An Exact Efficiency Formula for Holographic Heat Engines

Further consideration is given to the efficiency of black hole heat engines that perform mechanical work via the pdV terms present in the First Law of extended gravitational thermodynamics. It is noted that when the engine cycle is a rectangle with sides parallel to the (p,V) axes, the efficiency can be written simply in terms of the mass of the black hole evaluated at the corners. Since an arbitrary cycle can be approximated to any desired accuracy by a tiling of rectangles, a general geometrical algorithm for computing the efficiency follows. A simple generalization of the algorithm renders it applicable to more general classes of heat engine, beyond the black hole context.


Introduction
This paper concerns the efficiency of holographic heat engines, which were defined in ref. [1]. They are a natural concept in extended gravitational thermodynamics, which, in making dynamical the cosmological constant (Λ) in a theory of gravity, supplies 1 a pressure variable p = −Λ/8π and its conjugate volume V (see refs. [2][3][4][5][6][7][8][9][10][11][12][13]). The gravitational solution (a black hole, in the cases studied here) supplies the equation of state: The temperature T , entropy S, and other quantities can be computed [14][15][16][17], and there are relations between them. There's also a relation between the thermodynamic volume V and the horizon radius of the black hole [2]. The precise form of all these relations depends upon the type of black hole, and of course the parent theory of gravity under discussion. For example, refs. [18,19] study the efficiency in the situation when the parent gravity theory has Gauss-Bonnet and Born-Infeld sectors.  In this extended thermodynamics context, we work with a negative cosmological constant (defining a positive pressure), for which such physics has an holographic duality [20][21][22][23][24] to nongravitational field theories in one dimension fewer, at large N (where N is the rank of a field theory gauge group, or an analogue thereof). This is why our heat engines in this context are called "holographic". As pointed out in ref. [1], since changing Λ involves changing the N of the dual theory, the heat engine cycle is a kind of tour on the space of field theories rather than staying within one particular field theory 2 . It is currently unclear as to the precise application of these heat engines in this context or any other, but as defined above they are certainly rich and well-defined enough in their own right to 1 Here we are using geometrical units where G, c, , kB have been set to unity. 2 As pointed out in ref. [25] it also involves changing the size of the space the field theories live on.
warrant an exploration of their physics.
For now, focus on the cycle given in figure 2. In ref. [1], it is explained why such a choice is natural for static black holes, which we study here 3 where the subscripts refer to the quantities evaluated at the corners labeled (1,2,3,4). isochors are also adiabats for static black holes, and so the heat flows take place entirely along the top and bottom, with the upper isobar giving the net inflow of heat and the lower isobar giving the exhaust: where the specific heat at constant pressure In the previous work on various black hole examples, the resulting efficiency η = W/Q H was evaluated in a high temperature limit, since in general, the relations between S and T are such that the T -integral needed to evaluate Q H is difficult to perform exactly. Refs. [18,19] organize the computations of all needed quantities by working in terms of the horizon radius r + , treating it as an independent parameter in terms of which all quantities can be most easily written. The high temperature expansions of all quantities were then developed by working out the high temperature expansion of r + (T ). It is worth noting here that a next logical step could be to recast the Tintegrals in equations (1) as r + -integrals. Since C p (r + ) is (for some classes of black hole) a ratio of polynomials in r + , depending upon the nature of the ∂r + /∂T Jacobian this might result in simpler expressions for the efficiency 4 . Indeed, as can be seen by evaluating a few examples, this is correct.
In fact, there's a much simpler way of looking at the whole picture that results in simple expression for the efficiency that makes it easy to see exactly why this works. We will explore this next.

A Simple Efficiency Formula
The key point is that when black holes are the focus, it is the enthalpy H that takes center stage in the First Law of Thermodynamics, since it is identified with the mass M of the black hole [2].
So instead of writing the First Law in terms of the internal energy, dU = T dS − pdV , one writes: and since our heat flows are along isobars, for which dp = 0, we see immediately that our total heat flow T dS, regardless of which variable we care to write it in terms of, is simply the enthalpy change, which is just the change in the black hole mass M . Therefore, we have a remarkably simple way to write our efficiency formula entirely in terms of the black hole mass evaluated at the corners: The mass M is most naturally written as a function of r + and p. Depending upon which parameters of the cycle are prescribed (e.g. scheme 1 (T 1 , T 2 , p 1 , p 4 ) or scheme 2 (T 1 , T 4 , V 1 , V 3 ) in refs. [18,19]), the equation of state can be readily used to evaluate the r +,i and hence the mass values. In fact, if the parameters specified are all pressures and volumes (e.g. (p 1 , p 4 , V 1 , V 2 )), the equation of state is not needed at all, (since the volume V is a simple function of r + ) and the resulting exact efficiency formula is remarkably simple.

