  Thermal motion in complex fluids is a complicated stochastic process but
ubiquitously exhibits initial ballistic, intermediate sub-diffusive, and
long-time non-Gaussian diffusive motion, unless interrupted. Despite its
relevance to numerous dynamical processes of interest in modern science, a
unified, quantitative understanding of thermal motion in complex fluids remains
a long-standing problem. Here, we present a new transport equation and its
solutions, which yield a unified quantitative explanation of the mean square
displacement (MSD) and the non-Gaussian parameter (NGP) of various fluid
systems. We find the environment-coupled diffusion kernel and its time
correlation function are two essential quantities determining transport
dynamics of complex fluids. From our analysis, we construct a general, explicit
model of the complex fluid transport dynamics. This model quantitatively
explains not only the MSD and NGP, but also the time-dependent relaxation of
the displacement distribution for various systems. We introduce the concepts of
intrinsic disorder and extrinsic disorder that have distinct effects on
transport dynamics and different dependencies on temperature and density. This
work presents a new paradigm for quantitative understanding of transport and
transport-coupled processes in complex disordered media.
