Footnote 17 sounds confusing and probably wrong to me, but I haven’t thought it through. Macrostates should have some constraint that makes their total probability 1; you can’t have a macrostate containing a single very unlikely microstate.
(Edit: “wrong” seems a bit harsh on reflection but I dislike the vagueness about “cheating” a lot. The single-improbable-thing macrostate should just not typecheck instead of somehow being against the spirit of things.)
I would maybe say that your “average entropy” (what I’d call entropy) is always the average over every state, every single time, and (uniform) macrostates are just a handy conceptual shorthand for saying “I want all of these states to have equal p (equal -log p) and all of these to have zero p (infinite -log p)” without getting bogged down in why 0 log 0 is 0. A state is “in” a macrostate if it’s one of the states with nonzero p for that macrostate, but the sum is always over everything.
Footnote 17 sounds confusing and probably wrong to me, but I haven’t thought it through. Macrostates should have some constraint that makes their total probability 1; you can’t have a macrostate containing a single very unlikely microstate.
(Edit: “wrong” seems a bit harsh on reflection but I dislike the vagueness about “cheating” a lot. The single-improbable-thing macrostate should just not typecheck instead of somehow being against the spirit of things.)
I would maybe say that your “average entropy” (what I’d call entropy) is always the average over every state, every single time, and (uniform) macrostates are just a handy conceptual shorthand for saying “I want all of these states to have equal p (equal -log p) and all of these to have zero p (infinite -log p)” without getting bogged down in why 0 log 0 is 0. A state is “in” a macrostate if it’s one of the states with nonzero p for that macrostate, but the sum is always over everything.