Interview with Chiara Marletto, Part I: The Second Law of Thermodynamics [10/7/2020]
How constructor theory reconciles the irreversibility of the second law of thermodynamics with reversible laws of motion.
Read the original paper on constructor theory of thermodynamics here
Note—for an introduction to constructor theory, see my earlier post.
Logan: Thermodynamics is a branch of physics concerned with heat, work, entropy, and related concepts. It’s been formulated a few different ways throughout history. Using constructor theory, you’ve improved the so-called ‘axiomatic formulation’ of thermodynamics. Before we get into the constructor theoretic version of thermodynamics, can you briefly outline the axiomatic formulation?
Chiara: This requires a detour in general on thermodynamics and statistical mechanics. The first idea of thermodynamics was proposed right around the Industrial Revolution. During that time, people were concerned with the efficiency of heat engines, objects that could transform energy that was capable of doing work into other kinds of energy and vice versa. It became clear that there were two kinds of energies. One is work, which is a form of energy that can do something useful. The other is heat, which is a form of energy that is not useful. It’s a kind of ‘disordered’ energy, if you like.
When energy is converted into heat during a certain physical process–for example, when you brake on your bike–the heat that is produced is not useful to produce work. So part of the mechanical energy that you put into pushing your bike ahead gets lost once you perform the act of braking.
There are two laws that describe how these different energies–work and heat–get transformed into one another: the first law of thermodynamics and the second law of thermodynamics. The first law is about the fact that no matter what you do with heat and work, you can’t create energy out of no energy. It’s about the conservation of the overall energy in your system. The second law is about the fact that once you convert work into heat, it’s impossible to convert the heat back into useful work. The problem with this formulation of the second law is that it’s very vague. There are several formulations of the second law that amount more or less to the same thing as far as heat engines are concerned, but they mean very different things as far as physics is concerned.
So here is where the axiomatic approach to thermodynamics comes in. When people like Kelvin, Clausius, and Carnot thought about the second law in terms of heat and work, the concern was about how can we have laws that tell us about the efficiency of heat engines. So in that sense, the concern wasn’t about fundamental physics. But as it became clear that the second law was an idea of irreversibility, lots of physicists became more intrigued. We know that the other kinds of laws that describe the universe—the dynamical laws of motion—are time-reversal symmetric. The laws are the same if you run them forward or backward. So the idea of irreversibility–as required by the second law–is at odds with the laws that describe microscopic particles–say, the constituents of a given heat engine.
So the question is, how do we fit this idea of irreversibility that seems to apply at the level of heat and work with the fact that things that can perform work or can get heated up are ultimately made up of elementary particles that obey reversible laws?
There are different ways of going about trying to solve that problem. Statistical mechanics was initiated by people like Boltzmann and then Gibbs, and it’s one way of reconciling this irreversibility that seems to appear at the macroscopic level with the presence of reversible trajectory of microscopic constituents.
[Note—statistical mechanics derives the second law of thermodynamics from some underlying dynamical laws but relies on coarse-graining approximation schemes. Therefore, the second law in this framework cannot be a scale-independent principle.]
But there’s also a different approach to the second law that just tries to formulate the law in more fundamental terms such that, instead of trying to derive the irreversibility out of the reversibility, one makes some additional assumptions so that one can formulate the second law in an exact way that is compatible with time-reversal symmetric laws. This way, the law does not have to use approximations, nor would it depend on the scale of the object in question. This approach is called axiomatic thermodynamics, introduced by mathematician Carathéodory. Other people, like Plank, also tried to formulate the second law in similar ways.
In this axiomatic approach, instead of trying to derive the second law from microscopic laws, as statistical mechanics tries to do, you state the law as following from a set of axioms or assumptions about the physical world. These axioms I prefer to call principles. And then you try to explain why the second law as formulated in this way is actually compatible with time-reversible symmetric laws.
In this approach, the second law takes the form of a very simple statement: certain tasks are possible, so they can be performed in one direction to arbitrarily high accuracy–say, heating a volume of water from one temperature to another. In the axiomatic approach, the second law says that such a task is possible with side effects that thermodynamicists call mechanical means. But if you try to do the inverse–if you try to cool the water from one temperature to another–you can’t perform the task to the same accuracy with just these mechanical means. So you’d need other kinds of side effects to perform this inverse task.
Mechanical means are epitomized by a system where you’re lowering or raising a weight in a gravitational field. The reason why they’re called mechanical means is that they only involve systems that don’t dissipate, so they work like a perfect spring or flywheel–systems that don’t have any source of dissipation.
