Irreversibility & 2nd Law of Thermodynamics — Part 2

0:00 Charged capacitor via a very large resistance into another heat path. So if you close this circuit, a current will flow through the resistor, and it will give off heat to the heat path here until the capacitor is discharged. Now, this is a process which, by all standards and all authors, would be considered as a non-recoverable process in Planck's sense. Because after the capacitor has been discharged, there's no way in which you can take the heat that has been added to the heat bath and use it to recharge the capacitor. But at the same time, by making that resistance very, very, very large, I can make the current very, very, very small and approach a situation in which, at every instant of time, the system is very close to an equilibrium state. So that sequence of processes can have a limit in which the process would be called quasi-static, even though no matter how slowly I proceed with the process, it would always be non-recoverable. Here's another example. It might be maybe a bit simple and naive, but it's one that's been given by Planck himself You can consider a harmonic oscillator, say a pendulum, without friction, of course. And the state change to which that system goes is completely periodic. So out of itself, it always returns to any initial state in which you might have prepared it. So the process of that oscillator is completely recoverable, according to Planck. But he also admits that it's not quasi-static because, well, depending on the amplitude, that harmonic oscillator will not be very close to an equilibrium state during any course of its swings. How am I doing? Ten minutes. Ten minutes. Oh, dear. That is very good.

2:30 Well, I'll have to go somewhat faster then. Let me try to explain how these two different notions of reversibility came about. Let's first look at some quotations from Calvin. who is also known as William Thompson, of course before he was knighted or made a lord this is a paper from from 1851, although the reprint is from his collected works later. This is the first case in which anybody in the history of thermodynamics wants to point out that there is a tendency for systems to have a particular unidirectional form. Of course, writing this early, Kelvin doesn't use the notion of an increase of entropy because that notion hasn't invented yet, but he talks about a tendency towards the dissipation of mechanical energy. And I'm not going to read the entire quotation. It has some very curious aspects to it. But I would simply note that he says that he lays down a set of propositions regarding this dissipation of mechanical energy and claims that there are necessary consequences of the axiom that it's impossible by means of inanimate material agency to derive mechanical effects from any portion of matter surrounding us

5:00 below the temperature of the coolest surrounding object. And what are these consequences? Here are a number of them. If heat is created by a reversible process, so that the mechanical energy just spends maybe stored to its primitive collision, there's also a transference from a cold body to a hot body of a quantity of heat corresponding to that. On the other hand, if heat is created by an unreversible process, such as friction, and the full restoration of it to its primitive condition is impossible. These are actually the first cases in which terms like reversible occur in the physical literature. You can see this already from the fact that he doesn't use the now common phrase irreversible, but he talks of unreversible. But what's more important, you can see that Kelvin draws this very close connection between reversibility and the possibility to restore an initial collision and irreversible processes in which a full restoration is impossible. I will have no time to go into the question of whether Kelvin actually proves that there are necessary consequences But I cannot resist in giving you a picture of the consequences to which his conclusions lead to, the famous heat death of the universe, something to which Calvin explicitly draws attention in his paper. And this is a late 19th century depiction of the future fate of the universe because, well, at some point the sun will stop shining and we will be lost of our source of energy. and since every process that happens involves a dissipation which cannot be restored to its initial condition the fate of the human race will look something like this here you can see a mother still holding to her child

7:30 in this landscape full of icy conditions and this is somewhat later and there's nothing more left Skeletons. At the very same time... No, this is long. I have a quotation from Clausius, somewhere. Oh, here it is. And slightly later, from a contemporary English translation of Clausius' works, where he defines his notion of reversibility. He says, when an alteration of arrangements, which is what we would call a process, takes place such that force and counterforce are equal, this is actually showing that Clausius was more an Aristotelian thinker than a Newtonian then the alteration can take place in the reverse direction also under the influence of the same forces but if it occurs so that the overcoming force is greater than that which is overcome the transformation cannot take place in the opposite direction under the influence of the same forces that the transformation has occurred in the first case in a reversible manner, and in the second case in an irreversible manner. Now, force and counterforce being equal is like pushing a piston with a force which exactly is equal to the pressure of the gas

