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We investigate the energetics of a Brownian motor driven by positiondependent temperature, commonly known as the B{\"u}ttiker-Landauer motor.Overdamped models (M=0) predict that the motor can attain Carnot efficiency.However, the overdamped limit ($M\to 0$), contradicts the previous predictiondue to the kinetic energy contribution to the heat transfer. Using moleculardynamics simulation and numerical solution of the inertial Langevin equation,we confirm that the motor can never achieve Carnot efficiency and verify thatthe heat flow via kinetic energy diverges as $M^{-1/2}$ in the overdampedlimit. The reciprocal process of the motor, namely the B{\"u}ttiker-Landauerrefrigerator is also examined. In this case, the overdamped approach succeedsin predicting the heat transfer only when there is no temperature gradient. Itsfound that the Onsager symmetry between the motor and refrigerator does notsuffer from the singular behavior of the kinetic energy contribution.

Inertial effects in B{\"u}ttiker-Landauer Motor and Refrigerator at the Overdamped Limit