  The immense dependence of the glass transition temperature $T_g$ on molecular
weight $M$ is one of the most fundamentally and practically important features
of polymer glass formation. Here, we report on molecular dynamics simulation of
three model linear polymers of substantially different complexity demonstrating
that the 70-year-old canonical explanation of this dependence (a simple chain
end dilution effect) is likely incorrect at leading order. Our data shows that
end effects are present only in relatively stiff polymers and, furthermore,
that the magnitude of this end effect diminishes on cooling. Instead, we find
that $T_g(M)$ trends are instead dominated by shifts in $T_g$ throughout the
entire polymer chain rather than through a chain end effect. We show that these
data are consistent with a generic two-barrier model of $T_g$ and its
$M$-dependence, motivated by the Elastically Collective Nonlinear Langevin
Equation (ECNLE) theory. More broadly, this work indicates both a need to
reassess the canonical understanding of $T_g(M)$ in linear polymers (and
macromolecules at large) and an opportunity to reveal new glass formation
physics with renewed study of $M$ effects on $T_g$.
