-lap-like behaviour will be typical if the probability of picking out, when you pick out a member of the decomposition, of all the ones in which the system is not behaving in a thermodynamic light way, is very small.

55:00 Okay. Now, how do we deal with variables in this context? There are, because we've got three levels, we've got three sorts of variables. We've got functions of phase, little f. We've got functions over the macrostates, which I've called curly f. And we've got thermodynamic variables, which I've called straight f. And we need to have a relationship between the three as some sort of changing relationship. The relationship between f and curly f doesn't phase function. I mean, this is not all phase functions correspond to thermodynamic variables. The ones that do, plausibly, are continuous, and they're invariant under permutations of the microsystems. Now, if that is the case, even if the macrostate is very large, because it's made up of lots of cells in which you've got all these permutations, then the actual variation of F over the whole matrix state to be extremely small, so in fact this is approximately equal, and you can take this to be your definition without any real problem. Now this is the relationship between the statistical mechanical variable, currently F, and the thermodynamic variable, and we take, along a particular trajectory, we take fx0, which is f along the trajectory going through x0, to be simply the result of loss of measurements. But of course we've got no garbage decomposition, so there will only be one of those for each of the, all the trajectories in any particular member of the decomposition will be the same, So we'll have F kappa values, and then, of course, we can take F to be simply the average of those over the e-composition, and then the time average will turn out to be this.

57:30 Now, it doesn't have the problems of an infinite time average in the traditional sense, because it's actually simply a picking out the finite number of points. And if the system is behaving in a thermodynamic-like way, in that member of the decomposition, then f is simply the time average. OK. The last few minutes I've got, let me just relate this to Gibbs, because Gibbs appears in the title. Well, the usual hand-weighting is to say, OK, well, if the largest macro state is practically all the energy service or whatever, then there's really no problem relating the two, and in fact, in the limit of large N, these are the Gibbs entropy and the maximum value of the Boltzmann entropy, and in fact the time average on log of a member of decomposition, which is behaving a thermodynamic-like way, all tend towards each other. But we've abolished equilibrium, so how are we going to understand what the Gibbs entropy is? Well, my suggestion is that we abandon any discussion of the Gibbs entropy with a time-dependent probability density function, and we handle it in the following way. Let's think about it in terms of a gas in a box. Then what will be the normal standard story? If you had your box, you let it sit for a while, the Gibbs entropy in terms of a probability density function which is the equilibrium probability density function. If you put a partition in your box, then of course the probability density function would differ from the original one only in terms of the size of the box and you'd have two different entropies. Now, let's ask the following question

1:00:00 Let's suppose that X nought was a point which was such that, a phase point which you construe all the particles being one end of the box What would be the probability of being in a small region x naught. The answer, of course, would be whether it would depend on whether the partition was there or not. And so it would also depend, of course, on whether a partition had been there and had been removed. According to the standard interpretation, if the partition had been there had been removed, you would reuse the probability density function for what it was before, because you would then think it would be evolving. On the other hand, of course, if it had never been there in the first place, you would use that. Now, the question is, why should you use different probability density functions? How does the system know there was a partition there if you'd taken it away? The exact, the future development is precisely the same in each case. I mean, one gas will have a sort of false memory syndrome, But in terms of its actual future, it will be precisely the same. All the particles in the box were in one half could easily have occurred simply because the particles actually were in a rather peculiar fluctuation. So why don't we simply change the probability density function when we take the partition away? and this is then the picture we have I did some simulations this is this is the one of these problems were around those two dotted lines one of them is the with the gives that the false management people the maximum microstate the other one is the type that is the gives entropy and this is my this is the actual Boltzmann entropy that I simulated, and you can see they all sit very close to each other and simply changing the Gibbs entropy as you take the partition works perfectly well. That's everything apart from the conclusion, so I'll need to put that up.

