# Conjugate variables, in thermodynamics and elsewhere

When I wrote an explanation of cap and trade, I had a strong temptation to make a physics analogy, to an idea in thermodynamics. The trouble is nobody would understand the analogy, and I would be obliged to explain the physics instead of the economics. Well I’d still like to explain the physics, but in a separate article.

There are certain thermodynamic quantities that are considered to be paired with one another. For example, pressure and volume, or temperature and entropy. These pairs are called thermodynamic conjugate variables.

The concept of conjugate variables can be challenging for physics students to understand because the examples we use are unintuitive. The connection between pressure and volume is unclear, and most people don’t wholly understand what temperature or entropy even are. Therefore, I’d like to use a more down-to-earth example.

So, let’s consider a pool of water. The pool is described by two conjugate variables: the volume of water, and the height of the water.

I’m going to make a few observations, which I claim generalize to all thermodynamic conjugate variables.

First, note that height is determined by the volume of the water–and conversely the volume is determined by the height. It also depends on the size and shape of the pool, of course. But given a fixed pool, if we dump a certain volume of water into the pool, it will settle at a particular height. And if we fill the pool up to a certain height, that requires a certain volume of water.

Next, let’s consider what happens when we link two pools together. For instance, let’s say one pool of water is “the Atlantic Ocean” and the other pool is “The Mediterranean Sea”. When these two bodies of water are connected, water flows from one to the other, until both bodies of water have the same height. However, the two bodies of water do not have the same volume–obviously the Atlantic Ocean is more voluminous.

The volume of the Atlantic Ocean and the Mediterranean Sea is equal to the sum of their respective volumes. However, we would not say that the height is equal to the sum of their respective heights. In technical terms, height is an intensive variable, meaning that it doesn’t change when you add two bodies of water together. And volume is an extensive variable meaning that it gets added when you add two bodies of water together.

Strictly speaking, height and volume aren’t real thermodynamic conjugates, because we also require that the two quantities when multiplied together give us units of energy. (The real conjugate variables might be height and weight.) Still, it seems like there should be an expanded concept of conjugate variables. Maybe there is one and I just haven’t heard of it.

So, how does this apply to cap and trade? Cap and trade introduces a pair of conjugate variables: the number of allowances, and the price of each allowance.

In a cap and trade program, the government fixes the number of allowances, and this determines the price of each allowance (according to the demand curve). There’s also an alternative policy that, instead of fixing the number of allowances, fixes the price of each allowance. Under this alternative policy, the price of each allowance determines the number of allowances purchased.

We can also talk about what happens when you join together two cap and trade programs, such as the program in California, and the one in Quebec. The price of each allowance is an intensive property, and at equilibrium should be the same in both programs. The number of allowances is an extensive property, and gets added up.

The same can be said of pressure and volume, or of temperature and entropy. But those are somewhat harder to explain, despite being the most common examples.

1. says

Very interesting. How does this help?

I generally like physics analogies in economics. For example, virtual particles could be an analogy to bank money: banks can create money by creating an offsetting liability (money and anti-money), which self-destruct when they meet back at the bank (borrow repays the loan).

Sadly, my students know less about QM than economics, so the analogy doesn’t help me in the classroom.

2. Rob Grigjanis says

Larry Hamelin @1: I think that analogy does more harm than good. The popularized notion of virtual particle-antiparticle pairs popping in and out of existence by “borrowing energy from the vacuum” is nonsense, with no basis in quantum field theory.

3. Rob Grigjanis says

The connection between pressure and volume is unclear

I’m not sure what you mean by this. I’d guess most people have a fairly good grasp of the relationship, at least for gases. It’s not even much of an abstraction of generalized forces causing generalized displacements.

4. says

Arguing by analogy is a waste of time if it takes more effort to explain the analogy than to simply explain the topic of discussion. Cap and trade is not very difficult to understand, unless the concept is being deliberately obscured.

5. Rob Grigjanis says

Marcus @4: At the very least, I’d say that if the analogy teaches some basic physics, it’s worth the effort. The idea of conjugate variables is nigh ubiquitous in physics; classical mechanics, statistical mechanics, and quantum physics.

6. says

I pretty much agree that the analogy is not very useful to explain the economics. If anything, cap and trade is the easier thing to understand, and the analogy helps you understand the physics.

@Rob Grigjanis

I’m not sure what you mean by this. I’d guess most people have a fairly good grasp of the relationship, at least for gases.

Do they? I don’t recall this being the case when I TA’d intro physics… nine years ago. The thing that makes pressure and volume difficult to understand, IMHO, is that when we talk about two gasses exchanging volume, we’re not talking about gas transferring from one to the other. We’re imagining a thin membrane between the two ensembles, and imagining the membrane moving. Contrast to the volume/height example, where a transfer of volume is simply the flow of water. Volume/height is really more akin to number and chemical potential.