What motivated Gibbs's definition of Gibbs entropy? I have read and I think that I agree with the idea that if we have to choose probability distribution for an unknown system then it is a good idea to choose a distribution that has the least bias.

I understand why Boltzmann chose to assume equal probability distribution because for me, intuitively, it means the least bias possible for the guess of the distribution.

But apparently, when we do statistical mechanics we usually maximize Gibbs entropy subject to some constraints on average values of macroscopic parameters. I am interested in the intuition behind how one would develop this approach and maybe how one would go from Boltzmann's hypothesis to Gibbs hypothesis.

When I read similar questions online I get explanations using information theory and Shannon's entropy which are extremely interesting and give a lot of insight. But I am also wondering about how did people came up with this definition and how did they justify it before Shannon's papers.


closed as off-topic by stafusa, Aaron Stevens, ZeroTheHero, Andrew Steane, Gert Aug 18 at 23:39

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    $\begingroup$ If you question is about the actual history, History of Science and Mathematics would be a better place to ask $\endgroup$ – ACuriousMind Jul 20 at 20:09
  • $\begingroup$ @ACuriousMind Thank you, I will consider it. Maybe I should be more clear that I am also interested in "physical intuition" behind finding that concept. I am not really interested who exactly and when found it. $\endgroup$ – HydrodynamicsPlease Jul 20 at 20:10
  • $\begingroup$ I think Tolman and Kittel's books have excellent descriptions you might benefit from. The essential content of Boltzmann's hypothesis that all microstates are equally probable is that there's no a priori reason to expect anything else. Gibbs' hypothesis that the equilibrium state maximizes the entropy follows from the idea of ergoticity. If the system traverses all possible microstates through the course of its evolution, the equilibrium state will be the microstate which occurs most often. This means it's the microstate which maximizes the entropy -- the log count of microstates. $\endgroup$ – kevinkayaks Jul 20 at 20:48
  • $\begingroup$ @kevinkayaks thank you for your reply! can you please be more specific about which books you are talking about? thanks $\endgroup$ – HydrodynamicsPlease Jul 20 at 20:50
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    $\begingroup$ I'm voting to close this question as off-topic because it belongs in History of Science and Mathematics (as the question about the physical intuition has been answered in the comments). $\endgroup$ – stafusa Aug 1 at 16:54