Inattainability of Carnot efficiency in the Brownian heat engine
  We discuss the reversibility of Brownian heat engine. We perform asymptotic analysis of Kramers equation on B\"uttiker-Landauer system and show quantitatively that Carnot efficiency is inattainable even in a fully overdamping limit. The inattainability is attributed to the inevitable irreversible heat flow over the temperature boundary. 
Tsuyoshi Hondou and Ken Sekimoto