What Determines a Microstate's Identity?

[tim_lou]

It seems many textbooks do not provide a precise definition of microstate...

What exactly is a microstate? my guess is a specific arrangement of the objects in a system... but in what sense? energy arrangement? momentum arrangement? I guess the real question is what makes two states two DIFFERENT states?

Is microstate just a point in the phase space (a set of Hamiltonian coordinates and momentum)?

How do we know if one state is different than the other?

 

[StatMechGuy]

A specific microstate is some configuration for yoru ensemble of N particles in the canonical coordinates $$(\vec{p}_i, \vec{q}_i)_{i \in N}$$. The corresponding macrostate, in the statistical mechanics sense, are the set of all microstates that yield the same final total energy. Two microstates would be different if all the particles had their momenta and positions rearranged, but it would be an equivalent macrostates if the rearrangement led to the same energy.

It's like if you looked at all the points in the isoenergetic surface for a free particle $$p^2/2m = E$$. This forms a sphere of radius $$\sqrt{2 m E}$$ in momentum space, and any point on this surface is in the same macrostate, but a different microstates.

 

[tim_lou]

so you mean that a set of canonical coordinates specify a microstate? but what coordinates would qualify for defining a microstate?

I mean for instance, when we count the energy degeneracy in hydrogen atoms as different states, what coordinates are we using? radial coordinates $(r, \theta, \phi, p_r, p_\theta, p_\phi)_i$? what if I count the states using Cartesian coordinates? would the result be the same? (Of course, Cartesian coordinates should yield the same result... but my point is, what are the restrictions on the choice of coordinates)?