The second law of thermodynamics is one of the bedrocks of science, and it has become an integral part of pop culture, economics and politics. But it also sticks out due to its defeatism, and it is constantly mystified because “no one really knows what entropy really is” if you believe John von Neumann. In this question-answer-recap dialogue, I discuss with Matteo Polettini the role and meaning of the second law, mostly from a historic thermodynamic point of view, but also in view of contemporary approaches, its relation to society and the question what makes a good scientific law.

About Matteo Polettini: Matteo Polettini is a senior scientist at the University of Luxembourg, whose ideas have shaped the development of stochastic thermodynamics, chemical reaction networks and the foundations of thermodynamics. He also likes to think about philosophy of science, is a member of the scientific outreach community extemporanea, and occasionally blogs “about toothpaste”.

Required background: You should have heard about the second law and the entropy concept before.

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Matteo: Philipp invited me to contribute sporadically to his new blog with a reciprocal Q/A on the Second Law of Thermodynamics. I think it is best to start from the very popular saying that “the entropy of the universe never decreases”, which personally I find pointless. I would like to know from Philipp if he also agrees that this sentence is empty or even illogical.

Philipp: Maybe it is a good starting point to agree on the empirical fact that we can define a quantity, which we call “entropy”, that we always observe to increase. This is true for very different kinds of processes, and it is true here on earth as well as on the moon or elsewhere (as far as we can tell). Now, I see two big issues with the statement “the entropy of the universe never decreases”. First, what do we mean by “universe”? If we mean the Universe (captial “U”) in the cosmological sense, then one should honestly point out that nobody has yet succeeded to define its entropy, and it is not even clear whether this should be possible. That’s certainly a big open problem. Second, for systems where we have a good understanding of entropy, it turns out that its increase is always a consequence of choosing a very special kind of initial state (namely, a state of low entropy). Thus, in all systems where we understand the second law, its origin is found in the initial condition. Once one accepts this initial condition, the second law becomes a mere corollary, and the mind-boggling question remains “why this special initial state?” So in that sense I can see some “emptiness” in the second law, but I wouldn’t call it pointless or illogical.

M: What I take home is that in P’s view initial conditions are crucial in determining whether or not the second law holds, and that there is a difference between a Universe with capital “U” and (I assume) a universe with small “u”, whose nature though is not explained.

P: OK, I can try to explain the small “u” soon, but perhaps it is good if you first explain what you find pointless or illogical about the second law. Surely, that heat flows from hot to cold is a simple, useful and well confirmed law. So I guess, you are referring to something deeper and more subtle here. 

M: Take “heat flows from hot to cold”, which is a statement in the area of the Second Law. Consider the definition of “cold” and “hot”. The first thermometers were standardized (e.g. by Celsius and other coevals) by comparing the stabilization of columns of mercury against some phenomena that were assumed to be universal (e.g. the melting of ice and the boiling of water at sea level). So I argue that it is the other way around, it is the flow of heat (the expansion or contraction of mercury) that defines what is cold and what is hot, and not viceversa. Then, what is heat? To me, the First Law defines it to be the energy “lost” in a process, but how do you make sense of it if you don’t have a pre-conceived model for what is “hot” and what is “cold”? I understand this is a bit of a chicken-or-egg question, but that’s my pivot point, because actually I like science to be illogical (and also pointless). Apart from technicalities on which we will return if we discuss universe & Universe, my problem with statements like “the entropy of the universe never decreases” is that it pretends to solve all issues by deciding whether there was a chicken or an egg but it does not give you an idea on how to do it, by hiding the details under the carpet. So I reformulate, it’s not (just) illogical and empty, it’s also dishonest. But it’s not’ pointless.

P: These are interesting points. Let me check my understanding by using another example to illustrate them. The second law (maximization of entropy) is often used to predict that systems in thermal contact reach equilibrium, i.e., the same temperature characterized by an absence of heat flow—but to define any change of entropy in the first place, one needs a notion of temperature and heat flux, which are chosen such that the prediction turns out to be correct… So, you are saying that the second law, at least in its universal-entropy-increase formulation, is circular (= “illogical and empty”) and too simplistic (= “dishonest”), but it is still important (= “not pointless”).