1:02:30 I recommend throwing away the binary property of equilibrium, replacing it by continuous property. The requirement of the system, thermodynamic like behavior being typical, can be achieved in terms of an agotic decomposition. We're not worried about sets of measure zero. We accept the fact that sometimes systems won't behave, won't behave thermodynamically with, as it were, measure something larger, but small, but larger than zero. And, well, the rest is less than this. I've used ergodicity, but I would claim that I have not used it in a standard way and that I escape the normal strictures associated with it. Thank you. And we will have to move immediately to the next speaker, Jos Hufink, who will be talking about taking algebra too seriously, and the illusion of irreversible behavior in statistical dynamics. Statistical. Well, okay, yeah, it says statistical here, but they messed it up. Stochastic, at least. I was wondering about that. Well, I'm delighted to be here. I want to thank Roman for organizing this workshop. It's been very interesting so far. My title, of course, is a tribute to Craig's previous paper on taking thermodynamics too seriously. but I'm not in no way meant to criticize anything like that I actually want to talk about a particular approach to the foundations of statistical mechanics an approach which I will call stochastic dynamics and I'll try to sketch it very briefly in the short time I have and let me just say it moment that one of the aims of this particular approach is trying to solve one of the main foundational issues in statistical mechanics which is to say why or how

1:05:00 systems would show this behavior to works in equilibrium states if they're they're presently in a non-equilibrium state in other words to solve the problem of irreversibility and it's different approach from the ones that rely on a on the past hypothesis now the curious thing about this approach which as we will see relies very heavily on the on the properties of Markov processes because it models statistical mechanical systems as a statistical process in particular Markov process the peculiar thing is that there are different ways in which you want to approach the mathematics of stochastic processes and how that I'd like to to focus on is a probabilistic and an algebraic characterization of Markov processes, and although both of these characterizations are, of course, equally valid, it seems like one of them is very apt to lure you into the kind of illusion that you have actually derived irreversible behavior without having it put into the equations. And it's sort of, at least that's my impression of the situation, it sort of takes the step of getting back to approach to the whole issue, and to see that what you've actually done didn't solve the problem at all, that it was actually all the sort of illusion that you, the sort of inherent in this algebraic characterization. That's why I chose this particular title. so let me first try to to get on to the motivations of this approach I would actually like to stress that there were two main problems in the foundations of statistical mechanics which are relevant for this issue one of them is what I've already mentioned that we seem reversible behavior, an approach towards equilibrium, manifested in all the microscopic systems around us, and also demanded by the theory of thermodynamics.

1:07:30 And yet we seem to believe that at the microscopical level, the dynamics of all the systems we see, or present in the universe, seems to be reversible, time How can we reconcile those two issues? And there are many approaches to solve that, or at least approach that problem in the literature. There's a second problem that seems to be sort of less well studied, and that is the level of autonomy that we see in the behavior of microscopic systems. And I've sort of tried to illustrate it by writing down the Boltzmann equation, and don't worry about all the formula, all the symbols in this particular formula. So what it is here is f is just the density of particles in the gas, the density of particles with particular velocity. And the left-hand side of the Boltzmann equation tells you how that density changes over time. And there's a lot of stuff on the right-hand side, even in the special case that I've been studying here. But the important thing is, if you want to know how that density changes over time, all you need to know is what the density looks like at that particular time there's an autonomy involved in this law that says how things change characterized microscopically depends only on the characterization microscopically at that particular time in other words, although we know there are 10 to the power of 23 particles in the gas a lot, really a lot, of microscopical detail involved in the dynamics of the system. It seems like all that detail is irrelevant if we are just interested in predicting how the system behaves on the macroscopical levels.

1:10:00 There's an autonomy of the macroscopical variables in the sense that the way they change seem to depend only on the macroscopical variables as well. And that is a feature of semi-microscopic or macroscopical evolution equations that is, I think, much harder to explain by something like a past hypothesis. So, here's what the approach that I like to call stochastical dynamics looks like. What we simply do is, well, we forget about the idea that on the fundamental level our systems are described by Hamiltonian deterministic time-referential ethereum dynamics, and we replace them by an indeterminist evolution, namely by transition probabilities that in the moment will also behave. We'll have the Markov process so that they will define a Markov stochastical process. There are various ways in which you can arrive at such a thought, and you will find all of them in the literature. One of them is simply that we have to, that we arrive at the microscopical description of our system by coarse-graining, that is to say we forget about microscopical details in the internal state of a gas or any other system we're interested with. So we define a sort of partition on trace space, and we just keep track of in which cell of the partition the system is located. And now if we want to study evolution from one cell to another, we will find that that evolution is no longer deterministic, but we can only specify transition probabilities. So that's one approach. The other approach is to say, oh, now, it's not the internal degrees of freedom that are so complicated. Rather, it's the system that's in contact with an environment. And the environment is complicated. But apart from that, we sort of play the same strategy. We say, oh, we're only interested in the behavior of the system, not in the environment. So we trace out all the degrees of freedom of the environment.

