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Wikipedia (https://en.wikipedia.org/wiki/Maximum_entropy_probability_distribution#Distributions_with_measured_constants) describes in detail a theorem which gives the maximum entropy probability distributions subject to various constraints. They claim this theorem is originally due to Ludwig Boltzmann, but don't give a reference (instead they prove the theorem directly).

Does anyone know a reference to the original article/book/etc by Boltzmann where this theorem is first stated and proven?

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Have a look at the paper: Über die Beziehung zwischen dem zweiten Hauptsatze des mechanischen Wärmetheorie und der Wahrscheinlichkeitsrechnung. Chapter 5 seems to deal with the theorem you are refering to, but I am not sure.

The title of the paper "Über die Beziehung zwischen eines allgemeinen mechanischen Satzes zum zweiten Hauptsatze der Wärmetheorie" sounds also promising, but I could not find it online.

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