M: Yes, that could be, but I hope you disagree and that we can come back to this later… At the same time there are other statements of the second law of the kind “it is impossible for X under conditions Y to do Z”. What strikes me most is the specificity of this kind of statements, and the fact that the Second Law is framed in a negative way. I have two questions for you: how would you harmonize this kind of statement with the one above? Do you find any of the two more useful? 

P: We should definitely come back to the issue of circularity, where I indeed disagree. But I should blame myself for my first answer at the beginnning of our debate, where I mostly had statistical mechanics in mind (a curse of my job) instead of thermodynamics (which is a distinct theory). To confess: within thermodynamics nobody has found a general definition of entropy without the (local) equilibrium assumption so far. Therefore, you are right that the sentence “the entropy of the universe never decreases” becomes, taken at face value within thermodynamics, dishonest. So, I’d give you half a point here, but the question you just asked is actually perfect to argue why it is only half a point.

As you just hinted at, there are many formulations of the second law that are not even always equivalent. We had Clausius’ heat-flow-version and another example is the “impossibility to build a machine (operating between two fixed thermal reservoirs) surpassing Carnot efficiency”. So, how could such an operational formulation ever be related to the fact that entropy does not decrease? This was certainly far from obvious back in the 19th century, but here is the magic if you like: if you accept the same definition of entropy as in Clausius’ heat-flow-version, then also this formulation follows (straightforwardly) from the entropy-increase formulation (plus the first law).

Here is the thing: it is not temperature or heat or any other quantity that constantly increases, it is this particular combination of them that we call entropy. This non-trivial insight allows us to mathematically derive all the other formulations of the second law. Without it we had instead a long catalogue of second laws (plural!). You are right that the catchy sentence “the entropy of the universe never decreases” is too short to pay duty to all the subtleties, but its merit is to point to some possible deeper, underlying and unifying principle. This is very useful.

M: My takeaway message is: 1) statistical mechanics * and thermodynamics are distinct; 2) unless there is an authority (e.g. Clausius), there would be a plurality of “second laws” 3) there is some “magics” in all of this.

* I would include in this also any development of thermodynamics based on stochastic processes, but of course this is open to debate. 

P: Haha, well, if I talked about “authority” and “magic”, then in a mathematical sense. But OK, I guess it’s best if you explain your stand on the same question.

M: I would say that the catchy sentence is too short not just for the subtleties, but for anything at all, and that if you look deep inside (e.g. the very definition of “entropy”) you will find that you are just demanding to the whomever is in charge of any sort of experimental apparatus to take care of business [here I would have to build up on what is an “experiment” at all, especially in nonequilibrium thermodynamics, where I believe we have no experiments!]. It is neither descriptive neither prescriptive, unless you assume that your system behaves by the postulates of statistical mechanics. On the other hand, all sentences like “it is impossible that…” try to build the setup where that sentence makes more sense in a phenomenological way. I am also suspicious of this. The point is: if we believe that science is a universal language, how far do you have to go into describing detail-by-detail our devices, before making sense of it all?

P: Wait wait, I think I see your worry, but you are too quick here, and IMO exaggerating. Before getting lost in too general discussions, let me return to the heat exchange case. 

So, consider some bodies or substances A, B, C, … As you said, in order to speak about their temperature you need a reference system. Let’s call it X. So actually, when we talk about the temperature T_A of A we should write T_A(X) if you like, and the same with T_B(X), T_C(X), … Of course, our X-thermometer is chosen such that heat flows from A to X if T_A(X) > T_X (with T_X some fixed reference scale), so the second law is empty in this case. However, the point is that you can remove X and put, say, B and C into contact. Then, you will find that heat also flows from B to C if T_B(X) > T_C(X). 

And here is the point where I can no longer follow you if you say that this is not a prediction or that it is empty or circular. Of course, in the prescription above I have still left out details, for instance, I assumed that B is not pure hydrogen and C is not pure oxygen (otherwise ask a chemist). But without arguing about that also this situation would fit into the entropy increase paradigm, my point is this: 

You could make a long catalogue with all possible heat flow directions for all kinds of substances A, B, C, … as a function of their temperatures, volumes, etc., all with respect to some reference system X. That would be a very long catalogue. But you can also say: Wait, I realize some pattern, some regularity here. I can compress the entire catalogue into a much shorter description using the second law plus, perhaps, some other side constraints as mentioned above. But my point is: you can still compress it! 