10:00 that has to be overcome. And Clausius is well aware that, I mean, if they're equal, then nothing would happen. So he continues by saying that, strictly speaking, the overcoming force must always be more powerful than the force which it overcomes. Out of the excess of force, but as the excess of force, does not require us to have any assignable value. We may think of it as becoming continually smaller and smaller, so that its value may approach to not as nearly as we please. Hence, it may be seen that the case in which the transformation takes place reversibly is a limit which in reality is never reached, but to which we can approach as nearly as we please. Now, although the language is old-fashioned, the idea that is represented here is clearly that of quasi-static processes. And we also see that Clausius was well aware that you can only apply that notion in a limit in which the disturbance of equilibrium, so the non-equality of the overcoming and the overcome force, approaches to zero. On the other hand, let me remark one thing more. This is an English translation. If you look to the German original text, you will find that Clausius uses the word umkeerbaar here. Not very important, perhaps, at this point. But here is Max Planck in his famous book on thermodynamics. And since I assume that the audience knows German, here it is in the original language. The only way that can be made in a way, is irreversible. All the other processes are irreversible. So, if a process is irreversible, it is not possible to be able to do not from themselves, that is also in many mechanically processes, which are not irreversible, but it is required to be required, that there is no means to be required to do with all the nature of the regent, if the process is over,

12:30 And as I said in the beginning, there was always this question whether available in nature is a modal question or whether it expresses something actually out there. In Planck's case, you see, he talks about in der Natur vorhandenen recensien. And this is, well, available, but in almost the most actual sense you can give to that word. It's vorhanden. They are out there, he would almost say. Thank you. that I would like to make, is that Planck is really the author who has stressed that this particular notion of irreversibility, the one that he just defined, and which, as I said, expresses this kind of irrecoverability, that you can never restore the initial state of the total universe once an irreversible process has happened. elevates that to the central key meaning, the key issue in the second law of thermodynamics entropy increase is for Planck the sole criterion for irreversibility and he expresses the second law of thermodynamics as follows he says, every physical or chemical process occurring in nature proceeds in such a way that the sum of the entropies of all the process is increased. In the limit for a reversible process, this sum remains unchanged.

15:00 I would have liked to give you an argument of how Planck argues for this conclusion, but that is not a very simple matter, and it's a very complicated and delicate issue, in part because all these 11 additions of the four-laying on Uber Thermodynamic argument. And it's also important to realize that the English translation of this work is not very reliable. And among the most difficult and brutal errors in that translation, Planck uses both the The word reversible, as we have just seen, roughly in the same sense that Kelvin used it before, but he also uses the word umkeerbaar in the sense that Clausius has used it, namely as a quasi-static process. And now you may see what happens in the English translation. The word reversible is translated as reversible, and the word umkeerbaar is also translated as reversible. Planck tries to make a clear distinction between two different notions. In the English translation, it's just collapsed into a single notion, and you start to wonder how an author that has used a particular phrase all over his book suddenly decides on page 90 or so, suddenly to give a definition of that phrase. But Planck is not really so bad as his English translation would make him to be. But in any case, But I still have to comment that in all the versions of the argument that Planck gives, his argument, his derivation of this statement is really not very satisfactory at all. And, in fact, it always applies only to gases or fluids, in the most general case, systems in which, formally speaking, have just two independent state variables. And it's either a system like that or it's a larger system composed of such fluids. in which he at some point explicitly notes

17:30 that he excludes chemical changes from happening to that system. And yet we see him finding a, in formulating a conclusion in which he specifically includes chemical processes, although he has been excluding them from his entire consideration during the argument. so how this can happen is still quite unclear to me perhaps I should wrap up here the confusion which I illustrated in the course of my talk, the different definitions of reversibility which tend to be lumped together even in good textbooks in thermodynamics and also in many otherwise excellent expositions in philosophy of physics, I think it caused a great deal of confusion. It's curious to note that while the second law of thermodynamics seems to be a sort of prime example in the philosophy of physics literature as a law that expresses this error of time. Really, the main concern of thermodynamicists when they talk about irreversibility is always either that Clausius sense of umkeerbaarheid, which we would today call quasi-static processes, or the Calvin-Planck sense in which you talk about the irrecoverability of initial states. But as I said, that sense of irreversibility is not necessarily tied up with an arrow of time. I think it would be much sharper to say that it's related to a notion of a ravage of time rather than the arrow. Now, some final points that I haven't perhaps discussed during my talk. But, of course, for physicists, the real question, the real problem with thermodynamics is how it relates to a microphysics, either classical statistical mechanics

