  We provide molecular dynamics simulation of heat transport and thermal energy
diffusion in one-dimensional molecular chains with different interparticle pair
potentials at zero and non-zero temperature. We model the thermal conductivity
(TC) and energy diffusion in the coupled rotator chain and in the Lennard-Jones
chain either without or with the confining parabolic interatomic potential. The
considered chains without the confining potential have normal TC and energy
diffusion, while the corresponding chains with the confining potential are
characterized by anomalous (diverging with the system length) TC and
superdiffusion of energy. We confirm in such a way that, surprisingly, the
confinement makes both heat transport and energy diffusion anomalous in
low-dimensional phononic systems. We show that the chain, which has a finite
TC, is also characterized by the normal energy diffusion in the thermalized
chain (at non-zero temperature), while the superdiffusion of thermal energy
occurs in the thermalized chains with only anomalous TC. We present the
arguments, supported by our simulations, that the scaling relation between the
exponents in time dependence of the mean square displacement of thermal energy
distribution and in length dependence of the anomalous TC is not universal and
can be different, depending on the main mechanism of energy transport: by
weakly-scattered waves or by noninteracting colliding particles performing Levy
flights.