M: I may run too fast but I also feel that you are breaking the rules of the game, which were: (round) A asks question to B; B answers long; A sums up his takeaway short; (next round) same with A and B swapped. So now I will ask a question whose answer is your paragraph just above: Can you think of any setup where the second law in whatever form is obvious and that does not entail delegating to chemists?

My takeaway is that you always have to retrospectively re-create the conditions by which entropy made sense, or else that you have to imagine the (practically impossible) existence of a long database of more fundamental data. (I did not actually look into the paragraph with formulas because I almost hate them, as they are part of my daily job.)

P: Apologies for the interruption of the pattern. I think your summary that there must be an “authority” and that there is some “magic” provoked in me the desire for urgent clarification because this is not what I said. And now I am confused whether I have to answer something or ask something. 😉 So, I try to do both, and with the following I want to ensure that we do not talk past each other. 

I call (by my definition) a natural law “predictive”, “non-empty” and “non-circular” if this law allows you to write a computer programme that (a) outputs experimentally verifiable data within a well-defined class of setups and (b) if it is shorter than the programme “Print [data]” where [data] simply lists all the data for the mentioned class of setups. 

Now, my questions are: (1) Do you agree to this definition? If not, why not and what is your definition then? (2) If yes, do you think that any formulation of the second law allows you to compress data in this sense for some class of setups? (3) If yes again, which formulation allows the largest amount of compression in your view? 

M: (1) I may agree with this definition (although I have issues with the word “verifying”, but let’s keep this for another time); (2) An entangled quantum state No / Yes – Yes / No where if you say Yes I say No and viceversa. Therefore in this case No. The point I want to make is that in your program you would also have to specify the “class of setups”, and this basically amounts to describing them all. We develop theory together with its realm of application, and then we universalize, and this is unfair and undue. But that does not mean that theory is not powerful, but here it gets political…

For example, let’s make one of those silly thought experiments where somehow you encounter an extraterrestrial being called zE1Da in all removed from your context but that somehow speaks and understands all of your “natural” language up to some words and concepts, including the Second Law. You would still have to tell zE1Da how to construct an experiment, and that takes a lot of effort. Then, you have to construct a lot of other experiments and convince zE1Da that they can be captured by some common behaviour: a “law”?! But then you construct other experiments and the “law” does not hold, so you say: <<OK let’s stop here, there’s something we do not understand about these other systems, but as far as we are concerned the law holds>>! The impression that we achieve compression is due to the fact that we already encoded a lot of information in our histories as human beings. Our thought experiment falls because we cannot take for granted that “ideal gas” is already in zE1Da’s dictionary, and that it has the same meaning as for us human beings embedded (or, better, intertwined) in a very complex social environment.

If instead you said No to question 2), then I would have answered Yes, which allows me to go about question 3) I think old-style classical thermodynamics is by far the most inclusive theory because it is well-intertwined with society, even as if a mystical object; and that, by contrast, very strongly promoted and perfectly illuministic research programs (on which I work!) fail to meet any standards of experimental “verification” (here it comes again…).

P: So, you agree that it is a good idea to judge the usefulness of a theory by its ability to compress experimental data. But you say that the second law is useless in this sense because you need as many bits to describe the class of experimental setups as you would need to specify the experimental data itself. Then (not sure whether I can follow you here, so feel free to correct me), you also say the opposite to make the point that society and its history is so much entangled with the theory of thermodynamics that it would be impossible to judge which second law is more useful (if any were useful at all). 

M: I never say something and I always say its opposite! Jokes aside, I am OK up to some point with the compressibility idea, as far as we agree that what we can compress is only the class of processes that made up the theory up-to-obvious-similarities, and not one inch further than that. That can be very limited, and I feel that for historical reasons the Second Law is very limited in scope. So here is my question: to you, is there any substantial difference between the First Law and the Second Law of thermodynamics (beyond their statement)?

P: Uh, you are asking tough questions about which you have probably thought much longer than I. So, spontaneously my take on that question is the following. 

