Efficiency at maximum power and efficiency fluctuations in a linear Brownian heat engine model

We investigate stochastic thermodynamics of a two-particles Langevin system. Each particle is in contact with a heat bath at different temperatures $T_1$ and $T_2~(<T_1)$, respectively. Particles are trapped by a harmonic potential and driven by a linear external force. The system can act as an autonomous heat engine performing work against the external driving force. Linearity of the system enables us to examine thermodynamic properties of the engine analytically. We find that the efficiency of the engine at maximum power $\eta_{MP}$ is given by $\eta_{MP} = 1-\sqrt{T_2/T_1}$. This universal form has been known as a characteristic of endoreversible heat engines. Our result extends the universal behavior of $\eta_{MP}$ to non-endoreversible engines. We also obtain the large deviation function of the probability distribution for the stochastic efficiency in the overdamped limit. The large deviation function takes the minimum value at mean efficiency $\eta = \bar{\eta}$ and increases monotonically until it reaches plateaus when $\eta \leq \eta_L$ and $\eta \geq \eta_R$ with model dependent parameters $\eta_{R,L}$. It has been known for heat engines with a finite number of microscopic configurations with time-symmetric protocol that the probability of achieving the Carnot efficiency is minimum. Our result reveals that the least likeliness of the Carnot efficiency is not the generic property of heat engines.