Zipf law (which is a power law) is the maximum entropy distribution of a system of P particles in N boxes where P>>N. Its derivation is based on microcanonical ensemble in which the entropy is calculated for an isolated system. In the canonical ensemble the system is with contact with an external bath having a fix temperature T. The macroscopic quantities of the canonical ensemble are calculated from its partition function in which the probabilities decay exponentially with energy.
The question is: how is Zipf law a power law which can be obtained from exponential partition function?
http://www.entropy-book.com
Entropy: A concept that is not a physical quantity
https://www.researchgate.net/publication/230554936_Entropy_A_concept_that_is_not_a_physical_quantity
First, everything in the world is physical. Entropy is a quantity that represents the change in the world's uncertainty when an amount of energy is transferred between two bodies. The amount of the transferred energy is the heat Q.
The temperature T of the emitted body defines the "grade" of the heat. The higher the grade the higher the amount of work that can be extracted when the heat is absorbed in a given lower temperature body.
Work W may be viewed as heat emerged from infinitely high-temperature source (i.e. laser light). You can liken heat engine to waterfall were heights gap of the waterfall is analogue to the temperature gap of a heat engine. Therefore, when you replace Q by W in your Stirling engine you use two infinite temperatures. As you probably know, both zero and infinity are not true numbers and cannot be used in an arithmetical calculation. That is probably the reason for your erroneous conclusion.
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