Ok, I promised to explain the staircase I put up on Monday. I noticed something rather nice recently, and reported it (actually, two things) in a recent paper, here. It concerns those things I called “Holographic Heat Engines” which I introduced in a paper two years ago, and which I described in some detail in a previous post. You can go to that post in order to learn the details – there’s no point repeating it all again – but in short the context is an extension of gravitational thermodynamics where the cosmological constant is dynamical, therefore supplying a meaning to the pressure and the volume variables (p,V) that are normally missing in black hole thermodynamics… Once you have those, it seems obvious that you can start considering processes that do mechanical work (from the pdV term in the first law) and within a short while the idea of heat engines in which the black hole is the working substance comes along. Positive pressure corresponds to negative cosmological constant and so the term “holographic heat engines” is explained. (At least to those who know about holographic dualities.)
So you have a (p,V) plane, some heat flows, and an equation of state determined by the species of (asymptotically AdS) black hole you are working with. It’s like discovering a whole new family of fluids for which I know the equation of state (often exactly) and now I get to work out the properties of the heat engines I can define with them. That’s what this is.
Now, I suspect that this whole business is an answer waiting for a question. I can’t tell you what the question is. One place to look might be in the space of field theories that have such black holes as their holographic dual, but I’m the first to admit that I don’t yet know if the corresponding cycle on the space of those field theories is interesting for anything useful. But that’s ok. Until the killer app comes along for this toolbox, I’m happy to study it and find interesting things about it. Since I defined them back in 2014, there’s been some work (by me and others) on other examples and so forth. I refined, for example the process by which you can write down nice answers for the key object of study in this context, the efficiency, specifically for a special rectangular cycle (see figure) when the black holes are static. This works best in a high temperature limit where one can expand in 1/T and get some nice results. Have a look e.g. here.
The first cool thing I noticed recently is that there’s a really simple formula for the efficiency, which means one can know the answer exactly, and not just in a large T limit! It is this:
where [tex]M_i[/tex] is just the black hole mass evaluated at the corner labelled i in the figure above! That’s it. It’s really nice (and follows, in fact from the nice fact (Kastor et. al.) that in these extended thermodynamics systems, the enthalpy of the system is the black hole mass). You just take the difference along the top and divide by the difference along the bottom, and subtract it from unity.
So that would be nice enough (it gives exact results for systems studied in earlier papers if you care to revisit them), but then I noticed something else. I can define an algorithm for computing the efficiency of any cycle! Using the basic rectangle (and the formula) as a prototype, I can tile a cycle of any shape with a grid of them, getting as accurate I like by shrinking the cell sizes as much as I like. Since overlapping cycles add (see figure at the very top), the perimeter of my tiling is a good approximation to the cycle, and I just implement my formula, adding up all the mass differences along any exposed top parts of cells (giving total heat in), dividing the result by the sum of all of all mass differences along any exposed bottom parts of cells (total heat exhaust). Subtract that from unity and you’re done! The staircases below are an example of two discretizations, one more accurate than the other, of the triangular cycle in blue. The segments marked by red dots are the hot cells, and the green (along the bottom) captures the cold cells. (Click for larger view.)
Actually, this generalizes even more, giving you a nice way to do this for any heat engine (whether using black holes or not) if you know the adiabatic curves and a nice expression for the enthalpy. So this is a boon for the 19th century physicists who rally cared about these issues a lot – I’m a tad late, I guess. The key point is that with computers lying around everywhere, coding up this algorithm and computing the answer is easy to do, and gives a nicer (and quicker in some cases) way of computing the efficiency for a given complicated shape (where you’d instead do numerical integration of TdS along the curve). And the other contemporary advantage is that for black holes is that the enthalpy is the mass, and the mass is a nice thing to be able to compute (exact expressions are often known). Further, for static black holes, the adiabatic curves are just isochors. Hence the rectangle above.
Anyway, a much more careful discussion is in the paper, so have a look. Enjoy!