Inertial effects in Büttiker-Landauer motor and refrigerator at the overdamped limit

We investigate the energetics of a Brownian motor driven by position-dependent temperature, commonly known as the Büttiker-Landauer motor. Overdamped models (M=0) predict that the motor can attain Carnot efficiency. However, the overdamped limit (M-->0) contradicts the previous prediction due to the kinetic energy contribution to the heat transfer. Using molecular dynamics simulation and numerical solution of the inertial Langevin equation, we confirm that the motor can never achieve Carnot efficiency and verify that the heat flow via kinetic energy diverges as M{-1/2} in the overdamped limit. The reciprocal process of the motor, namely, the Büttiker-Landauer refrigerator, is also examined. In this case, the overdamped approach succeeds in predicting the heat transfer only when there is no temperature gradient. Its found that the Onsager symmetry between the motor and refrigerator does not suffer from the singular behavior of the kinetic energy contribution.