Irreversibility & 2nd Law of Thermodynamics — Part 1

0:00 As the theme of this conference is the notion of time and its philosophical problems, and since thermodynamics is often seen to be a crucial part of science in explaining some aspects of time, I'd like to make a few introductory remarks because the central connection of course between the study of time and the theory of thermodynamics lies in this so-called concept of the arrow of time. And this was a notion which was first introduced by, or a phrase first introduced by Arthur Eddington in his famous book on the nature of the physical world. But even at that time, if I may use that word, the notion of the error of time was very far from clear. In Eddington's book, it could stand for quite a number of different aspects of the notion of time. It could stand for the distinction between future and past. It could stand for the idea that physical processes have a certain direction to it, which is probably closest to thermodynamics. It could also stand for a notion of objective becoming, or what other people call the flow of time. The idea that there is a distinction between a central now point attached to some instance of time and that time is sort of flowing, that the future is approaching upon us and becoming the now and then changing it to the past and so on. And that is certainly not in an exhaustive list of the ways that Eddington used that phrase. In my talk, I will concentrate only on this notion of irreversibility, the idea that physical processes have a preferred direction in time.

2:30 And I will say nothing about this idea of an objective becoming. And the reason for that is that I actually firmly believe that that concept has nothing to do with thermodynamics. Although I'm aware that I'm contradicting here people like Prigogine. But the simple point remains that if you consider a thermodynamical process, say the running of a particular steam engine, it doesn't matter at all for the loss of thermodynamics, Whether that steam engine runs now, whether it was running in, say, 1865, when Commander Perry sailed into the harbors of Japan with his steamboat, or whether it's running 200 years from now. So the question, where this preferred now point is actually located in history, makes no difference for the laws of thermodynamics. The behavior of the steam engine should be the same according to those laws in all three cases. And from that example, one may conclude that the thermodynamics, the laws of thermodynamics are completely neutral with respect to the flow of time and that they therefore cannot have an important implication for this aspect of the error of time. So, as I said, I will concentrate on irreversibility. And I hope this is visible. This is sort of the outline of my talk. The main message I will try to present here is that But if you look at the notion of irreversibility, it is likewise for the notion of the error of time that there is a lot of confusion behind that simple phrase. It is used, I think, in at least three different meanings in text in thermodynamics and philosophy and in general physics texts. And the main message I would like to present is an attempt to disentangle these notions.

5:00 and I will try to illustrate that by also giving a very short history of the second law in the hands of Sadi Carnot, Rudolf Clausius, and Lord Calvin. So that is, well, the middle of the 19th century, say, up till 1870 or so. I will also say a few words about Planck, because Max Planck's Vorlesungen über thermodynamik is sort of the emblematic epitome of the laws of thermodynamics. That was the book that influenced all future generations of physicists. It has come to 11 editions from 1897 till 1950, even after that, long after Planck died. well not so long actually I will try to sketch how you can see all these different at least two different strands of the different meanings of reversibility and irreversibility joining that work I would very much like to continue and tell you something about the recent work of Elliot Lieb and Jakob Ingfason on the presentation of the second law of thermodynamics, because that is a work which is finally axiomatic and therefore formally clear. But I'm afraid I won't have much time to go into that. So I'm hoping you will forgive me if I skip that part. And then finally I will, of course, try to give you some conclusions. But let me first go through these three definitions you can have for irreversible processes. And the very first is one that's actually due to John Ehrman. It's a general definition that you can apply to any given physical theory to ask whether it has a certain symmetry in time built into it. And this is very general, so it doesn't necessarily apply to thermodynamics.

7:30 It doesn't necessarily is restricted to thermodynamics. You can apply it to any theory, at least in classical physics and probably also in quantum physics. The idea that you just have to focus on is that a physical theory will describe a system by a certain state, S, which is a member of a larger state space, gamma. And that physical state encapsules all there is to know about a system at that particular time. So, given the state at one moment in time tells you, well, everything you need to know about that physical system. a process is then something which happens in the course of time and you can represent that as a parameter, as a curve sorry, as a parameterized curve in the state space so, a little bit lower here is there a pointer or a laser pointer or something like that? Anyway, here is a very rough sketch of what you could take to be a phase space. Of course, in general, that phase space will have many, many, many dimensions, and I couldn't even draw it on the transparencies, but just for pictorial purposes, it's here, this two-dimensional sheet. and a process starting in some initial state and ending in some final state will be just some parameter, some curve in that phase space, which I also represented here as a set of states. Now, the first thing one would have to do in order to introduce a time symmetry is to introduce a so-called time-reversal operator on the phase space. And perhaps in order to avoid some confusion that may arise, the time-reversal operator is not an operator that reverses time, because nobody in physics or in the world will know how to change the direction of time, of course.

