Problem with microstates and corresponding volume in the phase space

Question

I've been struggling without any result on a formal problem about the relation between microscopic states of a system and the volume occupied by a macrostate in the phase space. I'm now very confused about everything.

I observed that:

• In general entropy is defined as $S = k_b \log(\Omega(E))$ where $\Omega(E)$ is the number of microstates that correspond to a given macrostate of energy E.
• In the microcanonical ensemble a macrostate's corresponding entropy can be calculated as $S = k_b \log(\Gamma(E))$ where $\Gamma(E)$ is the volume occupied by the macrostate in the phase space (i.e. the set of all possible $\vec q$ and $\vec p$ that satisfy $H(\vec q, \vec p, t) = E$.

Why can I use the volume instead of the number of microstates? How are $\Gamma(E)$ and $\Omega(E)$ related?

For example for an ideal gas in a box the condition that determines the volume in the phase space for a given macrostate of energy E (I'm looking only to the $p$s' volume) is $$V = \sum_{i = 1}^{3N} \frac{p_i^2}{2m} = E$$ which identifies a hypersphere of radius $\sqrt{2mE}$. There are infinite combinations of $\vec q$ and $\vec p$ that can satisfy the above equation so I would say that there are infinite microstates. Instead the volume is of course finite. How can I connect them?

Concluding, I read something about representative points and about a density function of these representative points but I couldn't understand well. What exactly are those representative points and how do they differ from "normal" points in the phase space like $(\vec q, \vec p)$?

Thank you in advance!

Answer 1

If microstates are discrete then they can be counted. So in information theory, for example, we calculate the entropy of a message by counting the number of equivalent messages and comparing that with the total number of possible messages.

However, if microstates are continuous then some other method of “counting” them is needed. This is typically done by defining a “density” of microstates and integrating this density function over a given volume of phase space.