Meaning of density of microstates in phase space

[Kaguro]

TL;DR Summary

A microstate is one of possible states of the system common to a macrostate. In 6N dimensional phase space every point is a possible state, then what is the meaning of density of states?

Hello all. I am studying stat mech from Pathria's book.
It says a system is completely described by all positions and momenta of all the N particles. This maybe represented by a single point in 6N-D gamma space. So, each point is a (micro)state.

Now if we restrict the system (N,V,E to E+ΔE), the there is an allowed region in which microstates may exist.
In this region, every single point is a different state (in classical mech., there's no Planck's constant and unlimited precission.) So what's even the meaning of asking the distribution of states? It's like asking density of real numbers!

For every point in this space, either it is one of the allowed state or it is not. Even if there's a Planck's constant and limited precision, still this density can be either a finite constant or 0. It sounds digital to me.

Please help.

 

[Kaguro]

Okay, I think I got it:
It doesn't matter if you consider a small volume of phase as a single microstate. So ([q0,q0+dq],[p0,p0+dp]) is a small patch and belongs to a single microstate. It doesn't matter exactly how small the dq and dp are. Planck's constant doesn't appear in Classical statistics.

Secondly, an ensemble is not the set of all the microstates. It is a set of a very large number of fictitious copies of the system. Large enough to cover all the microstates at least several times. Several members of the ensemble may be in a single microstate.

This density rho(probability distribution) is not the number of microstates per phase volume. That's always constant. It is the number of members of ensemble per phase volume. This density may change in non-equilibrium systems.

Each member of the ensemble has a representative point. These move around according to the Hamilton's equations. This causes shift in density. If the system is in equilibrium, then the number of such pts moving in is the same as the number moving out.

The density of these pts is constant if you move along with them. This is Liouville's Theorem.

If system is in equilibrium, then the density at a single region in phase space is also constant. That is the distribution of representative pts doesn't move around. This is a stronger condition than the previous one.

I understood all this from an excellent article:
https://web.stanford.edu/~peastman/statmech/statisticaldescription.html