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

TL;DR

This study examines the energetic limits of the B{"u}ttiker-Landauer motor and refrigerator, revealing that kinetic energy effects prevent the motor from reaching Carnot efficiency in the overdamped limit, contrary to previous predictions.

Contribution

It demonstrates through simulations and analysis that kinetic energy contributions cause divergence in heat transfer, challenging the overdamped approximation's validity for efficiency predictions.

Findings

Kinetic energy heat transfer diverges as M^{-1/2} in the overdamped limit.

The motor cannot attain Carnot efficiency due to kinetic energy effects.

The Onsager symmetry remains valid despite kinetic energy divergence.

Abstract

We investigate the energetics of a Brownian motor driven by position dependent 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 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 B{\"u}ttiker-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.