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I'm having difficulty understanding a statistical mechanics problem. I'm missing some basic understanding of counting the minimum energy states. My thought is that there are three states to choose from, $x$, $y$, and $z$. And energy is zero for the first two, the lowest energy, I just don't understand why that goes as $2^N$.

Problem: $N$ diatomic molecules are stuck on a metal surface of square symmetry. Each molecule can either lie flat on the surface in which case it must be aligned to one of two directions, $x$ and $y$, or it can stand up along the $z$ direction. There is an energy cost of energy > 0 associated with a molecule standing up, and zero energy for molecules lying flat along $x$ or $y$ directions.

(a) How many microstates have the smallest value of energy? What is the largest microstate energy?

I know this is the answer: The ground state energy of $E = Emin = 0$ is obtained for $2^N$ configurations. The largest microstate energy is $Ne$ is unique.

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  • $\begingroup$ You have two possible directions for each particle lying down on the surface: $x$ and $y$, hence the number of possibilities in the ground state is $2^N$ isn't it? $\endgroup$
    – gatsu
    Mar 10, 2014 at 11:59

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So lets start with the low energy configuration. To have the lowest energy all molecules must be lying in the x-y plane.

Each one has 2 directions it could be lying in (x or y), and there are $N$ particles, hence $2^N$ possible configurations.

(I always find picturing this with small number of molecules, say 2 or 3, for which counting the states is easy, helped me understand this kind of behaviour better before I could generalise it)

The largest microstate energy has nothing to do with the number of possible configurations (although as we'll show there is only one possibility)

As each particle can have a maximum energy of $e$ and will only have this when in the z direction, the maximum total energy is $N e$ corresponding to all molecules pointing in the z direction (hence why it is unique).

I think you may have just gotten a bit confused between the energy calculation and the counting of states, which are linked but very much not the same:)

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  • $\begingroup$ Thank you! I found my error and it was definitely in the counting, thank you again. $\endgroup$
    – Joseph
    Mar 11, 2014 at 2:39

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