So the statement of the second law in this axiomatic way is very promising because it’s self-contained and doesn’t require the law to be derived from reversible dynamics. But at the same time, there’s a gap: the law as formulated in this way doesn’t say what mechanical means are! It just says that there are tasks that are possible with the aid of mechanical means, and that if I try to use mechanical means to perform the reverse task, that’s impossible. But it’s not said what mechanical means are, so the usual thing is just to say that mechanical means are anything that is equivalent to raising or lowering a weight in a gravitational field. But this is not very fundamental and it’s very ad hoc. For the purposes of classical thermodynamics, this was perfectly fine and in fact quite powerful. But the issue that i had from the point of view of fundamental physics is that if one wants the second law of thermodynamics to be as general as other principles of physics—like, say, the principle of locality or the conservation of energy—you need to make this principle free of ad hoc assumptions and also scale-independent. The second law should apply not just to macroscopic objects like a weight being raised in a gravitational field but to any kind of physical object. Currently, that is the problem that my work on constructor theory is trying to address.
Logan: This scale-independence of the constructor theory formulation of thermodynamics is innovative because traditional formulations were never able to provide exact expressions of the laws or where exactly where they’d break down given that they’re currently scale-dependent.
Chiara: That’s correct. If you talk to a physicist, you will find more or less that they’re divided broadly into two camps. One camp thinks that the second law is not fundamental, meaning that it is approximate and only really holds for very specific situations that involve macroscopic objects. We can just say that they have to be sufficiently large with respect to a certain property. Now, what ‘specifically large’ means is never actually stated exactly, but it’s fair enough because this take on the second law is only relevant where you have big aggregates in certain regimes, and when you’re out of those regimes, the law doesn’t make sense.
So if you have a single atom interacting with another particle such as a photon, it wouldn't make sense to think of the second law as being applicable because all you have is the dynamical laws that describe the atom and the photon. And those dynamical laws are reversible, and therefore there’s no second law. This camp would be content with the second law being scale-dependent.
The other camp, the one to which I belong, is unsatisfied with the idea that the second law could depend on scale. Already, we have principles like the principle of locality–the fact that there’s no action at a distance– that holds at all scales. The principle of locality is true for our scale–if I try to communicate with you through a medium, there has to be a finite time between my utterance and your receiving of it due to the speed of light–and it’s true at the scale of single atoms. But in the case of the second law, this is not (according to some!) true. So although it’s a very powerful principle, it apparently stops being meaningful in certain situations–for example, when you don’t have a well-defined temperature, or when you have few particles. Constructor theory tries to bridge this gap and formulate the second law in the same way that, say, the principles of information are formulated.
Of course, that doesn’t mean we would invalidate the reversibility of dynamical laws. What it does mean is that there is a principle that holds irrespective of the scale, and that this principle can be shown to be compatible with time-reversible dynamical laws we know of. It’s subtle because the fact that there is irreversibility in a certain sense doesn’t necessarily mean that dynamical laws have to be irreversible as well. You can show compatibility between irreversible behavior with reversible dynamical laws just by assuming that there are different levels of explanation. The level of explanation with respect to irreversibility and the second law is disjoint from the one where you are observing reversibility of, say, particles.
The new element in the constructor theoretic approach is that it grounds this notion of mechanical means into physical properties of systems, specially counterfactual properties, and specifically information-theoretic properties. So that’s the way we get out of this problem.
Logan: An idea you introduced in your paper is that there is a distinction between a task being permitted and a process being possible. What does this distinction imply for our understanding of thermodynamics?
Chiara: This is a very nice question and I think you rightly point out the fact that the irreversibility is, as you say, not at the level of dynamic or trajectories, but at the level of tasks being possible or impossible. A dynamical law can be seen as a set of constraints on what trajectories in spacetime are allowed. The fact that these laws are time-reversal symmetric means that if you allow one trajectory, then you have to allow its time-reversed trajectory as well. For instance, if a particle can go from A to B according to some dynamical law, then that law must also allow for the trajectory in which the particle can go from B to A.
But the irreversibility in constructor theory is a different kind because it’s about tasks, not trajectories. A task is a transformation of a physical system from some state A to another state B. So in that sense, it looks like a trajectory, but actually it’s not. Because when we say that a task is possible, we don’t need the trajectory for the object to go from A to B to be allowed. We mean instead that the task of going from A to B can be performed to arbitrarily high accuracy by coupling the system with an environment. This environment can perform the task once and then–crucially–retains its ability to do so again. The fact that a task is possible means that you can bring it about with as many resources from the environment as you like with the requirement that when these resources are used, performing their ability to perform the task is maintained. So they work in a cycle.