First, on a purely formal level I think there is no difference. Both are simply axioms or postulates of thermodynamics, and I can not deduce one from the other. 

Second, thinking from the perspective of engineers trying to build a heat engine, the two seem to be inseparably entangled and mutually co-constitutive (to paraphrase Karen Barad). On the one hand, the second law quantifies the “resourcefulness” of energy in this context, which is only meaningful because energy is conserved (if you could create energy from nowhere it wouldn’t be a precious resource). On the other hand, the distinction between a heat and a work reservoir is—as far as I can tell—determined by the question whether there is a change in entropy in it or not. So, the two are so much intertwined that it is impossible to say which one comes “first”. 

Third, and perhaps we better do not pursue this further here, but coming from the perspective of statistical mechanics I see the first law as a consequence of the dynamics but the second law as a consequence of the initial conditions. And that is quite a substantial difference IMO. 

M: Yes, I also thought a lot about this question, but more from a historical perspective: the First Law is the direct descendant (in thermodynamics) of a pre-existent law of conservation of energy, while the Second Law is brand new of the XXth century, so I wonder if that has any implications…. What I take away from your answer is that we should put ourselves in the head of engineers building an engine, and that “resourcefulness” is a subjective concept whose specification whether it is the First or the Second Law to come first is what seems to be a sort of egg-and-chicken dilemma.

P: Alright, I think I finally found our source of discrepancy. You said that you agree with my definition based on compressibility only “up-to-obvious-similarities” (my italics). But there is no “obvious” in my definition. What you judge to be obvious would still go into the set [data] in the program “Print [data]”. This would make [data] obviously enormously long (what a great play on words).

Now, I like to know your stand on other laws is physics that are described by short phrases, such as “action equals reaction”, “nothing travels faster than light”, “the Universe expands”, “the wave function collapses”, etc. As far as I understand you, if you allow to include all the “obvious similarities”, they all could also qualify as empty, pointless or simplistic. Do you agree? Or do you see a substantial difference between the phrases above and “the entropy of the universe increases” (beyond its physical content, of course)?

M: (Let’s come back to the “obvious similarities” later)

I don’t think they are pointless or simplistic, but that their point and the reason behind their simplicity is to be found somewhere else than the fact that they capture some intrinsic property of “nature”. I challenge their universality and the self-consistency of their definitions, and I claim that the only difference among all laws is their persistence (which may not be forever). That is, I believe that to some extent all laws of physics describe objects whose very definition need the law itself in order to be defined.

Much like a language: if you take a dictionary, every word is defined in terms of other words. Are there “more fundamental words” in terms of which all other words can be defined, outside of mutual habits and institutional violence?

So: are we sure that all terms that define an “action” are experimentally independent of all terms that define a “reaction”? Even giving precise sense to what is the difference between inertial and gravitational mass took a long time… But some laws have a more ancient history and so they are loaded with positive examples that reinforce their status (somehow they are savalged from the strict rigour of the “scientific method”).

The proof of my concept is that, if you had to design an experiment today that claimed to have found a counterexample to “action-reaction”, you would think it more than twice before submitting it to your peers, unless you want to be banished from the scientific community forever. And you would have to dig far back into the history of science to get to the point of where this was established as a principle, and how and why… I am sure you would find that a lot of assumptions that made sense back then do not make sense anymore, and a lot of things that you do in your daily job have nothing to do with the principle at all. So, action-reaction is already not part of science, it’s part of mythology. Which, may I say, is not a bad thing!

The situation is completely different with “nothing travels faster than light”. I believe we are still in the time lapse where people hold some technical expertise on what experiments could challenge this claim. I can easily imagine that given the problems we have on Planet Earth in the coming years there will be smaller funds for HEP and astronomic programs that could detect any effect of anything possibly going faster than light. So in a very early tomorrow this will also be a myth.

With the Second Law, I have mixed feelings. It seems to me that it all started as a myth and that “it has never been modern”, to paraphrase one of my favourite thinkers. Before you start recapping my answer, a piece of question for you: can you think of one situation where thinking about the Second Law actually did something from a scientific standpoint (I recall you mentioning the Diesel engine at some point)?

