Yeah the author is conflating low entropy with a low number of microstates, which is consistent with the thermodynamic assumption that maximal entropy means a uniform distribution of microstates, but is confusing.
The purest mathematical justification for why low entropy means a low number of microstates probably comes from the fact that (classical) physical systems are a dynamical systems that preserve the measure induced by the standard metric of the phase space. The measure theoretic definition of entropy then implies the entropy of a partition of the phase space (i.e. a set of macrostates) is indeed the average of the logarithm of the number of microstates.
So indeed if W is the number of microstates in the current macrostate then entropy = log(W) (on average).
And using the typical set you can show that the probability that the average of log(W) over n samples is within 'epsilon' of the 'exact' entropy goes to 1 as n goes to infinity. This is the mathematical justification for the second law of thermodynamics.
The trick is that all of this is true no matter how you partition phase-space. Though that does mean that what is and isn't a high entropy state depends on your perspective.
>The trick is that all of this is true no matter how you partition phase-space. Though that does mean that what is and isn't a high entropy state depends on your perspective.
That seems to be correct.
>Yeah the author is conflating low entropy with a low number of microstates
Since entropy is found by counting micro-states (for example your third paragraph), that should be ok. What am I missing?
The equivalence is an important theorem (important enough to be engraved on Boltzmann's gravestone), and if you're switching back and forth in an explanation of what entropy is then you're skipping over some important details that answer what it means for something to be a 'low entropy state'.
The purest mathematical justification for why low entropy means a low number of microstates probably comes from the fact that (classical) physical systems are a dynamical systems that preserve the measure induced by the standard metric of the phase space. The measure theoretic definition of entropy then implies the entropy of a partition of the phase space (i.e. a set of macrostates) is indeed the average of the logarithm of the number of microstates.
So indeed if W is the number of microstates in the current macrostate then entropy = log(W) (on average).
And using the typical set you can show that the probability that the average of log(W) over n samples is within 'epsilon' of the 'exact' entropy goes to 1 as n goes to infinity. This is the mathematical justification for the second law of thermodynamics.
The trick is that all of this is true no matter how you partition phase-space. Though that does mean that what is and isn't a high entropy state depends on your perspective.