10:00 It's not an operator that reverses time. It's an operator which changes a state into a time-reverse state. And in the case of classical mechanics, of course, that would simply amount to taking the state with all the positions and the velocities of the constituent parts and then replacing the velocities by the minus velocities. the reversed directions. So if this would be a state of particles in which, say, the particles are moving forward, this would be a state in which the particles have the same positions but are moving backwards. And although nobody can reverse time in a laboratory, in some cases it is possible to prepare such a time-reverse state. Now, what I said applies to particles moving, but in general, physical systems may be more complicated, and then you would have to do something more, depending on the particular type of system, in order to specify this so-called time reversal operator. For example, if there are magnetic fields present, then you would also have to reverse those magnetic fields. So let us just assume in general that there is such a time reversal operator, then the time-reversed process I see I skipped an S here would be the collection of states that is traversed in the original process each of them reversed occurring in opposite ordering so I've changed the parameter T here to minus T And in this phase-space diagram, that is represented by this lower curve. So here is the initial state of the initial process, which goes to this final state. You can apply this mapping, this time-reversal operator, to each of those states. And then you take the process in which all these states occur in the opposite ordering. So this is the time-reversal process of this.

12:30 And now, the criterion to say whether a theory is symmetrical or whether the theory is time-reversal invariant, as it's usually called, is simply to say that whenever the original process is allowed by the theory, so is this time-reversed process. So the theory itself makes no distinction, states no preference between any two of this pair of processes. Either they're both allowed by the theory or they're both disallowed by the theory. In that case, the theory is time reversal invariant. Now, as I said, this is a very good and very precise definition, and it's applicable to almost any physical theory you can think of. except, perhaps, thermodynamics. But let me first extend this a little bit, because nowhere on this sheet, nowhere up to now, have I used the word reversibility, except in time reversal. But there is a natural tendency, certainly amongst philosophers, to use this criterion also for a definition of reversible processes. Because, as we have seen, a theory is not time reversal invariant. It means that there must be some process allowed by the theory, some process P, such that its time reversal is forbidden by that same theory. That's the mark by which you recognize a non-time reversal invariant theory. So if there are such theories allowed by the theory, then because the reversal is not allowed by the theory, there is a natural tendency, it's just a matter of simple language, you would always say, that you would call those processes irreversible. And so, it is done in the philosophical literature in general. Irreversible processes are those processes in a non-time reversal invariant theory

15:00 for which the reversal is forbidden. And, well, let me also stress this point. The power of the criterion of time reversal invariance is that whatever is reversible or not is decided by the formulation of the theory. You don't have to look at the physical world to see which processes are reversible in this sense or not. It's decided by the theory. Now, it's not important here whether you can also prepare or bring about these processes in the laboratory or on a larger scale in nature. For example, we all know that the motion of the planetary system has, well, all the planets evolve in a similar way around the sun. And you all go from east to west, I think. Maybe I'm mistaken here. But anyway, a time reversal of that process would be a process in which all the planets go the other way. And there is no way for us to bring about such a reverse motion of the planetary system. But that doesn't matter for this criterion, of course. The important point is that Newton's law would both allow a planetary system in which planets move one way, as well as a planetary system in which they move the other way. So it's not important what happens in nature. It's important what is allowed by the theory for this criterion. Now, let me come back to this second remark here, because, of course, I'm mainly interested in thermodynamics. And in that case, application of this criterion is not so simple as it looks. because, in general, to decide whether a process is allowed by a theory,

