Constraints between entropy production and its fluctuations in nonthermal engines

We analyze a mesoscopic conductor autonomously performing a thermodynamically useful task, such as cooling or producing electrical power, in a part of the system - the working substance - by exploiting another terminal or set of terminals - the resource - that contains a stationary nonthermal (nonequilibrium) distribution. Thanks to the nonthermal properties of the resource, on average no exchange of particles or energy with the working substance is required to fulfill the task. This resembles the action of a demon, as long as only average quantities are considered. Here, we go beyond a description based on average currents and investigate the role of fluctuations in such a system. We show that a minimum level of entropy fluctuations in the system is necessary, whenever one is exploiting a certain entropy production in the resource terminal to perform a useful task in the working substance. For concrete implementations of the demonic nonthermal engine in three- and four-terminal electronic conductors in the quantum Hall regime, we compare the resource fluctuations to the entropy production in the resource and to the useful engine output (produced power or cooling power).