20:00 or perhaps quantum statistical mechanics, or what future theory of physics there may be for the microphysical constituents of thermodynamical systems. We believe that, well, apart from electro-weak interactions, that all such physical theories will have the property of time reversal invariance. And so the question is, how does the second law of thermodynamics stand if we would try to explain it or perhaps reduce it to a more fundamental level of physical explanation? Well, I have no particular answers for that question, but I think there are two points to remember. Boltzmann famously produced an argument which, by many people, is seen as a first example, or at least a first candidate, for explaining second law in terms of microphysical considerations in his famous age theorem. which seems to be like the second law in the form that it says that there is a quantity H which can only change in, well, in this case, it can only decrease, entropy can only increase, but then one usually simply adds a minus sign to H to make it a candidate for entropy. I think, I believe that that theorem is not sufficient to derive a second law of thermodynamics, and it's also, there is really no need for a theorem that strong. And the arguments for that are, of course, well, we have seen that in thermodynamics, in classical thermodynamics, the emphasis is always on equilibrium states. So even in Clausius' formulation of the second law, there is a statement that says, In any adiabatic process, you can assign an entropy to the final state, if that's an equilibrium state. You can assign an entropy to the initial state, if that's an equilibrium state. But you cannot assign, in general, an entropy to an intermediate state, because they're usually not equilibrium states.

22:30 So, there is no need for a theorem that says that there is a quantity which is monotonically increasing in the course of time. You only have to find a quantity which is larger in the final state than in the initial state. So Boltzmann's theorem is stronger than needed for classical thermodynamics. And in fact, I think the major claim to fame is that it gives a candidate for a non-equilibrium version of entropy. But it's also not sufficient. And I haven't argued for that in any detail that might convince you. But Boltzmann's theorem is only derived for completely isolated systems, or what people often call closed systems. The second law of thermodynamics, instead, applies to all adiabatic processes. That is, processes in which you exclude any heat transfer between the system and its environment, but in which you may allow, say, the violent stirring that occurs in Jules' pedal wheel experiment. That's also an adiabatic process, and it's also an example of an irreversible process according to the second law. But it's not something which is covered by Boltzmann's H-theorem, because, as I said, that only considers isolated systems. So that theorem is, in a sense, both too strong and too weak to capture the second law. In a more perverse sense, one might therefore conclude it has nothing to do with it. The final question, which I still consider to be an open question, is whether classical physics, say Hamiltonian mechanics, could actually also incorporate processes which are irrecoverable in Planck's sense, Planck's or Kelvins. I wouldn't know that because, as I said, in order to decide whether its given process is irrecoverable, you need to consider also what happens to its environment. And even in a case, say, in which I would try to reverse the rotations of the planets in the solar system,

25:00 them. Of course, that could happen mechanically by appropriate collisions with other bodies. I could reverse the motions and make them go back to, say, a constellation of several centuries ago. But is that a complete recovery according to Planck? No, of course not, because now I've changed the momentum and the angular momentum of the bodies that performed the collisions. In order to have a complete recovery, according to Planck, one must also bring back the auxiliary apparatus to its initial condition. So I would have to somehow retrieve that momentum and angular momentum from the cricket bats that I used to kick the planets. Can that be done? Well, I assume so, but in order to find out really that is the case, you would have to go to a form of constrained Hamiltonian dynamics and see whether, say at some end point, you can retrieve all that angular momentum. And constrained Hamiltonian mechanics is a much more complicated theory than Hamiltonian mechanics of free systems. And whether or not such irrecoverable processes can be found in purely classical mechanical systems, I still think to be an open problem, but perhaps there may be somebody in the audience or elsewhere who knows the answer to this. So that's the final open question that I would like to end this talk with. Okay. Thank you. Thank you for this clear discussion of the different meanings of reversibility, irreversibility in thermodynamics. And we have about 15 minutes for discussion, so please be short in your questions. Thank you very much for these interesting distinctions. I make a short remark first to the second law