P: Haha, I feel honored that you remember a footnote in one of my papers. But indeed, if I understood the history right, for a long time the theory of thermodynamics had no impact on practical applications (it was developed around 100 years after the industrial evolution). Only at the end of the 19th century, Diesel realized that higher efficiencies require higher temperature differences according to Carnot’s theorem. So he designed the Diesel engine where the air is hot enough to self-ignite the fuel (in contrast to an Otto engine where you need a spark). 

M: I’m sorry to disappoint you, but I never read your papers (did you read mine?!). I skimmed through some of them, but most of what I know about your research and your scientific ethos comes from our personal interaction and from what other people in our field say about what you do. I never read your footnote, you told me about Diesel at a techno party somewhere around Berlin, and somehow it stuck to my mind. Which makes me come to the next question. The universalistic practices we predicate (the community of peers, independent reviewing etc.) are very well-principled, but in practice they are unattainable and consortiums of humans coalesce around partial truths. Should we salvage or savage our good-old scientific method?

P: That’s a bummer. I always read all your papers, in particular the appendix. I am shocked that you are so unfamiliar with my groundbreaking work. 

To answer your question (in the hope that you are reading that), I actually believe that we are no longer using the “good old scientific method”. Nowadays, institutions, agencies and other stakeholder actually create more incentives to bend the rules, to play unfair, to put truth second, or—if you want—to mystify things than before. In that sense, we should go back to the past. 

Of course, not everything was better in the past. For instance, in the past science was exclusively done by a very small circle of white males from an elite social class. Nowadays, this is a bit better. 

M: My takeaway: you are a Luddite! But I understand the nuances of your take given what you wrote in your essay.

P: I like to return to these “obvious similarities” to nail down things once and for all. My naive definition of a predictive or non-empty natural law left no space for anything obvious. For instance, if you have established that a 1kg piece of metal thermalizes, than realizing that a slightly larger 1.001kg piece of metal also thermalizes gives you a new data point, even if you find it obvious.

It is my impression, somehow, that you use all these interesting historical and cultural factors surrounding the second law too much against it. So, it is true that the second law concerns many daily life processes that appear intuitive and obvious to us humans, in contrast to predictions made by Newton’s law or Schrödinger’s equation, but that doesn’t make it less predictive from a formal point of view. 

Am I correct or wrong with this assessment? And have you thought about any other idea that would make the notion of a predictive/non-empty natural law rigorous and which, perhaps, includes these obvious similarities? 

M: You are wrong with the assessment that I am using historical and cultural considerations to bamboozle the discussion. I am not against the second law! I think it might serve some purposes. For example, it can serve as a guiding principle to inspect all clauses like “I found a way to make you rich and solve poverty in the world all in one” (as per a recent youtube ad by Elon Musk). Or like “I found a new source of energy that is intrinsically clean”. What is your response to such claims? If you are a scientist: doubt! That’s what the second law does: it embodies systematic doubt, which (I believe) is the only systematic principle of science. Many much more famous scientists (Ehrenfest, Einstein, etc.) claimed that the only law of science of all which would never be broken is the second law. But then, if we believe Popper’s “falsifiability”, the second law is outside of science itself! It’s more of a meta-principle.

Coming back to your example: “assuming that 1kg piece of metal “thermalizes” (that is, the device by which you probe it reaches a stable point), than realizing that a slightly larger 1.001kg piece of metal also thermalizes gives you a new data point, even if you find it obvious”.

It’s not obvious, and if you find out that there is some regularity between the data points between different metals or different conditions, you may have discovered something interesting. Then you have to go out there and you have to convince people that what you observe makes sense and might have a purpose. But how is the second law any useful in that? (Beyond the fact that you can employ it in your story-telling). 

P: Because that regularity is the second law! What else predicts thermalization? 

But alright, you are saying the second law is basically more useful outside than inside physics, which is an interesting point. Moreover, you say that a universally valid truth stands outside of science. Of course, my objection would be that you can still test the second law and try to falsify it a la Popper, but (I guess) your answer is then that I would loose all my reputation if I would publish any findings about “second law violations”. So your claim is that the scientific method stopped working properly as far as the second law is concerned. 

