# Can a system cooled to absolute zero become trapped in a local minimum of energy?

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))

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.

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 $$T=0$$.
• What about the state where all molecules have interatomic distance $x_2$? Does that macrostate not have a single microstate as well? Commented Dec 14, 2019 at 10:56
• @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. Commented Dec 14, 2019 at 11:45