  Microscopic dynamics of complex fluids in the early stage of spinodal
decomposition (SD) is strongly intertwined with the kinetics of structural
evolution, which makes a quantitative characterization challenging. In this
work, we use x-ray photon correlation spectroscopy to study the dynamics and
kinetics of a protein solution undergoing liquid-liquid phase separation
(LLPS). We demonstrate that in the early stage of SD, the structural relaxation
kinetics is up to 40 times slower than the dynamics and thus can be decoupled.
The kinetic decay rate is inversely proportional to time in the early stage,
followed by a nearly constant behavior during the coarsening stage. The
microscopic dynamics can be well described by hyper-diffusive ballistic motions
with a relaxation time exponentially growing with time in the early stage
followed by a power-law increase with fluctuations. These experimental results
are further supported by simulations based on the Cahn-Hilliard equation. The
established framework is applicable to other condensed matter and biological
systems undergoing phase transitions and may also inspire further theoretical
work.