M: There is no valid truth outside of science where the second law belongs (a strong tenet which served me no purpose—in fact I am not a Platonist at all!). My experience as an actor around the sciences, in part as expert and in part as practitioner—is that the second law is a powerful narrative. A narrative is no-one’s land, and you can use it for one purpose or for another. For me, the second law is weak, because it basically posits a common-sense reading of what can be achieved by present-day knowledge and technologies. It sets an institutional cut-off with respect to what can be achieved. Of course you loose your reputation by publishing “second law violations” (and rightly so!), but at the same time I would like to encourage whomever to explore the boundaries of the concept.

So here is my question: within the physical/mathematical framework we work on (the stochastic/quantum foundations of thermodynamics), how much freedom to interpret the “second law” are we allowed? If I make a quick survey of our papers, I realize that the second law is often deployed as a consistency check, not as a concept that can generate anything.

P: Well, the latter observation might be related to us being more interested in foundations than applications such as computing efficiencies (although I remember that you studied this quite deeply at some point, e.g., here or here). 

Anyway, turning to your main question, I think the scientific community in stochastic/quantum thermo nowadays allows a lot of freedom in the interpretation of the second law, sometimes more than what I’d call scientifically legitimate.

Let me illustrate this with the beloved expressions for the entropy production as the relative entropy between “forward and backward probabilities”. This is a very nice theorem, but it has become a plague nowadays as it was turned into the basic definition of entropy production. But by definition of the relative entropy, this makes entropy production non-negative and time asymmetric by fiat. This contradicts that entropy is a state function and that microphysics has time-reversal symmetry. Researchers seem to have forgotten that this definition was shown to be right only for specific Markov processes, which also clearly break time-reversal symmetry. 

So, I would subscribe to your statements about being pointless, illogical and dishonnest if it comes to that definition. Of course, I emphasize again that it is a beautiful theorem, but that theorem has assumptions (which already break time-reversal symmetry). Since it is nowadays possible to write papers by applying this tautological definition to anything you like, I conclude that there is (too) much freedom. 

M: I love your criticism of too many papers defining entropy production as log-ratio of positive to negative trajectories, but for a different reason: not because they don’t take it seriously that entropy is a state function, but because if that is the definition of entropy production, then the fluctuation relation is proven in one passage (I would hardly even call it a “theorem”! Mathematicians would (and do) laugh at us…). On entropy and entropy production being state functions: we might be standard bearers of two radically different approaches, one that builds up reality, and one that justifies it.

P: I agree to your remark about the fluctuation theorem, and perhaps we automatically marched into the direction of the question I always wanted to ask you. How do you think of the status of stochastic thermodynamics as a scientific theory? Do you think it is part of thermodynamics or part of statistical mechanics or both or neither? I also vaguely remember you stating at some point that you are convinced that thermodynamics should be taught using stochastic thermodynamics (but my memory can be unreliable). Furthermore, feel free to connect your thoughts to your last statement about builders and justifiers of reality… 

M: I must confess I am not at ease with the locution “stochastic thermodynamics”, I try to avoid it in my writings because on the one side I am more of a mathematical physicist, and on the other I think it has become a battlefield of cultural appropriation of a broad set of ideas which does not take into proper account all contributions and the fact that large portions of the theory are reframings of very well-known past ideas. When I used to go to conferences I noticed that some people are obsessed by who “owns” the fluctuation relation, and I also noticed that some communities are allergic to the term (e.g. people working with interacting particle systems). I think “Statistical Mechanics of Irreversible Phenomena” (as a nice recent book is titled) is a much more fitting description: it puts together the heritage of statistical mechanics and that of the understanding of phenomena where something flows. In fact I cannot think of any question that is specific of stochastic thermodynamics.

Maybe stochastic thermodynamics is more related to the kind of questions it asks, rather than to specific answers it gives (e.g. the focus on the problem of calculating efficiencies that you mention, a problem which by the way I think does not fit well into the XXIst century given that for complex systems it’s often very misleading to think of one input vs. one output). But then we have to pose the question of how effective the theory has been in going beyond what was already known from standard (linear irreversible) thermodynamics. Here I believe there are crucial unresolved issues, in particular as to how to interpret experiments. The ones I know are confirmations that some systems obey Brownian motion—and therefore all of its consequences, including properties of work and heat defined along a stochastic path (which, I want to point out, being a fractal requires some arbitrary cutoff, some calibration). But thermodynamics is more about calorimetry, about “seeing the system from outside”—and here I think we may be putting the cart before the horse. On the other hand I do believe there may be some potentially interesting outcomes of this way of thinking about things: for example I have always been impressed by the comparison of some of its microscopic predictions to some macroscopic data regarding nuclear power plants in PRL 105, 150603 (2010). So I don’t want to make any judgement, and I never had the chance to survey all of the literature about experimental confirmations of predictions of stochastic thermodynamics.

