Stochastic heat engine with the consideration of inertial effects and shortcuts to adiabaticity

When a Brownian particle in contact with a heat bath at a constant temperature is controlled by a time-dependent harmonic potential, its distribution function can be rigorously derived from the Kramers equation with the consideration of the inertial effect of the Brownian particle. Based on this rigorous solution and the concept of shortcuts to adiabaticity, we construct a stochastic heat engine by employing the time-dependent harmonic potential to manipulate the Brownian particle to complete a thermodynamic cycle. We find that the efficiency at maximum power of this stochastic heat engine is equal to $1-\sqrt{T_c/T_h}$ where $T_c$ and $T_h$ are the temperatures of the cold bath and the hot one in the thermodynamic cycle, respectively.