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Suppose I have a diatomic molecule with a potential function $V(x)$ depending on the distance between the atoms. Suppose also that this function has exactly two minima, one global at $x_1$ and one local $x_2$. (For clarity's sake, $V(x_1)<V(x_2)$)

If I cool this molecule (or alternatively, a box containing a bunch of them) down to absolute zero temperature, I would expect to find all the molecules are now frozen at an inter-atomic distance $x_1$. However, is it also possible that sometimes these molecules will become 'trapped' and instead freeze at the local minimum $x_2$? Or would such a thing never happen at absolute zero?

My thinking is that a system is at $T=0$ if it cannot give up any energy to its surrounding, and so getting trapped in a local minimum might make sense if the system can only change state in a continuous fashion.

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There are many states with given energy $E$ when some of the molecules are frozen with interatomic distance $x_1$ and others with $x_2$. There's only one state where all the molecules have interatomic distance $x_1$, so this is the case when entropy $S=0$. Since there's a state with $S=0$, by the third law of thermodynamics the other state with $S>0$ can't have $T=0$.

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  • $\begingroup$ What about the state where all molecules have interatomic distance $x_2$? Does that macrostate not have a single microstate as well? $\endgroup$
    – Bar Alon
    Commented Dec 14, 2019 at 10:56
  • $\begingroup$ @BarAlon that state has elevated energy. Such energy could also be realized e.g. in a state where some molecules are at $x_1$ while others are vibrating around $x_2$, giving the same average total energy. (Kinetic energy of vibration compensates lowered potential energy of $x_1$ molecules.) So entropy of this state is nonzero. $\endgroup$
    – Ruslan
    Commented Dec 14, 2019 at 11:45
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If the barrier between your two minima is high, yes you can be trapped into this state. However your system will not be considered to have reach an equilibrium state.

The typical example is glasses, where the system froze into a metastable non-equilibrium state and will stay there even if you cooling it down to zero temperature.

The tricky part is that if you allow your system to take an infinite amount of time to relax while you have cooling it to zero, then it will be into the lowest energy state. The way you are cooling your system will matter.

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