  Particle transport in complex environments such as the interior of living
cells is often (transiently) non-Fickian or anomalous, that is, it deviates
from the laws of Brownian motion. Such anomalies may be the result of
small-scale spatio-temporal heterogeneities in, or viscoelastic properties of,
the medium, molecular crowding, etc. Often the observed dynamics displays
multi-state characteristics, i.e. distinct modes of transport dynamically
interconverting between each other in a stochastic manner. Reliably
distinguishing between single- and multi-state dynamics is challenging and
requires a combination of distinct approaches. To complement the existing
methods relying on the analysis of the particle's mean squared displacement,
position- or displacement-autocorrelation function, and propagators, we here
focus on "scattering fingerprints" of multi-state dynamics. We develop a
theoretical framework for two-state scattering signatures -- the intermediate
scattering function and dynamic structure factor -- and apply it to the
analysis of simple model systems as well as particle-tracking experiments in
living cells. We consider inert tracer-particle motion as well as systems with
an internal structure and dynamics. Our results may generally be relevant for
the interpretation of state-of-the-art differential dynamic microscopy
experiments on complex particulate systems, as well as inelastic or
quasielastic neutron (incl. spin-echo) and X-ray scattering scattering probing
structural and dynamical properties of macromolecules, when the underlying
dynamics displays two-state transport.
