Generalized golden mean and the efficiency of thermal machines

TL;DR

This paper introduces a generalized golden mean to describe the efficiency of heat engines and refrigerators operating between two reservoirs, linking efficiency to irreversibility and extending the concept to three-reservoir systems.

Contribution

It proposes a novel efficiency formula based on the generalized golden mean, connecting thermodynamic irreversibility with mathematical constants, and extends analysis to multi-reservoir systems.

Findings

Efficiency equals the inverse of the generalized golden mean $\,\phi_p$.

Reversible systems recover the standard golden mean.

Extension to three-reservoir setups demonstrates broader applicability.

Abstract

We investigate generic heat engines and refrigerators operating between two heat reservoirs, for the condition when their efficiencies are equal to each other. It is shown that the corresponding value of efficiency is given as the inverse of the generalized golden mean, $\phi_p$, where the parameter $p$ depends on the degree of irreversibility of both engine and refrigerator. The reversible case ($p=1$) yields the efficiency in terms of the standard golden mean. We also extend the analysis to a three-heat-resrervoir setup.