  Recent studies unveiled the Fickian yet non-Gaussian (FNG) dynamics of many
soft matter systems and suggested this phenomenon as a general characteristic
of the diffusion in complex fluids. In particular, it was shown that the
distribution of particle displacements in Fickian diffusion is not necessarily
Gaussian, and thus the Einstein and Smoluchowski theory describing the Brownian
motion of individual objects in a fluid would not be applicable. In this
Letter, we investigate whether the FNG dynamics so far reported in gels,
granular materials, biological and active matter systems, is also a distinctive
feature of colloidal liquid crystals. To this end, we perform Brownian Dynamics
simulations of oblate and prolate colloidal particles in the nematic phase. We
detect a normal and Gaussian dynamics at short and long time scales, whereas,
at intermediate time scales, a non-Fickian and non-Gaussian dynamics is found.
Additionally, we revisit the nature of the decay of the self-van Hove
correlation function, $G_s(r,t)$, which is here approximated with an
ellipsoidal, rather than spherical, Gaussian distribution. The new expression
that we propose is able to correctly assess the Gaussian dynamics in inherently
anisotropic systems, like liquid crystals, where the standard Gaussian
approximation of $G_s(r,t)$ would fail.
