Energetics of a simple microscopic heat engine
  We model a microscopic heat engine as a particle hopping on a one-dimensional lattice in a periodic sawtooth potential, with or without load, assisted by the thermal kicks it gets from alternately placed hot and cold thermal baths. We find analytic expressions for current and rate of heat flow when the engine operates at steady state. Three regions are identified where the model acts either as a heat engine or as a refrigerator or as neither of the two. At quasistatic limit both efficiency of the engine and coefficient of performance of the refrigerator go to that for Carnot engine and Carnot refrigerator, respectively. We investigate efficiency of the engine at two operating conditions (at maximum power and at optimum value with respect to energy and time) and compare them with those of the endoreversible and Carnot engines. 
Mesfin Asfaw and Mulugeta Bekele