  As phenomena that necessarily emerge from the collective behavior of
interacting particles, phase transitions continue to be difficult to predict
using statistical thermodynamics. A recent proposal called the topological
hypothesis suggests that the existence of a phase transition could perhaps be
inferred from changes to the topology of the accessible part of the
configuration space. This paper instead suggests that such a topological change
is often associated with a dramatic change in the configuration space geometry,
and that the geometric change is the actual driver of the phase transition.
More precisely, a geometric change that brings about a discontinuity in the
mixing time required for an initial probability distribution on the
configuration space to reach steady-state is conjectured to be related to the
onset of a phase transition in the thermodynamic limit. This conjecture is
tested by evaluating the diffusion diameter and $\epsilon$-mixing time of the
configuration spaces of hard disk and hard sphere systems of increasing size.
Explicit geometries are constructed for the configuration spaces of these
systems, and numerical evidence suggests that a discontinuity in the
$\epsilon$-mixing time coincides with the solid-fluid phase transition in the
thermodynamic limit.

Statistical Mechanics;Materials Science