theoretical models for such stretched exponential behavior have been advanced, but their relation to and possible deduction from atomic-level Hamiltonians continue to remain obscure. Rigorous mathematical analysis of such connections or proving their strict absence would be a valuable contribution.

Mathematicians might be able to supply basic and extremely valuable insights to this field. For example, it would be important to establish or obtain the following results for some nontrivial class of intermolecular potential functions (e.g., pairwise additive spherical interactions):

  • Prove that the lowest overall potential energy, in the large system limit, is attained for a periodic spatial arrangement of molecules (a crystal).

  • Prove that if periodicity is strongly disallowed in some suitable sense, that amorphous molecular arrangements still exist as relative minima of the potential energy function.

  • Develop rigorous bounds on the rise in potential energy that results from imposition of the nonperiodicity constraint in the preceding result.

  • Enumerate the inequivalent potential energy minima, at least in the sense of exponential rise rate in the large system limit, or as rigorous bounds on this rise rate.

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