Microscopic theory of the Curzon-Ahlborn heat engine based on a Brownian particle

The Curzon-Ahlborn (CA) efficiency, as the efficiency at the maximum power (EMP) of the endoreversible Carnot engine, has significant impact on finite-time thermodynamics. However, the CA engine 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 a highly underdamped Brownian particle in a class of nonharmonic potentials. This theory gives microscopic interpretation of all assumptions made by Curzon and Ahlborn. In other words, we find a microscopic counterpart of the CA engine in stochastic thermodynamics. Also, based on this theory, we derive the explicit expression of the protocol associated with the maximum power for any given efficiency, and we obtain analytical results 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.

Figure 1

CA cycle based on a Brownian particle. (a) The cycle diagram in the space of the work parameter $\lambda$ and the effective temperature $\theta$. In the isothermal expansion (compression) process, the working substance remains at a constant effective temperature ${\theta }_H\left({\theta }_C\right)$, which is different from the temperature $T_H\left(T_C\right)$ of the hot (cold) heat bath. In the two adiabatic processes, the effective temperature $\theta$ of the working substance is proportional to the work parameter $\lambda$. (b) The control scheme of the work parameter $\lambda \left(t\right)$ and the evolution of the effective temperature $\theta \left(t\right)$ in a finite-time cycle. The work parameter $\lambda$ is varied exponentially with time in an isothermal process, and is quenched abruptly in an adiabatic process.

Figure 2

Distribution of the power (a) and the fluctuating efficiency deviation (b) of a Brownian CA engine for three different periods ${\tau }_{\mathrm{t}}=5,20,50$. (a) The vertical dotted line indicates the average power. (b) The vertical solid and dashed lines correspond to efficiency $\eta =0$ and $\eta ={\eta }_C$ (Carnot efficiency) respectively. The parameters are chosen as $T_C=300$, $T_H=600$, ${\mathrm{\Gamma }}_C=1$, ${\mathrm{\Gamma }}_H=1.2$.