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.

中文翻译：

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广义黄金分割与热机效率

我们研究了在两个热库之间运行的通用热机和冰箱，因为它们的效率彼此相等。结果表明，相应的效率值作为广义黄金平均数 $\phi_p$ 的倒数给出，其中参数 $p$ 取决于发动机和冰箱的不可逆程度。可逆情况 ($p=1$) 根据标准黄金平均数产生效率。我们还将分析扩展到三热库设置。