Classical Thermodynamics From "Intuitive" Symmetries? Part 1

In my last post, I promised to talk about Paul Samuelson's paper "Conserved Energy Without Work Or Heat". Now I will do so. Even earlier I had earlier promised a post about Disney Princesses. While I have a variety of observations, I haven't yet put them together into a theme.

First, a word about our author.

P Samuelson

Paul Samuelson is one of the father's of modern economics. Even more than with more famous economists like Keynes or Friedman, one can divide economics into a pre- and post-Samuelson quite easily. Paul Samuelson's most important work was his dissertation, Foundations of Economic Analysis. In that book, he provided perhaps the first completely mathematically clear explanation of what it is economists where doing. When an economist says that, for instance, a given tax is good or bad, what he or she means is that changing something given, the rest of the economy will eventually adjust and where it will settle will be better or worse than where it is now. A good example (with applications to finance) can be found here - and in innumerable other places! Samuelson made many other advances in almost every area of economics. Much can be said about his "scientific personality". He considered himself a child genius well into his 80's. He was immensely concerned with every aspect of scientific work, empirical, theoretical, philosophical, historical and pedagogical. Unlike many economists of his stature and influence, Samuelson almost never outright dismissed an economist or an economic theory, always describing them as containing nuggets of truth - of course, truths that he himself has formally obtained. His scientific ideal was that of classical thermodynamics, where the foundations were clear, the applications enormous and the empirical validity impeccable. Some have objected to his appreciation for classical thermodynamics, but most of these complaints are ill founded. To the extent he was inspired by classical thermodynamics, classical thermodynamics is inspiring. (Aside: it is not at all correct to believe that just because pre-Samuelsonian economists mostly used geometry and post-Samuelsonian economists mostly used algebraic notation that one was less mathematical. J.C. Maxwell's Theory of Heat was written almost entirely with geometry, and this example can be multiplied.)

Now on to the paper itself. This paper has multiple goals. One is a slick derivation of deep aspects of physics (esp. thermodynamics) from a few qualitative empirical regularities. One is to present the argument that one could have - in an alternate universe - completely missed the deep connections of thermodynamics to Newton's Laws. I am going to play loose with language, sometimes I will say "amount of heat" and other imperfectly defined things in order to bring these ideas closer to everyday experience. If this confuses anyone, then I will do another version which I am more careful (or they can read Samuelson's original).

(Aside 2: This idea, in a loose philosophical way, might be connected to "microfoundations" debate economists sometimes talk about. If microfoundations can be seen to be like statistical mechanics, then macroeconomics is like thermodynamics. This paper would then be an example where a simple empirical regularity is all that is needed to establish deep economic laws, rather than investigation of the deepest parts of the consumer's psyche. This argument is unimpressive, and Samuelson would never have dreamed of making it, but you can think about it if you like.)

Now, on to the deep parts of the paper. The main empirical principle of this paper is that if you put two hot things in contact, then they will equilibrate. A cold drink will turn warm in the hot summer afternoon. This principle is purely qualitative, but quantitative measures will fall out of it. Notation will simplify things. Our Principle is that "The Temperature of System 1 and the Temperature of System Two go to the Equilibrium Temperature in System 1 and the Equilibrium Temperature in System 2.". This is a mite cumbersome. Instead we say \( (t_1 ; t_2) \rightarrow (t_eq ; t_eq) \). We assume that \(t_eq\) is a function of the initial conditions.

Brief considerations as to measurability of heat are given, but these are minor enough that the assumption that the only important conclusion is that the final temperature is to be a function of the initial temperatures. This paper is written as an alternate history past Carnot, so we aren't interested in chemical, gravitational, electromagnetic, etc forces (more on aside 2: one of the many arguments against the above aside is that one couldn't do this without "microfounding" heat. Is this true? Discuss.). We consider the effects of heat by itself.

A thought experiment. Consider a bowl of hot soup kept in contact with a container of cool apple sauce. The bowl and the container are strong, they do not melt or flex because of the heat. They are kept in a insulating lunch bag, so that they don't lose heat to the environment. A sort of drawing of this situation:

What will happen? By our principle founded on common observation, the substances will come to be the same temperature. Of course, there is more to consider than just the temperature. I drew the above as if the soup and apple sauce were in equal volumes, but if I had enormously more soup or enormously more apple sauce, then the one with enormously more volume would barely notice the change due to the other. There might be other dependencies, but Samuelson follows Carnot wisdom that the main action can be captured considering only the interaction of volume and heat - and sometimes heat alone! Symbolically: $$(t_1,v_1 ; t_2,v_2) \rightarrow (t_eq,v_1; t_eq,v_2)$$ $$t_eq = f(t_1,v_1 ; t_2,v_2)$$ The first part is read "The Temperature and Volume of System 1 and the Temperature and Volume of System Two go to the Equilibrium Temperature and Original Volume of  System 1 and the Equilibrium Temperature and Original Volume in System 2.". Obviously, this doesn't depend on how the apple sauce and the soup are oriented, since the bag is being thrown around all day anyway. This implies that \(f(t_1,v_1 ; t_2,v_2)= f(t_2,v_2; t_1,v_1 )\). Today, I will concentrate solely on what can be concluded from experiments of this type alone, but in Part 2 I will introduce a more complex experiment involving pistons.

These experiments can be compounded arbitrarily. For instance, we can put four substances together:

And the above reasoning still applies. The arrangement can be manipulated so that red and yellow equilibriate while orange and burgandy equilibriate, then those two are placed next to each other and the whole system is allowed to equilibriate. Alternately, the arrangement can be manipulated so that red and orange equilibriate while yellow and burgandy equilibriate, then those two are placed next to each other and the whole system is allowed to equilibriate. Either way, the system comes to the same temperature. It is easy to arrange the volumes (actually, specific volumes) to be the same. In this case we have the simple symbolic expression for the above: \(f(f(t_1,t_2);f(t_3,t_4))=f(f(t_1,t_3);f(t_2,t_4))\). It is read "The equilibrium temperature of the equilibrium temperature of the first two substances touching the second two substances is equal to the equilibrium temperature of the equilibrium temperature of the first and third substances touching the second and fourth two substance.". You can start to see why we invented this notation!

Now we add a couple new assumptions. These assumptions are unobtrusive and intuitive, but they might be wrong and must be stated. We have already extensively discussed the existence and symmetry of the function \(f \), which takes the system set up and gives the equilibrium temperature. We also assume that the function \(f \) has the property that if two substances have the same temperature, then the equilibrium temperature is that temperature. (more on aside 2: if we were interested in "microfoundations" right now, then this would be a statement about the nature of an equilibrium - namely that it is an equilibrium!) Otherwise, purely mental divisions could make physical changes. Finally, we assume that if you perform the same experiment, but you make one of the substances hotter before hand, then you always get a hotter equilibrium. It is these properties of heat, perhaps, that lead to the idea that heat was an independent substance!

These assumptions give a remarkable conclusion, the existence of a sort of energy function! Perhaps a better name would be a caloric function (this does not mean, of course, that there is any such physical substance!). A brief verbal argument can be made. I said before that the above that the latter assumptions make heat seem like a substance, since if you add more of it, more comes. The amount of that substance is the caloric. The equilibrium temperature is derived by averaging the amount of caloric in both substances (notice averaging, not summing. This becomes clearer when the mathematical argument is fully expounded). What is remarkable is the role of symmetry in the proof. If the function were not symmetric, we'd have no reason to think that the equilibrium temperature could be found by averaging equilibrium temperatures of partitions. It was this fact that is the foundation of this theory! By choosing references, one can convert this equation into a standard internal energy function. We have begun the battle of recovering classical thermodynamics from simple symmetry arguments.

Phew! I have covered much ground and only barely scratched the surface of this paper! I need to go through and show how this caloric function is found, give more examples, and there's a an entire second experiment! These posts will come every Sunday. If there's particular aspects of this argument that interest you, drop me a comment. Before I go, however, there are a couple things I would like to highlight. First of all, notice that we derived an energy function, but I didn't check anything about its form. For instance, I didn't make any assurances that it was always positive. It is this reason I prefer to call it a caloric function, even if I risk misinterpretation that this somehow vindicates the physical concept of caloric. More importantly, I want to highlight that this argument makes no reference to Newton's or Schrodinger's laws. Physicists have a deep rooted appreciation for the laws of thermodynamics, and it is arguments like these that gives substance to those feelings. No matter what the universe is like, the laws of thermodynamics will apply as long as those fundamental symmetries are observed. This does not establish that the laws of thermodynamics are universal, but a non-physicist might wonder why they are believed to be and arguments like this will help the intuition in that regard.

Finally, a social observation. The laws of thermodynamics are wildly misused in bad science, bad philosophy and even bad politics. I have seen irresponsible writers on the internet imply and argue that they make action on global warming impossible, often with naive importations into economics or politics. What I hope is that when reading this, you absorb some of the actual intuition of this science, rather than the slogans such people use. When someone claims an application of thermodynamics, check first to see if even first principles - such as these - apply. If it is not obvious how, then they are not obviously right. See you next week!