27:30 if somebody says which is sometimes said that this defines the error of time but I think this is rather nonsense because what the law says is that the entropy increases in time so this proves that you presuppose already a kind of concept of time at least it could at most implicitly a little bit characterize but certainly not more This is the first remark. Second remark is that I wanted to say that Boltzmann in his right, you know, you said that there is a development in Planck. Of course, a development in Planck concerning this concept and then concerning his view on thermodynamics in general because at the beginning he says he hopes that the second law can be reduced to dynamical laws and so. Boltzmann. Yeah, Planck says at the beginning. And then later, I mean, his assistant was Zermelo, Ernst Zermelo. And Zermelo attacked Boltzmann and so on, but was then corrected by Boltzmann. So there's a long development, I agree with this point. But Boltzmann, I found always was more modest because he did never speak of reversibility. but if he takes this word then he says immediately what he means is very improbable recurrence of the state I mean of the microstate very improbable recurrence this is all what he claims and he says then this is very very compatible with Poincare's recurrence theorem and so on and so on so my last point does classical physics allow irreversible processes I would answer yes a point in case is dynamical chaos you have underlying dynamical laws which of course do not define any direction or allow any direction but the process itself and a lot of parameters for instance phase density and so on, they are non-recurrent of course so this is a nice case it. Thank you for your question. Let me first go back to your first remark. You said that it's often said that the second law of thermodynamics defines an error of time, and you said

30:00 this is nonsense. I agree with that, but of course, as I said earlier, what one actually of time is is a complicated issue because the same term can be used to cover so many different meanings I would say that if you understand by the second by the error of time the idea that there is a distinction between earlier and later events, so that time has an ordering according to that relation, then I'm in complete agreement that thermodynamics doesn't define that. Instead, every ground-laying consideration in which we use to derive the second law of thermodynamics, like a consideration of the Carnot heat engine, presupposes that you can order different sequences of a process in time. So that ordering relation is presupposed in thermodynamics and not derived in that. However, if you understand the error of time in a different sense, say, in the sense that some processes are allowed theory, well, the time-referred processes are not allowed by the theory, then I'm not so sure, because processes in which entropy increases, one would assume, given that entropy itself is non-refersel invariant, it is reversal invariant, that therefore the reverse process would be one in which entropy decreases. So the error is in the process, not in time. Exactly. That I agree completely. Now, your question about Planck and Boltzmann is more complicated for me to answer. I'm actually not sure that... I don't know any work of Planck in which he originally claimed that the second law of thermodynamics could be derived from dynamics alone.

32:30 He has, I hope... a lot of work in the future and something like that. I have given a citation in my abstract, but there is such a thing, yeah. In a letter, it is in a letter, it is not in the book. I see, that's interesting. But immediately after this he says, but I do not go so far as my friend, As my assistant, Zermelo, who claims that there is an incompatibility between dynamical and statistical law. But then, well, the final point I want to make is that the recurrence objection, which, well, introduced by Poincaré and used to good rhetorical purpose by Zermelo, is another issue from reversibility, I think. that in the late 90s of the 19th century, that was the subject of discussions, also by Boltzmann. But the reversibility objection already goes back to Loeschmidt. And it is true, as you said, that Boltzmann never uses the word reversibility, as far as I can remember. But his phrase is usually to say, He talks about any attempt to derive the second law of thermodynamics in the form of a circular integral of heat divided by temperature is less than zero. And usually one understands this as saying any attempt to derive an increase of entropy, like in its h-tune. And so that is the context in which I would place this discussion. Well, this follows up on Boltzmann and Planck. I think what Planck said in 1913, that he thought that you have the molecular disorder of Boltzmann as the way to found statistical laws on dynamical laws by a supplementary condition. Now, that was, of course, against all the people who read Boltzmann more and more in a probabilistic fashion, which was not so much accessible to Planck, project of 1913 were given up. But my question is this, Planck here is presented as somebody who

35:00 makes very tiny distinctions that fluctuate between umkepa and irreversible, but something that somehow always remains since the 1880s is this idea, you know, there are two principles in the world, basically energy conservation, and there's the second law, and there's two worlds, the irreversible world where the principle of least action is not valid and there is this world where everything as a goal of scientific research can be formulated by way of an action principle. So how does your presentation of Planck as somebody making very many distinctions even floating ones reconcile with what he emphasized for all his life that there were so to say two parts in the world? I don't actually see such a distinction. Yes, Planck makes very tiny distinctions when he needs to, and also he makes sometimes, I have to say, logical errors in his thinking where he should not have done that. But this idea that there is a huge overpowering principle that all the processes in the world are irreversible and that there is a subclass in which, well, as I quoted, the ideal cases in which processes are reversible. That is something that you can find in all the editions of the four-lasin and uber-thermodynamic. So I think that is a very constant threat in his life. I'm not sure if that answers your question. But, of course, the project in which you try to derive irreversibility from microscopic disorder, molecular and molar chaos, as it's sometimes called, is also one of the many, many approaches that was instigated by Boltzmann. And this is one in which you would blame all the initial conditions of the gas, or you would retrace the irreversibility to the initial conditions of the gas.