1:12:30 And what we end up with is an evolution law that's only stochastical and indeterministic. We cannot say how the system evolves if we trace out all the variables of the environment. Another approach is to say, well, let's forget about how I just argued in getting to this indeterministical transition probabilities. let's simply suppose that this is a new type of fundamental law so that we can actually forget about deterministic Hamiltonian evolutions and say well maybe nature is just like this, this is the most fundamental description we can get of evolution laws. And you can find that idea, too, in literature. I think Michael Mackie has been the most prominent voice, but even he is, as far as I understand, not particularly claimed that this is the only correct way to go. And then there are people like Rod Ray Streeter who actually took this phrase in the title of one of these books, who seems to be rather neutral about what is the reason for studying this type of approach, and just, well, you know, you can just study it for its mathematical interest without saying why it's interesting. Okay. In all cases, what people argue for is that the transition probabilities have a Markov property, so that we end up with a Markov process. And the real question is whether that will lead us to a description of the irreversible pages. Now, I have a quote here, maybe it's too long for this particular purpose, but it's from Oliver Penrose, one of the people who I had on the page of the slide. And what you see here in particular is, let me say, in present theory, that is the approach

1:15:00 The additional postulate which is needed to derive irreversibility is the Markov postulate, meaning that a stochastical process is in fact a Markov process. So, what he's arguing here is that once you have adopted the Markov property for a stochastic process, then you're able to derive the approach to equilibrium, and without it, you couldn't do it. Well, here is what a stochastic process is from a probabilistic background. It's actually nothing else than one particular probability measure on some complicated phase space, of course, with a set of measurable subsets defined on that. And the thing we're interested in here is a family, an infinite family, actually, of different random variables, parameterized by some parameter t, which we interpret as time. So these are, say, physical quantities, microscopic quantities, at particular times. And in order to make this more tractable, what people always introduce is a hierarchy of distribution functions for these random variables to take on particular values. And so you can ask, what is the probability that But at time t1, this quantity had this value, and at some other time it had another value, and at another time it had still another value. And from these you can obviously construct conditional probabilities all in the usual way. There's nothing tricky going on here. and you have distributions like that for every number of time instance you like so you get a

1:17:30 whole hierarchy of these equations the Markov property is now really very simple suppose all these time instants are ordered for one reason. The idea is that the value that the random variable takes at the last instance happens to be independent of all the variables it took on previous instance except the very last. And this means that all the complicated conditional probabilities you can think of in this hierarchy in fact all reduce to two time conditional probabilities that's all you need to know plus the the single time distributions and these will characterize the stochastic process completely once you know that it has the market and that's another condition that is important, and in fact, in some sense, not quite so trivial. It's what's called homogeneity, meaning simply that these conditional probabilities do not depend on the specific values of the time instance, but only at the time difference. So the conditional probabilities are time translation invariants. Now all this is very beautiful and I might also add that it's it's straightforward to show that Markov processes by this particular equation called the Comcor of Chapman which has a sort of reproducing kernel here. And let me just flash that back. And I've gone to suggest how you can think of a specific process from an algebraic point of view, or perhaps algebraic is not exactly the right word.

