> Which we don’t know precisely. Entropy is about not knowing.
No, it is not about not knowing. This is an instance of the intuition from Shannon’s entropy does not translate to statistical Physics.
It is about the number of possible microstates, which is completely different. In Physics, entropy is a property of a bit of matter, it is not related to the observer or their knowledge. We can measure the enthalpy change of a material sample and work out its entropy without knowing a thing about its structure.
> Minus infinity. Entropy in classical statistical mechanics is proportional to the logarithm of the volume in phase space.
No, 0. In this case, there is a single state with p=1 and and S = - k Σ p ln(p) = 0.
This is the same if you consider the phase space because then it is reduced to a single point (you need a bit of distribution theory to prove it rigorously but it is somewhat intuitive).
The probability p of an microstate is always between 0 and 1, therefore p ln(p) is always negative and S is always positive.
You get the same using Boltzmann’s approach, in which case Ω = 1 and S = k ln(Ω) is also 0.
> (You need an appropriate extension of Shannon’s entropy to continuous distributions.)
Gibbs’ entropy.
> Or you may study statistical mechanics
Indeed.