A simple model for Carnot heat engines

We discuss a model consisting of two reservoirs, each with $N$ possible ball locations, at heights $Eh$ and $El<Eh$ in a gravitational field. The two reservoirs contain $nh$ and $nl$ weight 1 balls. Empty locations are treated as weight 0 balls. The reservoirs are shaken so that all possible ball configurations are equally likely to occur. A cycle consists of exchanging a ball randomly chosen from the higher reservoir and a ball randomly chosen from the lower reservoir. We relate this system to a heat engine and show that the efficiency, which is defined as the ratio of the average work produced to the average energy lost by the higher reservoir, is $1−El/Eh$⁠. When $nl$ is comparable to $nh$⁠, the efficiency is found to coincide with the maximum efficiency $1−Tl/Th$⁠, where the temperatures $Tl$ and $Th$ are defined from a simple expression for the entropy. We also discuss the evaluation of fluctuations and the history of the Carnot discovery.