  Abstract The Curzon-Ahlborn (CA) efficiency, as the efficiency at the maximum
power (EMP) of the endoreversible Carnot engine, has a significant impact on
finite-time thermodynamics. However, the CA engine model is based on many
assumptions. In the past few decades, although a lot of efforts have been made,
a microscopic theory of the CA engine is still lacking. By adopting the method
of the stochastic differential equation of energy, we formulate a microscopic
theory of the CA engine realized with an underdamped Brownian particle in a
class of non-harmonic potentials. This theory gives microscopic interpretation
of all assumptions made by Curzon and Ahlborn, and thus puts the results about
CA engine on a solid foundation. Also, based on this theory, we obtain
analytical expressions of the power and the efficiency statistics for the
Brownian CA engine. Our research brings new perspectives to experimental
studies of finite-time microscopic heat engines featured with fluctuations.