About the overlap with thermodynamics. Thermodynamics is a very broad arena, it involves applied physicists, engineers, boiler technicians, etc. In a sense most of them know much more thermodynamics than I do! Given that I can hardly take care of my thermostat… I went to some more engineering-oriented conferences, and I realized that even within engineering there is a spread of positions, from applied to theoretical, and the disconnect between themselves (and even more so with “us”) is huge. To me it is clear that every community needs its spiritual experts and a piece of mystery. The idea that thermodynamics is “solved” once and for all and that one just needs to find the proper “The Stochastic Thermodynamics of this and that” paper is detrimental to our community and not respectful of how other communities go about it. So I definitely do not vouch for stochastic thermodynamics becoming THE way to teach thermodynamics.

But still I think it can be a useful tool to teach something. Now I more think of stochastic thermodynamics as a sort of bandwagon where anyone who somehow believes into her/his piece of truth can jump on and contribute to some overall “flow of knowledge” (I steal this expression from some memory of reading “Against Method” by Feyerabend) that has no particular purpose other than reproducing itself and belonging to the world. For example: stochastic processes had their apex thanks to (and together with) financial markets. Independently of their application, are they a culturally interesting concept? If so, we can use the bandwagon of stochastic thermodynamics to bring on the tradition of their understanding. Another example: we now understand the overall framework of large deviations much much better than we did at the time when we were toying with equilibrium phase transitions (remember that time when it was all about critical exponents of more and more complicated models?!). So I do believe stochastic thermodynamics (if we insist on calling it so) can be a tool to convey important ideas within (and even outside!) of science, but I would not dare to say it covers thermodynamics. It is inspired by it.

P: So, there seem to be five main messages if I counted correctly. First, there are many different schools or directions of thermodynamics across a wide range of scholars. Second, stochastic thermodynamics is to some extend a rebranding of old ideas. Third, it is currently a battlefield, where researchers try to claim priority or try to dominate the discussion. Fourth, it is actually unclear what the main questions or answers of stochastic thermodynamics are. For this reason, you would prefer to call it “statistical mechanics of irreversible phenomena”. Fifth, stochastic thermodynamics is nevertheless useful technically and also as a cultural bandwagon, on which you can jump to develop an identity for you as a researcher and to delimit the circle you belong to. 

I think these are fair points, and I much like the bandwagon picture. I only find the designation “statistical mechanics of irreversible phenomena” too broad to be useful in this case. 

M: The bandwagon idea is not mine unfortunately, it is (once again!) of CE Shannon. But let’s come to your own understanding of the field. It seems to me that there exists a quantum thermodynamics community that wants to build up the world from its fundamental pieces and, in the process, thermodynamics comes about mostly as an incident due to the fact that the program fails for technical reasons (see e.g. “eigenstate thermalization etc.”). These are the people who put the “Hamiltonian” before all. And then there are the people who do the opposite, they live without Hamiltonians but rather they have Markov generators and just infer thermodynamic properties of the world without never confronting it (I belong to this second kind). Where do you belong here? And how do you place stochastic thermodynamics in this arena?

P: Starting with the last question, I would place stochastic thermodynamics (ST) in the corner of thermodynamics, because in my understanding ST is about the thermodynamic consequences of Markov processes obeying local detailed balance. That is, ST starts from a time asymmetric picture where the second law is postulated. The Markovian description is then something what people in ancient times probably would have called “constitutive equations” (such as Fourier’s law, Ohm’s law, etc.). Those laws are essential to compute anything dynamically and they contain a bunch of phenomenological parameters (such as conductivities in ancient times and transition rates nowadays) that need to be matched to experiments. 

