  We analyze multiple new issues concerning activated relaxation in glassy hard
sphere fluids and molecular and polymer liquids based on the Elastically
Collective Nonlinear Langevin Equation (ECNLE) theory. By invoking a high
temperature reference state, a near universality of the apparent dynamic
localization length scale is predicted for liquids of widely varying fragility,
a result that is relevant to recent simulation studies and quasi-elastic
neutron scattering measurements. In contrast, in the same format strongly
non-universal behavior is found for the activation barrier that controls long
time relaxation. Two measures of cooperativity in ECNLE theory are analyzed. A
particle-level total displacement associated with the alpha relaxation event is
found to be only of order 1-2 particle diameters and weakly increases with
cooling. In contrast, an alternative cooperativity length is defined as the
spatial scale required to recover the full barrier and bulk alpha time. This
length scale grows strongly with cooling due to the emergence in the deeply
supercooled regime of collective long range elastic fluctuations required to
allow local hopping. It becomes very large as the laboratory Tg is approached,
though is relatively modest at degrees of supercooling accessible with
molecular dynamics simulation. The alpha time is found to be exponentially
related to this cooperativity length over an enormous number of decades of
relaxation time that span the lightly to deeply supercooled regimes. Moreover,
the effective barrier height increases almost linearly with the growing
cooperativity length scale. An alternative calculation of the collective
elastic barrier based on a literal continuum mechanics approach is shown to
result in very little change of the theoretical results for bulk properties,
but leads to a much smaller and less temperature-sensitive cooperativity length
scale.
