Irreversible Brownian heat engine
  We model a Brownian heat engine as a Brownian particle that hops in a periodic ratchet potential where the ratchet potential is coupled with a linearly decreasing background temperature. It is shown that the efficiency of such Brownian heat engine is far from Carnot efficiency even at quaistatic limit. At quasistatic limit, the efficiency of the heat engine approaches the efficiency of endoreversible engine $\eta=1-\sqrt{{T_{c}/T_{h}}}$ \cite{c18}. On the other hand, the maximum power efficiency of the engine approaches $\eta^{MAX}=1-({T_{c}/T_{h}})^{1\over 4}$. Moreover, the dependence of the current as well as the efficiency on the model parameters is explored analytically by omitting the heat exchange via the kinetic energy. In this case we show that the optimized efficiency always lies between the efficiently at quaistatic limit and the efficiency at maximum power. On the other hand, the efficiency at maximum power is always less than the optimized efficiency since the fast motion of the particle comes at the expense of the energy cost. The role of time on the performance of the motor is also explored via numerical simulations. Our numerical results depict that the time $t$ as well as the external load dictate the direction of the particle velocity. Moreover the performance of the heat engine improves with time. At large $t$ (steady state),the velocity, the efficiency and the coefficient of performance of the refrigerator attain their maximum value. 
Mesfin Asfaw Taye
cond-mat.stat-mech