I think the analogy between thermodynamics and economics is really helpful for improving my thermodynamics intuitions. I have heard people allude to there being an analogy here before, but I didn’t actually find anywhere that discusses the version of the analogy which I came up with when I sat down and thought about it a few days ago. The best part of this post is a cute argument for the formulas describing efficiency of heat engines and refrigerators. This post probably won’t be useful to you if you don’t already know basic thermodynamics.

Some papers about similar analogies include this one and this one; the analogy I’ll describe here is a bit simpler. Schroeder’s excellent “Thermal Physics” (which I learned most of my stat mech from) has a qualitative version of this analogy that I ran across after writing this; he doesn’t go as far as describing temperature as an inverse price, which means he misses out on my cute argument about heat engine efficiency.

In this post I’ll just establish the analogy for energy, temperature, entropy, and limits on efficiency of heat engines and refrigerators. You could continue this analogy to more general thermodynamics concepts like pressure or chemical potential; I might write that up at some point in the future.

In thermodynamics, you have objects that have various amounts of internal thermal energy. If you put them in contact with each other, then they will transmit thermal energy from the higher temperature object to the lower temperature object, because systems naturally evolve towards states of higher entropy, and temperature is the inverse of the derivative of entropy with respect to energy.

Here’s the analogy: I’m the US president, looking at the economies of different US states. Each economy has some utility function with respect to some good that can’t be created or destroyed, just bought and sold; let’s use gold as our example. The price of gold in a city is the derivative of utility with respect to quantity of gold. If you allow trade in gold between two cities, then the city with the lower gold price will sell gold to the city with the higher gold price.

(In economics, sometimes you talk about markets in terms of quantities per unit time. In this analogy, we’re not doing that–in the cases we’re talking about, when there’s a price difference, there’s some trade for a while and then trade ceases.)

The correspondences:

  • utility <-> entropy
  • gold <-> energy
  • conservation of gold <-> conservation of energy
  • inverse of the price of gold (eg, how much gold can you buy for $1) <-> temperature
  • when allowed to trade, the place with lower price sells to the place with higher price <-> when in thermal contact, the object with higher temperature transmits thermal energy to the object with lower temperature
  • This is kind of out of scope, because you can’t actually derive it just with statistical mechanics without more assumptions than I’m otherwise making here, but the rate at which trade occurs is going to be higher if the price difference is greater.

There’s a bit of subtlety about the definition of utility (eg, what units are we measuring in?). I think that it basically works out as long as you measure changes in utility in dollars, which corresponds to the people in these markets having other goods that they can buy with money.

In thermal physics, you’re only “able” to do certain things to your system. Eg you can insert or remove insulators between objects “for free”–without having to do work. In this analogy, because I am a strict constitutional literalist, all I’m allowed to do is regulate interstate commerce–I can add tariffs or subsidies to trade between states, but I can’t directly take people’s stuff: all I can do is set up restrictions on various consensual trades, and then see if people still voluntarily make them.

The equivalent of the second law of thermodynamics is that there’s no way for me to lower total utility. If initially, all trade is banned, then I can increase total utility by allowing trade. But after I’ve done this, the price will equilibrate, and I won’t be able to decrease utility by banning trade again; the only effect of banning trade is making it so that if the prices diverge in those two states, I’m then preventing there from being as much utility as there could have been.

For example, suppose that I want a bunch of gold for myself, and I know that New York and California have big markets in gold at different prices. Specifically, in NY, the gold price is $5 and in CA it’s $1. How much money can I make by allowing a limited amount of trade between these states? Well, the best I can do is to put a slightly-less-than-$4 tax on CA selling to NY. At this price, for every kilogram of gold that leaves CA, I’ll get 0.8kg and NY will get 0.2kg. It’s impossible for me to get more of the gold than that–any larger tax would make it no longer worth it for CA to sell. So the amount of wasted energy is just the ratio of the two gold prices.

This is completely analogous to the maximum efficiency of a heat engine. In a heat engine, you have a bunch of hot stuff and a bunch of cold stuff, and you want to extract work from the hot stuff by letting it transfer energy to the cold stuff. The maximum possible efficiency here is given by the same formula.

Clearly, my gold taxation will be more efficient if the prices of gold in the two places are as great as possible. Ideally the price of gold would be basically zero in one place and basically infinite the other; then I could take basically all of the gold for myself. This is analogous to the fact that a heat engine is most efficient when the hot reservoir is much hotter than the cold reservoir.

What about if I want to make a refrigerator? A refrigerator is when you expend energy in order to force heat to go from a cold place to a hot place. Under what conditions will this be most efficient? I’ll first provide the analogy to a refrigerator as a hint, and then I’ll provide the answer.

Wanting to spend energy in order to force heat from a cold to a hot place is like trying to increase the price of gold in a place with a high gold price by causing them to sell to a place with lower gold price. Eg to continue the previous example, where the price was $5 in NY and $1 in CA, we’re trying to get NY to sell to CA.

So what’s the efficiency?

Well, all we can do is offer a slightly-more-than-$4 subsidy to NY buying from CA. So for every kilogram of gold moved to CA, we had to spend four kilograms. That’s 20% efficiency. In contrast, if the prices are $9 in Nevada and $10 in Arizona, we only have to spend $1 per kg moved, which is 90% efficient. So back to refrigerators, that means that they’re most efficient when the inside and outside temperatures are as similar as possible.

Other tidbits:

  • You can have a state which has a preference against having some good, eg nuclear waste. So this state will sell it to anyone who wants to buy it, no matter how low their price is. And they’ll even pay a state to take it, if that state has a negative preference that’s weaker. This is analogous to negative temperatures (though remember that temperature is analogous to the inverse of price, not to price itself).
  • Heat capacity is the change in temperature per unit energy. This is analogous to the change in price per unit quantity of gold purchased. Something with a large heat capacity doesn’t change its temperature very rapidly in response to an energy change; a high “heat capacity” state doesn’t change its market price of gold very much as gold flows in or out.

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