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Consider the time-dependent problem of mass transfer of a solute between twophases and assume that the solute is initially distributed in one phase (phase2) and absent from the other (phase 1). We desire the fraction of total solutepresent in phase 1 as a function of time, ${\cal S}(t)$, which we call the {\itspreadability}, since it is a measure of the spreadability of diffusioninformation as a function of time. We derive exact direct-space formulas for${\cal S}(t)$ in any Euclidean space dimension $d$ in terms of theautocovariance function as well as corresponding Fourier representations of${\cal S}(t)$ in terms of the spectral density. We derive closed-form generalformulas for the short- and long-time behaviors of the spreadability in termsof crucial small- and large-scale microstructural information, respectively.The long-time behavior of ${\cal S}(t)$ enables one to distinguish the entirespectrum of microstructures that span from hyperuniform to nonhyperuniformmedia. For hyperuniform media, disordered or not, we show that the "excess"spreadability, ${\cal S}(\infty)-{\cal S}(t)$, decays to its long-time behaviorexponentially faster than that of any nonhyperuniform two-phase medium, the"slowest" being antihyperuniform media. The stealthy hyperuniform class ischaracterized by an excess spreadability with the fastest decay rate among alltranslationally invariant microstructures. Moreover, we establish a remarkableconnection between the spreadability and an outstanding problem in discretegeometry, namely, microstructures with "fast" spreadabilities are also thosethat can be derived from efficient "coverings" of space. We also identifyheretofore unnoticed remarkable links between the spreadability ${\cal S}(t)$and NMR pulsed field gradient spin-echo amplitude as well as diffusion MRImeasurements.

Diffusion Spreadability as a Probe of the Microstructure of Complex Media Across Length Scales