1:20:00 more or less a semigroup group. And basically what you notice in this case is that it's very attractive to think of these conditional probabilities as evolution operators. You can think of them as matrices to begin with, and then if you look at from a sort of Chapman equation, you see that what it actually looks like is a matrix multiplication. So this looks like it's an evolution from t1 to t2, this is an evolution from t2 to t3, this is an evolution from t1 to t3, and summing over the middle variable just means that you take matrix multiplication of those two matrices and so this is what it looks like in operator notation and it's very suggestive it's just the group property of ordinary evolution operators with the proviso that all these time differences have to be positive, so it's not really the group property, it's a semi-group property. And so, putting these things together, there's even a further step that becomes very attractive, if almost inevitable, and that is that once you think of these these conditional probabilities as if they were evolutions, transition probabilities, which is the change in the magnitude which already suggested there are evolutions. You also attempted to think of the one variable probability distribution now as the analogy of a state. So this is a state that evolves by this evolution operator it gives you a state at a later time and that is a way of thinking which we all know from various other theories of physics you can think of quantum mechanics whether this is unitary evolution or you can think of ordinary classical deterministic

1:22:30 and there are obviously various advantages with that approach. You can bring in the whole machinery of linear algebra to your great advantage. You can study the spectral properties of these matrices and they will tell you lots of interesting things. You can bring in the Spectral Theorem, of course, and Perron-Colbenius Theorem. Lots of interesting things to say just because you can help yourself with all that mathematics. How am I doing? Ten minutes. Ten minutes. Okay. state. In particular, let me say this, a question of course is, is there going to be something like an equilibrium state? Well, it's very natural to think of an equilibrium state as a stationary state under this evolution, so an equilibrium state is that state which is of the evolution. Hopefully there is one, and hopefully there is a unique one. That is not always very easy to guarantee, but you can think of the conditions that have to be imposed to guarantee them. Now, let me start. The question is, can we derive, or is there some kind of approach to equilibrium in this particular theory, just using the assumptions I just mentioned? Well, there are various theorems you can write down, and the most general is this one. If the relative Shannon entropy, in this way, and you think of two one-time probability distributions, the analogies of states, and you ask yourself how they evolve in the course of time, given the same transition probabilities, then it's possible to show that at a later time this relative entropy increases compared

1:25:00 to what it was at an initial time. And this relative entropy is a measure that shows how, roughly speaking, how similar probability distributions are. And so what it says is that, at least in all cases in which the inequality is a proper inequality, it says that typically probability distributions tend to become more and more alike if they both evolve under this stochastic transition probability. which is roughly already some kind of irreversible behavior, right? It tends to become more like it. Now, much stronger theorems can be derived if you impose additional conditions. For instance, conditions that will guarantee that there is an equilibrium state and that it is unique. And then you can, under these conditions, also show that probability distribution will evolve towards that equilibrium state. And all of these things are very desirable if you think you want to explain the approach to equilibrium using this particular triangle. And yet, I would like to argue now that all of this is an illusion. but let me first try to formulate what you can find in the literature think of that theorem i just showed you about the relative entropy it had a greater than or equal sign and you can ask yourself well when is the equal sign non-applicable so that there is really an increase of similarity. Well, it happens to be the case that a condition is just the case when the transition operator is non-invertible.

1:27:30 So it doesn't have an inverse as a matrix, not a matrix. And so, it seems like the non-invertibility the transition probabilities is just the right condition for showing the irreversibility of the behavior. But, the important thing to note is that non-inversibility is just a property of matrices. If we ask ourselves, does a particular matrix have an inverse matrix? And that happens to be the case. But if you look back through the probabilistic characterization, you will see that these conditional probabilities had only a tenuous relation with matrices discarded. If you think of them as conditional probabilities, So, because this is how probability of time zero evolves into a probability of time t, then there is always a proper way to inverse that relation. But the proper way to do it is not by taking the inverse matrix, it's by using Bayes' theorem. So, there's always another conditional probability defined by Bayes' theorem that will tell you how the original probability is calculated from the later probability. But, of course, defining this sort of inversion by Bayes' theorem is something which is completely different from taking the matrix from it. So the non-existence of the matrix doesn't mean that we can't give the transition, backward or consistent probabilities from the later to the earlier times, these conditional probabilities are guaranteed to exist. And in fact, under the Markov's tradition, they look also for a semi-group, a semi-group who's backwards. And whatever you suppose about the forward

1:30:00 You can also be modelled with a backward theorem, and for the backward theorem, you will also see that the theorem about relative density is true. So if you study how this distribution behaves in the other directions, they will also be They will also approach the difference. So, do you happen to really show anything at Loa? Um, and to show perhaps this is...