= Scope of Boltzmann's entropy formula
The strings in <Boltzmann entropy of a two-box occupancy> are complete <microstates> only within the binary toy model. They retain a <coarse-grained variable> for each particle and omit its exact <position> within a box and its <momentum>. An actual classical <gas> instead occupies continuous <phase space>; obtaining a dimensionless count from a <phase space> volume requires a specified cell convention and the appropriate treatment of particle identity. A <quantum state> count must respect the allowed <quantum states> and <quantum statistics>. Thus $2^N$ is not a general count of <microstates> for a <gas> obeying <quantum ideal-gas statistics>. https://ocw.mit.edu/courses/8-333-statistical-mechanics-i-statistical-mechanics-of-particles-fall-2013/0cdb670a825c6aaea7e229ac40776cbf_MIT8_333F13_Lec13.pdf[Kardar, MIT 8.333 (2013), lecture 13, §§IV.D–E: gas phase space and particle counting]{ref}
For a discrete <probability distribution> with unequal weights, use the existing <Gibbs entropy> article; its equal-weight case gives <Boltzmann's entropy formula>. <Shannon entropy> uses the corresponding information measure, with the logarithm base setting its units. A count alone cannot replace the specified <probability distribution>, as the <Boltzmann distribution> examples demonstrate. https://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node56.html[MIT, §7.3: statistical entropy for general microstate probabilities]{ref}
Overwhelming multiplicity explains why <thermodynamic equilibrium> is overwhelmingly probable in macroscopic systems under the appropriate statistical assumptions. It does not make low-<entropy> <thermal fluctuations> logically impossible. Nor does counting alone derive irreversible macroscopic evolution: one also needs dynamics, an initial preparation, and a reason the system samples the allowed <microstates> on the timescale of interest. https://ocw.mit.edu/courses/8-333-statistical-mechanics-i-statistical-mechanics-of-particles-fall-2013/0cb99318c2ed889316cdcff839759e8d_MIT8_333F13_Lec12.pdf[Kardar, lecture 12, §§IV.A–B: statistical equilibrium and the limits of counting arguments]{ref}
Historically, Boltzmann's 1877 paper counted microscopic arrangements, called complexions, and related the <natural logarithm> of their number to <entropy>. The Sharp and Matschinsky translation appeared in \i[Entropy] 17 (2015), pages 1971–2009, DOI 10.3390/e17041971. Its notation differs from ours: Boltzmann's permutability measure $\Omega$ is already logarithmic, whereas here $\Omega$ is the count. https://gilles.montambaux.com/files/histoire-physique/Boltzmann-1877-entropy.pdf[Boltzmann (1877), translated by Kim Sharp and Franz Matschinsky (2015), §III after equation 34 and §V]{ref} <Josiah Willard Gibbs>'s \i[Elementary Principles in Statistical Mechanics] (1902) developed the <statistical ensemble> framework; Chapter XIV compares several statistical definitions of <entropy> with <thermodynamics>. https://en.wikisource.org/wiki/Elementary_Principles_in_Statistical_Mechanics[Gibbs (1902), title page and preface]{ref} https://en.wikisource.org/wiki/Elementary_Principles_in_Statistical_Mechanics/Chapter_XIV[Gibbs (1902), Chapter XIV: discussion of thermodynamic analogies]{ref}
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