Heat engines for scale invariant systems dual to black holes

Abstract

According to holography, a black hole is dual to a thermal state in a strongly coupled quantum system. One of the best-known examples of holography is the Anti-de Sitter/Conformal Field Theory (AdS/CFT) correspondence. Despite extensive work on holographic thermodynamics, heat engines for CFT thermal states have not been explored. We construct reversible heat engines where the working substance consists of a static thermal equilibrium state of a CFT. For thermal states dual to an asymptotically AdS black hole, this yields a realization of Johnson's holographic heat engines. We compute the efficiency for a number of idealized heat engines, such as the Carnot, Brayton, Otto, Diesel, and Stirling cycles. The efficiency of most heat engines can be derived from the CFT equation of state, which follows from scale invariance, and we compare them to the efficiencies for an ideal gas. However, the Stirling efficiency for a generic CFT is uniquely determined in terms of its characteristic temperature and volume only in the high-temperature or large-volume regime. We derive an exact expression for the Stirling efficiency for CFT states dual to AdS-Schwarzschild black holes and compare the subleading corrections in the high-temperature regime with those in a generic CFT.

Figures (6)

• Figure 1: Efficiency vs. compression ratio. This plot shows the efficiency $\eta$ as a function of compression ratio $v$ for Otto (blue), Diesel (green), and Stirling (red) engines in $D=4$. The solid lines correspond to a monatomic ideal gas ($f=3$) and dashed lines to general CFTs (for Stirling: CFT on a plane). The fixed temperature ratio for Stirling is $t\equiv T_{\text{h}}/T_{\text{c}} = 2$ and the fixed cutoff ratio for Diesel is $V_3/V_2 = 1.5$.
• Figure 2: Efficiency vs. temperature ratio. This plots shows the efficiency $\eta$ as a function of $t\equiv T_{\text{h}}/T_{\text{c}}$ for Stirling (red) and Carnot (blue) engines in $D=4$. The solid lines correspond to a monatomic ideal gas ($f=3$) and the dashed line to a CFT on a plane. The fixed compression ratio for Stirling is $v\equiv V_2/V_1= 2$.
• Figure 3: Pressure-volume diagrams for holographic CFT heat cycles. These plots are heat cycles for thermal CFT working substances dual to AdS-Schwarzschild black holes. The number of CFT spacetime dimensions is $D=4$. In all figures dotted lines correspond to isotherms, short dashed lines to adiabats, long dashed lines to isobars, and dotdashed lines to isochores. The red curves indicate the cycle, the numbers at the vertices denote the ordering of the cycle, and the arrows the direction of the cycle. $T_{\text{h}}$ and $T_{\text{c}}$ are the temperatures of the hot and cold reservoirs, respectively. The panels represent (a) the Carnot cycle (isotherm-adiabat-isotherm-adiabat), (b) Brayton cycle (adiabat-isobar-adiabat-isobar), (c) Otto cycle (adiabat-isochore-adiabat-isochore), (d) Diesel cycle (adiabat-isobar-adiabat-isochore), (e) rectangle cycle (isochore-isobar-isochore-isobar), and (f) Stirling cycle (isotherm-isochore-isotherm-isochore).
• Figure 4: Pressure-volume diagrams for ideal gas heat cycles. These plots are heat cycles for (a) Carnot (isotherm-adiabat-isotherm-adiabat) and (b) Stirling (isotherm-isochore-isotherm-isochore) engines with a working substance consisting of a monatomic ($\gamma=5/3$) ideal gas in $D=4$ spacetime dimensions. The dotted lines correspond to isotherms, short dashed lines to adiabats, and dotdashed lines to isochores. The red curves indicate the cycle, the numbers at the vertices denote the ordering of the cycle, and the arrows the direction of the cycle. $T_{\text{h}}$ and $T_{\text{c}}$ are the temperatures of the hot and cold reservoirs, respectively.
• Figure 5: Temperature-entropy diagrams for holographic CFT heat cycles. These plots are heat cycles for thermal CFT working substances dual to AdS-Schwarzschild black holes. The number of CFT spacetime dimensions is $D=4$. In all figures dotted lines correspond to isotherms, short dashed lines to adiabats, long dashed lines to isobars, and dotdashed lines to isochores. The red curves indicate the cycle, the numbers at the vertices denote the ordering of the cycle, and the arrows the direction of the cycle. $T_{\text{h}}$ and $T_{\text{c}}$ are the temperatures of the hot and cold reservoirs, respectively. The panels represent (a) the Carnot cycle (isotherm-adiabat-isotherm-adiabat), (b) Brayton cycle (adiabat-isobar-adiabat-isobar), (c) Otto cycle (adiabat-isochore-adiabat-isochore), (d) Diesel cycle (adiabat-isobar-adiabat-isochore), (e) rectangle cycle (isochore-isobar-isochore-isobar), and (f) Stirling cycle (isotherm-isochore-isotherm-isochore).