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I often came across completely different definitions of a macrostate, or at least they feel different. One of the definitions were supported by an example which I would like to state here:

Suppose N=3 distinguishable particles are distributed across 4 energy levels with energies 0, $\epsilon$, $2\epsilon$, $3\epsilon$. Letting $N_i$ denote the number of particles in the $i$th energy level with $0\leq i\leq 4$, there are multiple ways of arranging the particles to yield the total energy $E=6\epsilon$. For example: $N_0=2$ and $N_3=1$, $N_0=N_1=N_2=1$, or $N_1=3$ all yield the same energy $6\epsilon$.

The statement is now that each of these different specifications of occupation defines a different macrostate. And this is were I want to disagree. To my understanding, each of these configurations is one and the same macrostate, since they all specify the same energy of the system. This is at least how I learnt it; a macrostate is specified by its energy, s.t. different energy corresponds to a different macrostate, regardless of the occupation of the different levels (they become important when counting the number of microstates for the given macrostate).

All of this then confuses me even more when coming to the thermodynamic probability $w$, which gives the number of microstates using combinatorics, answering the question in how many ways $N_j$ particles out of a total $N$ can be placed in the $j$th energy level. What I struggle with to understand is how we specify these different $N_j$. Can anybody provide a somehow easy (and relevant, meaning no coin tossing experiments, but more in the direction of some basic quantum mechanical system) example?

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