  For over 30 years, mode-coupling theory (MCT) has been the de facto theoretic
description of dense fluids and the liquid-glass transition. MCT, however, is
limited by its ad hoc construction and lacks a mechanism to institute
corrections. We use recent results from a new theoretical framework--developed
from first principles via a self-consistent perturbation expansion in terms of
an effective two-body potential--to numerically explore the kinetics of systems
of classical particles, specifically hard spheres obeying Smoluchowski
dynamics. We present here a full solution to the kinetic equation governing the
density-density time correlation function and show that the function exhibits
the characteristic two-step decay of supercooled fluids and an
ergodic-nonergodic transition to a dynamically-arrested state. Unlike many
previous numerical studies and experiments, we have access to the full time and
wavenumber range of the correlation function and can track the solution
unprecedentedly close to the transition, covering nearly 15 decades of time.
Using asymptotic approximation techniques developed for MCT, we fit the
solution to predicted forms and extract critical parameters. Our solution shows
a transition at packing fraction $\eta^*=0.60149761(10)$--consistent with
previous static solutions under this theory and with comparable colloidal
suspension experiments--and the behavior in the $\beta$-relaxation regime is
fit to power-law decays with critical exponents $a=0.375(3)$ and $b=0.8887(4)$,
and with $\lambda=0.5587(18)$. For the $\alpha$-relaxation of the ergodic
phase, we find a power-law divergence of the time scale $\tau_{\alpha}$ as we
approach the transition. Through these results, we establish that this new
theory is able to reproduce the salient features of MCT, but has the advantages
of a first principles derivation and a clear mechanism for making systematic
improvements.
