  We investigate the anomalous dynamics in smectic phases of short host rods
where, counter-intuitively, long guest rod-shaped particles diffusive faster
than the short host ones, due to their precise size mismatch. In addition to
the previously reported mean-square displacement, we analyze the time evolution
of the Self-van Hove functions G(r,t), as this probability density function
uncovers intrinsic heterogeneous dynamics. Through this analysis, we show that
the dynamics of the host particles parallel to the director becomes
non-gaussian and therefore heterogeneous after the nematic-to-smectic-A phase
transition, even though it exhibits a nearly diffusive behavior according to
its mean-square displacement. In contrast, the non-commensurate guest particles
display Gaussian dynamics of the parallel motion, up to the transition to the
smectic-B phase. Thus, we show that the Self-van Hove function is a very
sensitive probe to account for the instantaneous and heterogeneous dynamics of
our system, and should be more widely considered as a quantitative and
complementary approach of the classical mean-square displacement
characterization in diffusion processes.