So to give a clear example, consider the work fluctuation theorem. Crook’s derivation of it is a result of ST, but the derivations of Bochkov, Kuzovlev and Jarzynski are results of statistical mechanics because the latter start from a microscopic time symmetric description. Of course, both match under the right assumptions as it should be, otherwise we would be in big troubles. 

Personally, I very much belong to the Hamiltonian stat mech community, if you want, simply because I love to study how emergence (“more is different”) comes about. Of course, also within ST you can study emergent phenomena, for instance, how to get deterministic (ancient) thermodynamics from ST. However, I can’t help, but it feels like cheating when I start from a Markov process. Maybe I should look for a psychiatrist. xD

Now, what quantum thermodynamics is about is a good question. First of all, I would say it is a misnomer as there is nothing to “quantize” in thermodynamics, but it’s a catchy misnomer. Next, I believe that part of the community really does thermodynamics because they start again from a “constitutive equation”, namely a quantum master equation (with the caveat that it is much less clear from which one to start compared to classical ST). However, another big part is interested in deriving the master equation and other notions microscopically, which I would then call part of quantum stat mech. 

Finally, I need to ask you to clarify your mentioning of a failing program for technical reasons in particular with reference to the eigenstate thermalization hypothesis. So, let me lean out of the window a bit, and proclaim that the eigenstate thermalization hypothesis is IMO perhaps the biggest thing we have discovered in stat mech since Boltzmann’s S = kB*log(W). 

M: My takeaways: 0) There is nothing to quantize in thermodynamics (I very much agree on this); 1) ST (or, if you prefer, the thermodynamics based on Markov processes) is a minor piece of a much broader science that is thermodynamics; 2) In ST the second law is a postulate (but I wonder: is this normative state of the second law actually different in thermodynamics?); 3) Something needs to be matched to experiments; 4) The “more is different” philosophy is helpful in thinking about systems from bottom-up; however it is not prescriptive, and as far as I understand the proposal by Phil Anderson was strongly rooted on his own understanding of quantum field theory and equilibrium statistical mechanics. As regards the eigenstate thermalization hypothesis: I agree that it is a fascinating set of ideas. I would just argue that it does not actually predict a-priori the behaviour of any system, at best it explains it a-posteriori.

P: This seems to bring us back to the definition of a “predictive law” that we already discussed above. Perhaps we have a fundamentally different understanding of science or prediction? 

So, to follow the example, the eigenstate thermalization hypothesis is a very precise statement (prediction?) about the matrix elements of thermodynamically relevant observables. This might be hard to test directly in a lab, but you can directly test it numerically. Moreover, it explains (predicts?) why you can use the conventional ensembles of statistical mechanics at first place, which are subsequently used to derive critical behaviour, fluctuation theorems, master equations, linear response theory, etc. How much more prediction do you want to have? 

I don’t even believe the relevance of the microcanonical ensemble! Some time ago I opened up about this in a funny paper arXiv:1307.2057. I don’t think in this respect “quantum” solves the conceptual problems of “classical” (at best it makes them worse, if not fatal). But let’s not go into this…

To me there are at least three dimensions to your question: 1) How is the program of building up the world of open systems from the tenets of underlying isolated systems justified? 2) Has matching to experiment happened and in what sense (and what do numerical confirmations confirm)? 3) Are prediction and description ever separated in science?

On top of that I tend to mix up positions from three philosophical points of view: A) the critical rationalist approach that there is a scientific method with strict rules (e.g. “falsifiability”), maybe the most popular among scientists when they have to describe their own work—despite the fact that rational criticists also claim that knowledge is not achievable to any degree; B) the anarchist approach that “anything goes”; C) the socio-anthropological approach that contextualizes the mix of science & society.

It would be an interesting exercise to use the ETH as a testing ground for all these 3*3 intersections. Maybe this is too much for the moment. So I would like to conclude with a very personal question to you, that you should not answer—so that we close the cycle while leaving it open.

I think we are sharing these thoughts not only because we know each other from a long time but also because we share a sort of “political drive” about the conditions upon which scientific research is conducted/financed/evaluated etc. If you are in the field it is very easy to see the political influences into science; it is more subtle to see the scientific into the political. To what degree are your political stands on where academia should move (that you expressed very well in your blog) influenced by your own stands on how nature should be thought about?

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