17:30 in the usual theories of physics, one decides this by looking at the equations of motion. There is a certain initial state, and the process is allowed if and only if solving the equations of motion from a given initial state will give you a process that obeys those equations. But in thermodynamics, we don't have equations of motion. And that is a curious fact in which thermodynamics actually stands out and perhaps I should have elaborated this more on the transparencies but the point is that in thermodynamics or usually in classical thermodynamics at least one starts out by considering a system in equilibrium so you say you have a gas in a certain volume with a certain temperature and a certain pressure and as long as you leave that container of gas alone state doesn't change. The only way in which thermodynamics describes processes is when somebody or something from the outside performs an intervention on the system. So either somebody comes along and starts to push a piston and changing the volume of the gas, or you might stir it with a stirring device, or you might remove a partition, or you might start to heat it, whatever you do. But whatever happens to that system during such a process is not decided by dynamical laws of the system alone. It's decided by an intervention from outside. such an intervention happen or not? That's not decided by the theory. That's just decided by the experimenter. And so thermodynamics doesn't give you the evolution of a system starting out just from its initial condition. And that's a complication for which the original criterion doesn't help you so well.

20:00 It's a very powerful criterion, and it's extremely well-fitted for the dynamical theories of classical physics. It's not straightforward to apply it to thermodynamics. But as I said, this is just one definition. me now present a second definition, which I will show later in perhaps more detail, is something which is due to, especially to Lord Calvin and Max Planck, although this is not literal quotation but this is a definition you can extract from many modern but classical thermodynamics texts you can say that the process is reversible if and only if it can be completely undone that is to say if you can find another process which will restore the initial state of your system and also restore all the auxiliary apparatuses with which it had interacted during the course of the process just considered. Again, I can try to give a sort of quasi-formal formulation of this criterion, but you will have noted here that I'm not only talking about the system, I'm also talking about its environment, so to say, or the auxiliary systems with which the system has interacted during the process. So let's say we have a system in some state S and some environment to which I attribute some formal state Z. And the criterion would then look like it as follows. If there is a process in which the state of the system changes to some other state,

22:30 and the environment is also changed into another state, then that process would be called reversible, just in case you can find another process, P prime, which sort of undoes the original process. So it takes both of the final states and brings them back to the initial state, both of the system and the environment with which we started originally. This type of criterion differs quite strongly from the previous one. let me not go through the notes in the order in which they are written. But first of all, you should note here that this second process, which is what I call the recovery process, doesn't have to go through all the intermediate stages that the original process, P, has gone through. So it's different from the criterion but even if we ignore for the moment that here we have a system in the environment and here we have just a system without the environment, in the original definition, this would be the reversal of that process. And every intermediate step in the original process in the reversal process, in the reversed process, only with a reversal operator mapping between those two states. But suppose I wouldn't have this process, but I would have another one that has the same initial and final states. That would not be a time reversal according to the first definition, but it would be a recovery process according to the second definition, because here we only focus on the initial and the final state, and it doesn't matter what happened in between. So that's an obvious and important distinction.

25:00 The second point, although this may not be so obvious from what I said above. I said here, well, I said this process would be reversible if you can find another process, P prime. And I've written if there is another process, P prime. And the question arises, of course, what do you mean by that? Does that mean that you can construct such a process or that it happens in the real world, or would it just matter whether such a process is allowed by the theory? In the latter case, we would have an analogy with the first definition, because, as I said there, it's the theory which decides which processes are reversible and which not. But, as we will see, when I give you some quotations from the literature, the phrase used by, especially by Planck, and similar words are used by Kelvin, is not that such processes should be allowed by the theory, but that they're somehow available in nature. And this is, of course, a curious phrase, in which may have different readings. It may mean simply that they are out there in nature for us to use, but there's also this ugly or multi-interpretable abble at the end of available which sort of suggests that it is a modal notion and I'm not quite sure what to make of it I'm sure that there are different interpretations of what available in nature would mean I think that in the case of Planck and Kelvin, it simply means that either they are out there in nature or, at the most, that we could construct them by simple operations from agencies which are out there in nature.

