On A Local Carnot Engine

Starting from a master equation in a quantum Hamilton form we study analytically a nonequilibrium system which is coupled locally to two heat bathes at different temperatures. Based on a lattice gas description an evolution equation for the averaged density in the presence of a temperature gradient is derived. Firstly, the case is analysed where a particle is removed from a heat bath at a fixed temperature and is traced back to the bath at another temperature. The stationary solution and the relaxation time is discussed. Secondly, a collective hopping process between different heat bathes is studied leading to an evolution equation which offers a bilinear coupling between density and temperature gradient contrary to the conventional approach. Whereas in case of a linear decreasing static temperature field the relaxtion time offers a continuous spectrum it results a discrete spectrum for a quadratically decreasing temperature profile.