  Consider the time-dependent problem of mass transfer of a solute between two
phases and assume that the solute is initially distributed in one phase (phase
2) and absent from the other (phase 1). We desire the fraction of total solute
present in phase 1 as a function of time, ${\cal S}(t)$, which we call the {\it
spreadability}, since it is a measure of the spreadability of diffusion
information as a function of time. We derive exact direct-space formulas for
${\cal S}(t)$ in any Euclidean space dimension $d$ in terms of the
autocovariance function as well as corresponding Fourier representations of
${\cal S}(t)$ in terms of the spectral density. We derive closed-form general
formulas for the short- and long-time behaviors of the spreadability in terms
of crucial small- and large-scale microstructural information, respectively.
The long-time behavior of ${\cal S}(t)$ enables one to distinguish the entire
spectrum of microstructures that span from hyperuniform to nonhyperuniform
media. For hyperuniform media, disordered or not, we show that the "excess"
spreadability, ${\cal S}(\infty)-{\cal S}(t)$, decays to its long-time behavior
exponentially faster than that of any nonhyperuniform two-phase medium, the
"slowest" being antihyperuniform media. The stealthy hyperuniform class is
characterized by an excess spreadability with the fastest decay rate among all
translationally invariant microstructures. Moreover, we establish a remarkable
connection between the spreadability and an outstanding problem in discrete
geometry, namely, microstructures with "fast" spreadabilities are also those
that can be derived from efficient "coverings" of space. We also identify
heretofore unnoticed remarkable links between the spreadability ${\cal S}(t)$
and NMR pulsed field gradient spin-echo amplitude as well as diffusion MRI
measurements.
