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In statistical mechanics the macro-state of a system corresponds to a whole region in the microscopical phase-space of that same system, classically, that means that an

I've seen different derivations of the gibbs entropy, one in which a region in the microscopic phase-space corresponding to a macro-state of a system is divided into subregions of constant energy, and another where regions of constant energy get divided into subregions that correspond to different macro-states.

Although the procedure is aways to find the configuration of micro-states in that region that maximize the number of ways in which the micro-states can be organized into the subregions, I don't see how that makes sense in either case.

In the case where the outer region corresponds to a macro-state, does organizing the micro-states into subregions of constant energy means changing the hamiltonian? Because given a hamiltonian, a point in phase-space has a definite energy, right? Also, how does it even make sense to rearrange the micro-states into the subregions? For example, given a hamiltonian if all micro-states but one have the same energy, there is still only one way I can arrange those micro-states in that configuration, because the hamiltonian dictates that configuration.

I don't even know how to think about the other approach..

My other question comes later, and it has to do with the "transformation" of the expression that represents the number of ways you can organize the micro-states into the subregions

$$ \frac{N!}{\prod\limits_in_i!} $$

into the expression that later gets simplified into the gibbs entropy

$$ k_bln\left(\frac{N!}{\prod\limits_in_i!}\right). $$

Is that an empirical adaptation? Or is there a logical step involved? I'm okay if it's just an empirical thing, but I would like to know what that "thing" is.

EDIT: In the expressions above the N represents the number of micro-states in the outer region, and the n's represent the number of micro-states in each subregion. I know that those number are all infinite classically, but this derivation is an adaptation of a quantum mechanics derivation where those number are finite because the states of the system are discrete. In this case those numbers are just representations.

**infinity**of micro-states relate to a single macro-state. Similarly, given a hamiltonian, a whole surface in the microscopical phase space of a system can correspond to a single energy level (the surfaces of constant energy).I've seen different derivations of the gibbs entropy, one in which a region in the microscopic phase-space corresponding to a macro-state of a system is divided into subregions of constant energy, and another where regions of constant energy get divided into subregions that correspond to different macro-states.

Although the procedure is aways to find the configuration of micro-states in that region that maximize the number of ways in which the micro-states can be organized into the subregions, I don't see how that makes sense in either case.

In the case where the outer region corresponds to a macro-state, does organizing the micro-states into subregions of constant energy means changing the hamiltonian? Because given a hamiltonian, a point in phase-space has a definite energy, right? Also, how does it even make sense to rearrange the micro-states into the subregions? For example, given a hamiltonian if all micro-states but one have the same energy, there is still only one way I can arrange those micro-states in that configuration, because the hamiltonian dictates that configuration.

I don't even know how to think about the other approach..

My other question comes later, and it has to do with the "transformation" of the expression that represents the number of ways you can organize the micro-states into the subregions

$$ \frac{N!}{\prod\limits_in_i!} $$

into the expression that later gets simplified into the gibbs entropy

$$ k_bln\left(\frac{N!}{\prod\limits_in_i!}\right). $$

Is that an empirical adaptation? Or is there a logical step involved? I'm okay if it's just an empirical thing, but I would like to know what that "thing" is.

EDIT: In the expressions above the N represents the number of micro-states in the outer region, and the n's represent the number of micro-states in each subregion. I know that those number are all infinite classically, but this derivation is an adaptation of a quantum mechanics derivation where those number are finite because the states of the system are discrete. In this case those numbers are just representations.

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