I always thought of −∑x∈microstatesP(x)logP(x) as the exact / “real” definition of entropy, and log(number of microstates) as the specialization of that “exact” formula to the case where each microstate is equally probable (a case which is rarely exactly true but often a good approximation). So I found it a bit funny that you only mention the second formula, not the first. I guess you were keeping it simple? Or do you not share that perspective?
This post was very specifically about a Boltzmann-style approach. I’d also generally consider the Gibbs/Shannon formula to be the “real” definition of entropy, and usually think of Boltzmann as the special case where the microstate distribution is constrained uniform. But a big point of this post was to be like “look, we can get surprisingly a lot (though not all) of thermo/stat mech even without actually bringing in any actual statistics, just restricting ourselves to the Boltzmann notion of entropy”.
Of course if one insists on some of the assumptions you did not need, namely doing the standard microcanonical ensemble approach, it trivializes everything and no second law comes out.
In microcanonical ensemble the system is isolated, meaning its energy is fixed. Microstates are partitioned into a macrostates by their energy (stronger version of your assumption of macro being a function of micro), so they don’t switch into a different macrostate. If you take them to be energy eigenstates, the microstates don’t evolve either.
I don’t endorse the idea of a macrostate secretly being in a certain microstate. They are different things, preparing a microstate takes a lot more effort.
I always thought of −∑x∈microstatesP(x)logP(x) as the exact / “real” definition of entropy, and log(number of microstates) as the specialization of that “exact” formula to the case where each microstate is equally probable (a case which is rarely exactly true but often a good approximation). So I found it a bit funny that you only mention the second formula, not the first. I guess you were keeping it simple? Or do you not share that perspective?
This post was very specifically about a Boltzmann-style approach. I’d also generally consider the Gibbs/Shannon formula to be the “real” definition of entropy, and usually think of Boltzmann as the special case where the microstate distribution is constrained uniform. But a big point of this post was to be like “look, we can get surprisingly a lot (though not all) of thermo/stat mech even without actually bringing in any actual statistics, just restricting ourselves to the Boltzmann notion of entropy”.
Of course if one insists on some of the assumptions you did not need, namely doing the standard microcanonical ensemble approach, it trivializes everything and no second law comes out.
In microcanonical ensemble the system is isolated, meaning its energy is fixed. Microstates are partitioned into a macrostates by their energy (stronger version of your assumption of macro being a function of micro), so they don’t switch into a different macrostate. If you take them to be energy eigenstates, the microstates don’t evolve either.
I don’t endorse the idea of a macrostate secretly being in a certain microstate. They are different things, preparing a microstate takes a lot more effort.