  The origin of stretched exponential relaxation in supercooled glass-forming
liquids is one of the central questions regarding the anomalous dynamics of
these fluids. The dominant explanation for this phenomenon has long been the
proposition that spatial averaging over a heterogeneous distribution of locally
exponential relaxation processes leads to stretching. Here we perform
simulations of model polymeric and small-molecule glass-formers in the
isoconfigurational ensemble to show that stretching instead emerges from a
combination of spatial averaging and locally nonexponential relaxation. Results
indicate that localities in the fluid exhibiting faster-than-average relaxation
tend to exhibit locally stretched relaxation, whereas slower-than-average
relaxing domains exhibit compressed exponential relaxation. We show that local
stretching is predicted by loose local caging, as measured by the Debye-Waller
factor, and vice versa. This phenomenology in the local relaxation of
in-equilibrium glasses parallels the dynamics of out of equilibrium under-dense
and over-dense glasses, which likewise exhibit an asymmetry in their degree of
stretching vs compression. On the basis of these results, we hypothesize that
local stretching and compression in equilibrium glass-forming liquids results
from evolution of particle mobilities over a single local relaxation time, with
slower particles tending towards acceleration and vice versa. In addition to
providing new insight into the origins of stretched relaxation, these results
have implications for the interpretation of stretching exponents as measured
via metrologies such as dielectric spectroscopy: measured stretching exponents
cannot universally be interpreted as a direct measure of the breadth of an
underlying distribution of relaxation times.