37:30 And, of course, in that case, you would have irreversible processes, well, you would have irreversible processes in some sense occurring thanks to this particular choice of initial conditions, while at the same time having the time-reversal invariant theory. also, of course, imply, and this is, of course, a notion which is not in the least contradictory to many claims by Boltzmann, that the second law of thermodynamics would not be an exact law because, in principle, there are deviations from it allowed by microphysical mechanics. And these exceptions to the second law were exceptions that were not explicitly recognized when he wrote down the AIDS theorem. In that case, even Boltzmann still thought that the second law of dynamics would be an absolute law and that he could derive this absolute validity from underlying mechanics. Thank you very much for your nice remarks, but my question would be if it's possible to transform or transfer your results on human culture, because you relate on chemical resources, and this is only a matter of nature or a matter of physical world. But the cultural erosion could be reversed by human work. Cultural erosion? Yeah, cultural erosion we have all over Europe. And I think if we improve the culture and the cultural resources by human work, we could recover from this erosion, you know, maybe. I think I agree with that. I mean, I would see no reason from thermodynamics to imply that that is impossible. Instead, I haven't addressed that in any way, but there is, of course, a well-known tendency to extrapolate insights from physical theory and thermodynamics in particular to the world of human culture

40:00 Even, well, there have been many discussions on the question of whether the second law of thermodynamics is incompatible with evolution theory and so on and so on. And I think that one should be very, very careful in that case. Thermodynamics in its physical formulation has to do with work and heat and temperature and entropy is defined in those concepts. it's not easy to associate an entropy with culture or a social entropy has been used in some textbooks. That is at best a metaphor which, well, if it's used wisely then it's used prudently and you cannot derive very much of it but often the discussion goes into a flight of fancy that leaves the metaphor so tiny that really it has nothing to do anymore with thermodynamics. But I agree that the prediction of what happens to culture, culture is such a complicated issue anyway that you cannot derive its future fate from thermodynamics. I just wonder whether you could pin down a bit more what you might mean by an arrow of time and also this question about whether the ravages of time don't actually define an arrow of time you were emphasizing quite a bit the Kano processes but in fact if you look around the universe there's virtually no Kano processes taking place major irreversible processes. Of course, a lot of this is to do with the effect of gravitation, which is notoriously difficult to bring into thermodynamics discussions. I just wonder, I mean, it does seem to me, and certainly Stephen Hawking takes this view, that there is some real sense in which something that you might call entropy is increasing in the universe, and that is what gives us our psychological sense of the arrow of time, that we're moving from the past into the future? Well, I would... Again,

42:30 the idea that the fact that the notion of entropy is sometimes used rather liberally is not exclusive to all these connections with cultural or social theory. The notion that there is an entropy of the universe has been debated a lot, but is also at the same time still very controversial, especially if you would take it to be an entropy in the same sense of thermodynamics, in which, as I said, the basic idea is always that you can perform interventions on your system. You can add heat to it, you can extract heat from it, you can let the system perform work on something else, and all of these conceptions are really not appropriate for the universe in its totality. And I admit that the idea of a notion of an entropy of the universe was already introduced by Clausius, and Kelvin, of course, also didn't use the entropy of the universe, but had no problem in applying thermodynamical considerations to the universe as a whole. And I think that, well, the conceptual implications of such an exercise are still very, very profound. This is not to say, of course, that there are not major and entropy-increasing processes occurring, say, in stellar evolution, But that's still not the entire universe. Now, considering your other question, the question of exactly what relations are between this notion of a ravages of time and the error of time, I admit that that's a very intriguing question, but the easy answer for me is unfortunately that I say I have no clear answers to offer at this point. It's not so clear to me that there is a, well of course intuitively there is an intimate connection to the two, but if you would take say the

45:00 the formal definition of John Ehrman for a time-reversal invariant theory. As I said there, it's not clear to me that having a non-time-reversal invariant theory would imply anything like a ravages of time, and it's also not clear to me that a time-reversal invariant theory might still envisage some processes in which irrecoverability would be impossible. So there would be a form of revages of time. Maybe not very helpful, but to me these are still open questions. So let's thank the speaker again. And thank you for your attention.