Heat engine in the three-dimensional spacetime

TL;DR

This paper constructs and analyzes a heat engine model based on three-dimensional charged BTZ black holes, addressing unique thermodynamic challenges and deriving an analytical efficiency expression that supports a specific thermodynamic approach.

Contribution

It introduces a novel thermodynamic parameter to resolve subtleties in the heat engine analysis of 3D BTZ black holes and provides the first analytical efficiency formula for this system.

Findings

Derived an analytical efficiency expression for the 3D charged BTZ black hole heat engine.

Confirmed the correctness of the efficiency formula through exact calculations.

Compared the heat engine efficiency with Carnot cycle results in three-dimensional spacetime.

Abstract

We define a kind of heat engine via three-dimensional charged BTZ black holes. This case is quite subtle and needs to be more careful. The heat flow along the isochores does not equal to zero since the specific heat $C_V\neq0$ and this point completely differs from the cases discussed before whose isochores and adiabats are identical. So one cannot simply apply the paradigm in the former literatures. However, if one introduces a new thermodynamic parameter associated with the renormalization length scale, the above problem can be solved. We obtain the analytical efficiency expression of the three-dimensional charged BTZ black hole heat engine for two different schemes. Moreover, we double check with the exact formula. Our result presents the first specific example for the sound correctness of the exact efficiency formula. We argue that the three-dimensional charged BTZ black hole can be viewed as a toy model for further investigation of holographic heat engine. Furthermore, we compare our result with that of the Carnot cycle and extend the former result to three-dimensional spacetime. In this sense, the result in this paper would be complementary to those obtained in four-dimensional spacetime or ever higher. Last but not the least, the heat engine efficiency discussed in this paper may serve as a criterion to discriminate the two thermodynamic approaches introduced in Ref.[29] and our result seems to support the approach which introduces a new thermodynamic parameter $R=r_0$.