  We apply microcanonical ensemble considerations to suggest that, whenever it
may thermalize, a general disorder-free many-body Hamiltonian of a typical
atomic system has solid-like eigenstates at low energies and fluid-type (and
gaseous, plasma) eigenstates associated with energy densities exceeding those
present in the melting (and, respectively, higher energy) transition(s). In
particular, the lowest energy density at which the eigenstates of such a clean
many body atomic system undergo a non-analytic change is that of the melting
(or freezing) transition. We invoke this observation to analyze the evolution
of a liquid upon supercooling (i.e., cooling rapidly enough to avoid
solidification below the freezing temperature). Expanding the wavefunction of a
supercooled liquid in the complete eigenbasis of the many-body Hamiltonian,
only the higher energy liquid-type eigenstates contribute significantly to
measurable hydrodynamic relaxations (e.g., those probed by viscosity) while
static thermodynamic observables become weighted averages over both solid- and
liquid-type eigenstates. Consequently, when extrapolated to low temperatures,
hydrodynamic relaxation times of deeply supercooled liquids (i.e., glasses) may
seem to diverge at nearly the same temperature at which the extrapolated
entropy of the supercooled liquid becomes that of the solid. In this formal
quantum framework, the increasingly sluggish (and spatially heterogeneous)
dynamics in supercooled liquids as their temperature is lowered stems from the
existence of the single non-analytic change of the eigenstates of the clean
many-body Hamiltonian at the equilibrium melting transition present in low
energy solid-type eigenstates. We derive a single (possibly computable)
dimensionless parameter fit to the viscosity and suggest other testable
predictions of our approach.