An example of a possible task is, say, performing a NOT gate on a bit–going from 0 to 1. That task is possible because you can build more and more sophisticated computers that perform that task to arbitrarily high accuracy. And a task is impossible when there is something that intervenes and stops the environment from performing the task to arbitrarily high accuracy and retaining the ability to cause it again.
Consider the conservation of energy. When we say that energy is conserved, and we consider the task of going from zero energy to nonzero energy, that task is impossible because if you try to do that with an environment that stays unchanged, you would violate the principle of conservation of energy in the sense that you would be creating some energy out of zero energy with no side effects on the environment. The environment would have to change–it would have to provide some energy to allow this task to be performed.
Therefore, the fact that the trajectory of going from state A to state B is allowed does not mean that the task of transforming the system from being in state A to being in state B is possible.
While under time-reversal symmetric laws the fact that the trajectory is allowed in one direction but not the other is not permitted, it’s still logically possible that under these laws you have some task that is possible in one direction to arbitrarily high accuracy–for instance, heating up some water with mechanical means only–but if you try to perform the reverse task with only those means, you can’t, and so this inverse task is impossible. If you try to simply reverse the dynamical trajectory corresponding with the task at hand, you will not get a process that performs the reverse task to the same arbitrarily high accuracy. So the fact that the dynamical laws are time-reversal symmetric doesn’t help you perform the task both in its forward and backward directions. This is how you can reconcile irreversibility of the second law with time-reversal symmetry of dynamical laws of motion.
Logan: You use concepts such as ‘work media’ and ‘heat media’ in expressing the laws of thermodynamics in a constructor theoretic way. Could you explain what these are?
Chiara: Let me clarify one thing first. The idea that I just said–that a task could be possible in one direction but not in the other, despite the reversibility of the underlying dynamical laws–is an idea that was already discussed widely among physicists, but not necessarily in this constructor theoretic form in which possible and impossible are well-defined. Plank, for example, already had this idea of possibility and impossibility being compatible with time-reversal symmetry.
Constructor theory allows for a more general and rigorous definition of ‘possible’: the second law can be formulated such that a certain task is possible in one direction with mechanical means only but is impossible in the backward direction with the same mechanical means only. Constructor theory helps us define mechanical means in a general way, so it took what had been an ad hoc notion and helps us express it in a general, scale-independent way and independent of the underlying dynamical laws.
Work media are the way I call the generalization of mechanical means. In classical thermodynamics, these are objects that can perform useful work on other objects–and they never dissipate. They are, for example, a perfect spring–when compressed it doesn’t heat up, it can be decompressed again, etc. A flywheel is another example. An atom that can be prepared in a sharp energy state that can be excited or excited without dissipating is yet another. All are ideal scenarios. You can define work media such as these by first asking: ask what is it about these objects that makes them behave in the same way? What is the feature that makes them all able to be used to perform work, store energy, or transfer energy without dissipating?
The answer is that they have in common certain counterfactual properties–that is, properties about the ability to perform a certain task. The way we define work media is to give a set of tasks that are possible on them, and in this sense work media are very similar to information media, which is some other class of media that I defined in my work with David Deutsch about information theory. For example, we defined all the objects that can contain information as objects that can have a set of states that can be permuted and copied. So it’s all about the possibility or impossibility of tasks. And this definition is very general because it’s scale-independent and doesn’t require us to specify a particular dynamical law.
For work media, we can do a similar thing. We can define a class of objects that have certain tasks being possible on them, and in this way provide a foundation for the second law being formed in terms of a given task being possible in one direction with side effects via mechanical means only, and at the same time the impossibility of the reverse task with the same side effects. So that is work media.
Heat media are systems that have at least a couple of states, A and B, with the property that it’s possible to perform the task of going from A to B by having a side effect on a work medium only, but if you try to perform the task of going with side effects on work media only, that’s impossible. An example of a heat medium is a volume of water with different temperatures available. If you try to heat up this volume, you can do so by isolating this volume in a container and stirring very hard. This is mechanical means being used to heat up the water. But if you try to cool it–that is, perform the reverse task–by using the same mechanical means (stirring), it is impossible.
This was already well-known in classical thermodynamics, but in constructor theory we can now define heat media as a generalization of these objects that have states–sometimes called thermal states.
The key difference in the constructor theory approach is that you can talk about this idea of dissipation and so on without using the usual approximations that thermodynamics does. The key to all of this is the idea of work media, since its definition in constructor theory is general and scale-independent and doesn’t rely on any dynamical laws. This makes the original axiomatic approach much more powerful than it would have otherwise been.
Note — view/listen to original interview here.