  Transport and diffusion of heat in one dimensional (1D) nonlinear systems
which {\it conserve momentum} is typically thought to proceed anomalously.
Notable exceptions, however, exist of which the rotator model is a prominent
case. Therefore, the quest arises to identify the origin of manifest anomalous
transport in those low dimensional systems. Here, we develop the theory for
both, momentum/heat diffusion and its corresponding momentum/heat transport. We
demonstrate that the second temporal derivative of the mean squared deviation
of the momentum spread is proportional to the equilibrium correlation of the
total momentum flux. This result in turn relates, via the integrated momentum
flux correlation, to an effective viscosity, or equivalently, to the underlying
momentum diffusivity. We put forward the intriguing hypothesis that a
fluid-like momentum dynamics with a {\it finite viscosity} causes {\it normal}
heat transport; its corollary being that superdiffusive momentum diffusion with
an intrinsic {\it diverging viscosity} in turn yields {\it anomalous} heat
transport. This very hypothesis is corroborated over wide extended time scales
by use of precise molecular dynamics simulations. The numerical validation of
the hypothesis involves three distinct archetype classes of nonlinear
pair-interaction potentials: (i) a globally bounded pair interaction (the noted
rotator model), (ii) unbounded interactions acting at large distances (the
Fermi-Pasta-Ulam $\beta$ model, or the rotator model amended with harmonic pair
interactions) and (iii), a pair interaction potential being unbounded at short
distances while displaying an asymptotic free part (Lennard-Jones model).