27:30 I will come back to that. The last thing I noted here is that it's not so clear how to assign states to auxiliary systems. And that is really a difficult issue in the entire theory of thermodynamics. I'm applying here, I'm assuming that there are auxiliary apparatuses, like the pistons with which I compressed my gas, some other machine that receives work done by the gas in its steam engine cycle. But what are the auxiliary apparatuses? What are the characteristics? What are the states? What is the theory that describes them? And this could be especially important and complicated when the auxiliary systems contain living beings. It's not straightforward to say what the states are. And, in fact, most of thermodynamics is very silent about the nature of the auxiliary systems and, more particularly, doesn't assume that thermodynamic concepts apply to them. And if that is the case, then we have a real problem, because then we have one theory describing the states of the thermodynamic system, And we might have another theory, which is as yet unspecified, describing the states of the environment. So that's why I stress that the states of the environment here are just formal. Because we don't really have a good theory to describe them in general. but perhaps I just left out the point that may be the most important of the notes I wanted to make this notion of irreversibility doesn't really address the symmetry on the time reversal as we already seen because you don't have to It means something else, it means really that if a process is irreversible, then there is no way, there's no process you can execute on the system once this process has taken place, which will allow you to recover the initial state, this one, of both the system and the environment.

30:00 If an irreversible process happened here, you have lost something. You cannot go back to the initial state anymore. Something has happened to the system which is impossible to retrieve. And that notion, I think, and this will also be my conclusion, has not so much to do with an error of time, but it's much more aptly captured by a phrase like the references of time. I'm sure that also in... I believe the notion of references of time comes from Shakespeare. I'm sure that in other European languages there are other phrases which capture these sort of irretrievable changes that happen to systems in the course of time in Dutch week we say that the tooth of time have the teeth of time which take away youth and lead to decay and corruption I'm not sure whether what phrase you use in German after son But having distinguished these two definitions, one from a philosophy of physics background and one from going back to classical texts of Kelvin and also Planck, I should say that there is a third definition. and actually I believe this is the most common one in thermodynamical textbooks. Let me open here with a citation from Bridgman, a citation that may look quite confusing

32:30 and strange in the beginning. Bridgman says, it's almost always emphasized that thermodynamics is concerned with reversible processes states, and that it can have nothing to do with irreversible processes or systems out of equilibrium. I said on the first occasion that citation may look strange because, as I opened my talk, many philosophers and physicists think that irreversibility is almost a defining criterion of classical thermodynamics, that it is the first and perhaps the only thing, no, not the only, but the first general and well-known theory of thermodynamics that implies that there is something like irreversibility going on in physical processes. And here is Bridgman, not all that long ago, saying that he believes that it's almost always emphasized that thermodynamics has nothing to do with that notion. And Bridgman is, of course, a foremost thermodynamicist himself. So, yet I think the quotation becomes more understandable if you remember, or at least recognize, that the term reversibility is actually used by most thermodynamicists, and Bridgman included, to stand for another type of processes. processes that proceed so delicately and so slowly that you can, up to a negligible error, consider the system as remaining in equilibrium during all the time of the process. It's much like the way you would maneuver a cup of tea which has been filled to the brim and bring it from the table to your chair, say. You move so delicately and so slowly that you don't spill anything. There are no changes in the surface of the liquid here. Here's another quotation that expresses the same idea, perhaps more clearly. It's from Max Born, who also did, apart from his work in quantum mechanics,

35:00 also did some important work in thermodynamics before that time. He writes, one conducts the process infinitely slowly so that a state at every moment can be regarded as an equilibrium state. One should call these processes quasi-static, which is a phrase introduced by Carateo Luri. However, Boyne continues, employs the word reversible, because in general they have the property that they can be reversed. Now I might take issue with this last addition to Born's sentence, but this is really what happens. One employs the term reversibility to processes which are much better known as quasi-static, because that's what they are. They are almost always up to a small error in equilibrium. Now, you will probably recognize that it's not so easy to see any connection between that notion of reversibility and, of course, its denial, irreversibility, with the previous two definitions. And a little more analysis brings this out, because here's, say, a phase space of thermodynamics, it consists only of equilibrium states, because as I said, and Bridgman just confirmed, the states of thermodynamics, classical thermodynamics at least, are always equilibrium states. And so I can have here an initial state, S1, and I can have here a candidate final state, S2. And what I would have to do to bring the system from S1 to S2 is to make an intervention by my piston or by heating the system or doing whatever I need to do to it. In doing that, I will bring it out of equilibrium. So this is just a formal representation. but I would have to bring it out of equilibrium and undergo through some changes, and then hopefully it will settle to a new equilibrium state later on.

