The FRW Universe as a van der Waals-like Thermodynamic Heat Engine

TL;DR

The paper addresses how the Friedmann-Robertson-Walker (FRW) universe, endowed with thermodynamics on its apparent horizon, can be treated as a van der Waals-like heat engine. It derives a thermodynamic equation of state $P = T/(2 R_A) + 1/(8 \pi R_A^2)$ using the first law with Misner–Sharp energy and expresses horizon quantities in terms of the horizon radius $R_A$ and volume $V$. The authors analyze two cycles, Carnot and rectangular, to compute heats, works, and efficiencies; they show the Carnot efficiency $\eta_C = 1 - T_C/T_H$ and a rectangular-cycle efficiency $\eta = (1 - P_4/P_1) \big[1 - (R_{A2}-R_{A1})/(2 P_1 (V_2 - V_1))\big]^{-1}$, finding $0 < \eta < \eta_C < 1$ with $\eta/\eta_C \to 1$ as the pressure grows large. This work demonstrates that the FRW universe can function as a thermodynamic heat engine within standard second-law bounds, offering a bridge between cosmic expansion and finite-volume energy conversion with potential implications for cosmological thermodynamics and evolution.

Abstract

It is well known that the Friedmann-Robertson-Walker (FRW) universe is a dynamical spacetime, and it has thermodynamics embodied on the apparent horizon. Notably, it also possesses a van der Waals-like equation of state, enabling us to consider thermodynamic cycles and explore the FRW universe's potential as a heat engine. In this paper, we investigate two types of cycles--the Carnot cycle and the rectangular cycle--based on the phase diagram derived from the equation of state, to study the heat engine characteristics of the FRW universe. Furthermore, we calculate the work done and assess the corresponding efficiencies, illustrating the efficiency diagram for the FRW universe's heat engine. We observe that the efficiency of the rectangular cycle consistently remains below unity and never exceeds the Carnot efficiency--the thermodynamic upper limit. This finding aligns with traditional thermodynamic principles governing heat engines.

Figures (3)

• Figure 1: $P$-$V$ diagram of the thermodynamic cycle for a Carnot heat engine in the context of the FRW universe: Paths $1\rightarrow2$ and $3\rightarrow4$ are isothermal. Paths $2\rightarrow3$ and $4\rightarrow1$ are isochoric or adiabatic.
• Figure 2: $P$-$V$ diagram of the thermodynamic cycle for a rectangle heat engine in the context of the FRW universe: Paths $1\rightarrow2$ and $3\rightarrow4$ are isobaric. Paths $2\rightarrow3$ and $4\rightarrow1$ are isochoric or adiabatic.
• Figure 3: Specific example with the parameters chosen as $P_4=1$, $R_{A1}=1$, $R_{A2}=4$.