两相介质扩散可扩展性长时间渐近行为的精确确定
Precise Determination of the Long-Time Asymptotics of the Diffusion Spreadability of Two-Phase Media
AI总结：
通过引入高阶修正项和解析性质，提出改进算法精确提取两相介质扩散可扩展性的长时标度指数α，并构造两点Padé近似逼近全时域行为。
摘要：
时间依赖的扩散可扩展性 $\mathcal{S}(t)$ 是一种强大的动力学探针，用于跨长度尺度表征两相非均匀介质的微观结构 [Torquato, S., \emph{Phys. Rev. E.}, 104 054102 (2021)]。研究表明，当谱密度在波数 $|\mathbf{k}|$ 趋于零时呈现幂律形式 $\tilde{\chi}_{_V}(\mathbf{k})\sim |\mathbf{k}|^\alpha$，归一化过剩可扩展性 $\mathscr{s}^{ex}(t)$ [正比于 $\mathcal{S}(\infty)-\mathcal{S}(t)$] 在长时间极限 $t\to\infty$ 下按 $\mathscr{s}^{ex}(t)\sim t^{-\frac{d+\alpha}{2}}$ 标度，从而可以确定无穷波长标度指数 $\alpha$。先前已设计出一种能够从长时间可扩展性数据中可靠提取指数 $\alpha$ 的算法 [Wang, H., Torquato, S., \emph{Phys. Rev. Appl.}, 17 034022 (2022)]。本文通过引入长时间渐近的高阶修正项并利用 $\tilde{\chi}_{_V}(k)$ 在原点的解析性质，进一步改进该过程以获得更精确的 $\alpha$。我们通过分析超均匀 ($\alpha>0$)、典型非超均匀 ($\alpha=0$) 和反超均匀 ($-d < \alpha <0$) 的两相介质模型来展示我们的过程。此外，通过将 $\mathscr{s}^{ex}(t)$ 的大$t$渐近展开与小$t$展开相结合，我们设计了一种两点Padé近似，仅用少数参数即可逼近所有$t$下的 $\mathscr{s}^{ex}(t)$。我们的发现有助于根据数值可扩展性数据或NMR弛豫测量获得的实验数据，跨长度尺度表征两相介质的微观结构。我们的工作也可应用于具有目标可扩展性行为的两相微结构的逆设计。
英文摘要：
The time-dependent diffusion spreadability $\mathcal{S}(t)$ is a powerful dynamical probe of the microstructure of two-phase heterogeneous media across length scales [Torquato, S., \emph{Phys. Rev. E.}, 104 054102 (2021)]. It has been shown that when the spectral density takes the power-law form $\tildeχ_{_V}(\mathbf{k})\sim |\mathbf{k}|^α$ as the wavenumber $|\mathbf{k}|$ tends to zero, the normalized excess spreadability $\mathscr{s}^{ex}(t)$ [proportional to $\mathcal{S}(\infty)-\mathcal{S}(t)$] scales as $\mathscr{s}^{ex}(t)\sim t^{-\frac{d+α}{2}}$ in the long-time limit $t\to\infty$, enabling one to determine the infinite-wavelength scaling exponent $α$. An algorithm that allows one to reliably extract the exponent $α$ from long-time spreadability data was previously devised [Wang, H., Torquato, S., \emph{Phys. Rev. Appl.}, 17 034022 (2022)]. In this paper, we further improve this procedure to obtain $α$ even more accurately by incorporating higher-order correction terms to the long-time asymptotics and by utilizing analyticity properties of $\tildeχ_{_V}(k)$ at the origin. We illustrate our procedure by analyzing hyperuniform ($α> 0$), typical nonhyperuniform ($α=0$), and antihyperuniform ($-d < α<0$) models of two-phase media. In addition, by combining the large-$t$ asymptotic expansion of $\mathscr{s}^{ex}(t)$ with the small-$t$ expansion, we have devised a two-point Padé approximant to approximate $\mathscr{s}^{ex}(t)$ for all $t$ with just a few parameters. Our findings facilitate the characterization of the microstructure of two-phase media across length scales as obtained from numerical spreadability data or experimental data obtained from NMR relaxation measurements. Our work can also be applied in the inverse design of two-phase microstructures with targeted spreadability behaviors.
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