Examples
A nice class of examples is the static black hole of charge q in D dimensions that has mass formula: where for α = 0 we are in Einstein-Maxwell gravity, and for non-zero α we are in the Gauss-Bonnet generalization (see refs. [18,30] for details and background). The relationship between T, r + and p is given by: There is no need to work at high temperature, as our formula (3) is exact. In fact, it is pleasing to see how the high temperature expansion is corrected by it. In ref. [18], for a given chosen (large) temperature, the expansions begin to break down once α gets too large. Using the formula to compute the exact efficiency is straightforward, and if (for example) T 1 , T 2 , p 1 and p 4 are specified parameters, one can invert equation (5) to compute the radii r +,1 = r +,4 and r +,2 = r +,3 , and compare to the result for the large T procedures done in ref. [18]. Such an example is presented in figure 3. As α increases, the deviation from accuracy (the upper curve is the large T result) grows since the large T expansion under-estimates the heat and over-estimates the work, as can be determined by a direct comparison of those quantities with the exact result.

The Efficiency for Arbitrary Cycles
Working out the efficiency of more general shapes of cycle is in principle difficult, since the specific heat along the cycle is in general quite complicated, even for static black holes. Computing the heat flows would be quite a challenge. A remarkable and immediate consequence of our result (3) is that it can be used as the basis for an algorithm for computing the efficiency of a cycle of arbitrary shape to any desired accuracy. Here's how: Cycles are additive, in the sense indicated in figure 4, and any closed shape in the (p, V ) plane can be approximated by tiling with a regular lattice of rectangles. will give Q H and summing all the cold cell mass differences (evaluated at the bottom edges) will yield Q C , and the efficiency follows:

A Triangular Example
A triangular cycle was chosen to illustrate the algorithm in action. It has two edges parallel to the axes and one sloping edge in cycling from (p, V ) values (2,30) to (1,30) to (1,10) and returning.
One can choose to break it into a staircase-like rectangular lattice tiling that has N rectangles (not to be confused with the N of section 1) approximating the sloped edge, with better accuracy at larger N , as in figure 5. A simple procedure was coded for the computation of the required r + and p values for each cell, summing the contributions to Q H according to our formula (6). In this case Q C comes from the bottom edge as the overall mass difference along the base. Einstein-Maxwell black holes were used (i.e., α = 0), choosing the case D = 5 in equation (4) for the mass. Figure 6 shows the rapid convergence toward the result as N is increased.

Closing Remarks
A remarkably simple expression (6) for the efficiency of a rectangular cycle was presented, simplifying earlier presented expressions and allowing it to be computed exactly. The whole formula is in terms of the black hole mass evaluated at each of the four corners of the cycle. Since the mass is often readily computable in a closed form expression, this is an extremely compact result.
Section 2.1 compared the results to the high temperature expansions obtained in earlier work.
While it might seem relevant only to a special class of cycle, the expression turns out to be quite powerful, being the seed of an algorithm for computing the efficiency of cycles of arbitrary shape using a well-defined geometrical procedure. Such shapes would in general be hard to compute the efficiency for by e.g. numerical integration methods along the path. The presented example (the triangle of section 3.1) shows how rapidly the algorithm converges to a result for the efficiency.
In fact, the procedure extends to wider classes of heat engine, whether a black hole is used as a working substance or not. The key object to be able to compute with is the basic unit cell needed to tile a cycle of arbitrary shape. More generally, the unit cell is actually a Brayton/Joule cycle, composed of two isobars and two adiabats. (In this paper it is a rectangle, since for static black holes, adiabats are isochors). So all the heat flows are taken into account on the top and bottom edges. Since they are isobars, the heat flows can be written entirely in terms of the enthalpy. So for any system for which one can readily compute the enthalpy H (and black holes turn out to be such a system since H is simply the mass M ), and for which one knows the adiabatic curves, the same geometrical algorithm can be used to compute the efficiency of an arbitrary cycle in the manner described in section 3. It would be interesting to study some examples of this application.