37:30 I can repeat that a number of times in such a way as to approach my desired final equilibrium state. But if the changes out of equilibrium are large, then that process would not be a quasi-static process, and so I can consider a sequence of such processes, in which the changes are made smaller and smaller and smaller, and in the end, hopefully, that sequence of processes would approach to a curve in equilibrium space. And it's really that curve, the limit of a sequence of processes, which is the adiabatic process of Karateodori. So here you meet another important distinction. Quasi-staticity is not a characterization of a process, it's a characterization of a limit of processes. It's just a curve in this equilibrium phase space. Born said these processes can also be reversed. Well, that is true in the following sense. you can take another system starting out in state S2 and have this as your desired final state and perform a sequence of interventions on this system which will take it to this state. And these interventions will in general be quite different from the Green's ones I considered here above. So here's another intervention which brings the system out of equilibrium it return to another equilibrium state and then another intervention and then another intervention and again i can consider a sequence of such processes in which the disturbances by the intervention become smaller and smaller and smaller and approach the same curve in equilibrium space but now traversed from S2 to S1. And yes, that is another quasi-static process, and yes, in the same sense, as the limit, and yes, it goes from S2 to S1, but does it have anything to do

40:00 with reversing processes in the course of time? I would say no. be the case, and usually is the case, that all these green processes irreversible in the previous two senses. It can also be the case that all the black processes, in taking a sequence of such processes which approach this curve, are also irreversible. In the sense for example, that they both involve an increase in entropy in the system plus its environment. So we can have a sequence of irreversible processes which approaches this curve, and we can have a sequence of irreversible processes which approaches the same curve to first in the other direction. And so, well, you can see the confusion starting to build up here. one can have a process which is called reversible by Borg, approached by irreversible processes, and of which the reversal of that irreversible process is also a reversible process. This is one of the reasons why thermodynamics is often so obscure to logical people. All this is just caused by a confusion of language because the quasi-static curve is actually not a process at all. It's a sequence of processes and it's a limit of a sequence of processes and as such, it doesn't occur in space and time and it makes no sense to say whether it's reversible or not in the previous two senses. Here are perhaps a few examples that might help to see the distinction between the last two definitions, which are common in thermodynamics, and see how they are independent.

42:30 I can have, say, a gas closed by a piston in a container, and it's in a heat bath at a constant temperature, and I can very, very, very slowly let it expand by taking out the piston very slowly. Now, what happens, of course, is that its volume increases, its temperature would tend to drop if it wasn't in contact with the heat bath, But the gas will perform work on the outside. At the same time, heat will flow into the system from the heat bath. And I can do this very slowly until the system has reached another size of its volume. Perhaps in the form of a limiting procedure. i could also let the whole thing go backwards i can compress the gas equally slowly and then work will be done on the gas and heat will be expelled into the heat bath and so this would be a process which in in typical thermodynamics textbooks would be explained as one that is both quasi-static and recoverable that is the the the plank kelvin sense of reversibility You can go from one volume to another volume and back again. And during the recovery process, all the work that's been done in the original process is brought back to the system. And all the heat that's been entering the system in the first process is expelled back into the heat bath. So you get a complete recovery of the system and its auxiliary systems. Here is a processor which typically would be regarded as non-quasi-static and also as non-recoverable in Planck's and Kelvin's sense This is Jules' famous pedal wheel experiment in which you have a stirring device in, well, not a gas but a liquid I think he used a fish clue and the pedal wheel is driven by some weights in the gravitational field So you can calculate or determine, measure very precisely how much work has been done on the system.

45:00 And after that has been done, well, the entropy of the system has changed, and there's no entropy decrease in the environment. So something unrecoverable has happened to the system. There's no way, according to the second law of thermodynamics, to go back to the initial state. And it's also, of course, a non-quasi-static process because the pedal wheel brings about disturbances and turbulences in the fluid which take it very far from equilibrium. Now, these first two examples would be just like one would hope them to be in the case of a close connection between the notion of recoverability, that is the Planck-Kelvin sense of reversibility, and the quasi-static notion of reversibility because they say oh they can go together and the absence can also go together but there are also examples in which only one of the two