LAUSR

Significance Many disordered fluid systems exhibit anomalous transport dynamics, which do not obey Einstein's theory of Brownian motion or other currently available theories. Here, we present a new transport equation governing thermal motion of complex fluidic systems, which provides a unified, quantitative explanation of the mean-square displacement, the non-Gaussian parameter, and the displacement distribution of various complex fluids. The applicability of our theory is demonstrated for molecular dynamics simulation results of supercooled water and dense hard disc fluids and for experimental results of colloidal beads diffusing on lipid tubes. This work suggests previously unexplored directions for quantitative investigation into transport and transport-coupled processes in complex disordered media, including living cells. Thermal motion in complex fluids is a complicated stochastic process but ubiquitously exhibits initial ballistic, intermediate subdiffusive, and long-time 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 challenging problem. Here, we present a transport equation and its solutions, which yield a unified quantitative explanation of the mean-square displacement (MSD), the non-Gaussian parameter (NGP), and the displacement distribution of complex fluids. In our approach, the environment-coupled diffusion kernel and its time correlation function (TCF) are the essential quantities that determine transport dynamics and characterize mobility fluctuation of complex fluids; their time profiles are directly extractable from a model-free analysis of the MSD and NGP or, with greater computational expense, from the two-point and four-point velocity autocorrelation functions. We construct a general, explicit model of the diffusion kernel, comprising one unbound-mode and multiple bound-mode components, which provides an excellent approximate description of transport dynamics of various complex fluidic systems such as supercooled water, colloidal beads diffusing on lipid tubes, and dense hard disk fluid. We also 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 an unexplored direction for quantitative understanding of transport and transport-coupled processes in complex disordered media.