I put a set of new results out on to the arxiv recently. They were fun to work out. They represent some of my continued fascination with holographic heat engines, those things I came up with back in 2014 that I think I’ve written about here before (here and here). For various reasons (that I’ve explained in various papers) I like to think of them as an answer waiting for the right question, and I’ve been refining my understanding of them in various projects, trying to get clues to what the question or questions might be.
As I’ve said elsewhere, I seem to have got into the habit of using 21st Century techniques to tackle problems of a 19th Century flavour! The title of the paper is “Approaching the Carnot limit at finite power: An exact solution”. As you may know, the Carnot engine, whose efficiency is the best a heat engine can do (for specified temperatures of exchange with the hot and cold reservoirs), is itself not a useful practical engine. It is a perfectly reversible engine and as such takes infinite time to run a cycle. A zero power engine is not much practical use. So you might wonder how close a real engine can come to the Carnot efficiency… the answer should be that it can come arbitrarily close, but most engines don’t, and so people who care about this sort of thing spend a lot of time thinking about how to design special engines that can come close. And there are various arguments you can make for how to do it in various special systems and so forth. It’s all very interesting and there’s been some important work done.
What I realized recently is that my old friends the holographic heat engines are a very good tool for tackling this problem. Part of the reason is that the underlying working substance that I’ve been using is a black hole (or, if you prefer, is defined by a black hole), and such things are often captured as exact solutions to Einstein’s equations. As a result, the resulting thermodynamic system can be written down as a set of exact equations. Furthermore, the prototype engine cycle I’ve defined, while simple enough in the original paper, was made even simpler in a followup paper I wrote that showed how to define the efficiency of the engine entirely in terms of the mass of the black hole, evaluated at various points on the cycle. So various analyses of the engine that could only be done in various limits, like expanding away from high temperature, volume, or pressure, could be done exactly.
That exact formula for the efficiency is exactly what I needed, since I wanted to explore a region of the p-V plane where the usual approximations would not do. For various reasons I was curious about the properties of the engine at intermediate temperatures where (for a class of black holes with charge q) there is a non-trivial phase structure (a phase structure I discovered with colllaborators way back in 1999, so some of this was a trip down memory lane!). In particular, there is a second order critical point (which, incidentally, had always struck me as an answer waiting for the right question too) near which I wanted to run my system as a heat engine, to see the effect on the efficiency. Since the specific heat has markedly different behavior near such a point, you might imagine that this might show up in an interesting way in the efficiency, somehow. (There were a bunch of other reasons why that point was of interest to me as a place to run an engine, but this is the one that’s easier to explain.)
So to cut a long story short, I found a way (actually, a family of ways) of designing an engine that gets as close to Carnot efficiency as you desire while doing finite work in finite time. The key is the charge parameter q on the black holes. There is a new limit (as far as I can tell) of use here: large q. If you stay near the critical point (in a natural manner that I outline – its location is a function of q), as q gets larger and larger, the efficiency of the engine gets closer and closer to the Carnot efficiency! It actually converges rather swiftly especially if you run the cycle right through the critical point.
Actually it turned out that for various reasons, some of the bona fide, card-carrying stat mech researchers had thought about driving a working substance to criticality as a means of tackling the Carnot problem, using careful argumentation in various regimes of validity, so I found myself in good company. The cool thing is that I have an exact system to play with, and so can track all the heat flows and so forth at will, using my black-hole-powered exactly formula. In fact, one paper (by Campisi and Fazio) had a system of N quantum Otto cycles working together, and argued that by tuning to criticality they can get the power to scale the right way with N as they approach Carnot. In a way, my q here is a bit like their N. It is a system size parameter too, although the details are (it seems) quite different (I designed things so that my work has no q dependence, which makes things simpler). In the gravitational language, the neighbourhood of the critical point singles out a particular family of black holes whose volume scales as q cubed while the temperature goes inversely with q. (They’re sort of trying to be like flat space black holes.) Is there a non-gravitational version of this story, perhaps dual to this? This all feels like a strongly coupled system of an interesting sort to me. Definitely worth better understanding.
There is actually (at least) one more result in the paper that is very pleasing to me, but I think I’ll wait to tell you about it some other time. Or you can just read the paper. It’s very short, so it won’t take long.