A simple quantum heat engine

A simple urn model is presented that may describe heat engines employing as working agents spin-1/2 particles (e.g., electrons) or two-level atoms. In this model the cold reservoir is an urn located at altitude $\epsilon_{c}$ that contains $N$ balls of total weight $n_{c}$. Likewise, the hot reservoir at altitude $\epsilon_{h}$ contains $N$ balls of total weight $n_{h}$. A cycle consists of exchanging two randomly selected balls between the reservoirs. Elementary considerations show that the heat-engine efficiency (ratio of the average work performed divided by the average hot-reservoir energy consumption) is $\eta=1-\epsilon_{c}/\epsilon_{h}$. The case where the two reservoirs have negative temperatures is symmetrical to the case where the two reservoirs have positive temperatures in the sense that heat engines become heat pumps and conversely. When the cold reservoir has positive temperature while the hot reservoir has negative temperature the maximum efficiency is unity. When many cold and hot sub-reservoirs (of the same temperature signs) are employed the Carnot efficiency may be reached. For some parameter values the work performed does not fluctuate from cycle to cycle