标题： 交叉有限尺寸标度理论及其在渗流中的应用 显示英文标题

标题： Crossover Finite-Size Scaling Theory and Its Applications in Percolation

摘要： 临界相变（$t=0$）的有限尺寸标度（FSS）表明，在大小为$|t|\sim L^{-1/\nu}$的窗口内，任何可观测量$Q$在线性尺寸为$L$的系统中的标度行为在$Q(t,L)=L^{Y_Q}\tilde{Q}(tL^{1/\nu})$时渐近遵循一个标度形式，其中$\nu$是关联长度指数，$Y_Q$是FSS指数，${\tilde Q}(x)$是缩放的临界距离的函数$x \equiv tL^{1/\nu}$。 我们系统地研究了在广泛可观测量下${\tilde Q}(|x|\to\infty)$的渐近标度行为，通过要求 FSS 和无限系统临界行为在交叉临界区域与$t \to 0$和$|x|\to\infty$相匹配。 这种交叉 FSS 理论预测，当临界性以较慢的速度接近时，随着$|t|\sim L^{-\lambda}$和$\lambda <1/\nu$，FSS 变为$\lambda$依赖性，并且可以推导出指数。 作为应用，考虑了爆炸性渗流和高维渗流。 对于前者，显示在无限系统临界性下广泛观察到的异常现象$t=0$可以归因于基于事件的集合中伪临界点附近的标准 FSS 行为的混合效应。 对于后者，如果使用自由边界条件，在无限系统临界点和伪临界点处的 FSS 指数是不同的，并且通过使用交叉 FSS 理论相互关联。 从这些观察中，渗流系统的 FSS 分为三类。 进行了大量模拟以确认这些预测。 显示英文摘要

摘要： Finite-size scaling (FSS) for a critical phase transition ($t=0$) states that within a window of size $|t|\sim L^{-1/\nu}$, the scaling behavior of any observable $Q$ in a system of linear size $L$ asymptotically follows a scaling form as $Q(t,L)=L^{Y_Q}\tilde{Q}(tL^{1/\nu})$, where $\nu$ is the correlation-length exponent, $Y_Q$ is an FSS exponent and ${\tilde Q}(x)$ is a function of the scaled distance-to-criticality $x \equiv tL^{1/\nu}$. We systematically study the asymptotic scaling behavior of ${\tilde Q}(|x|\to\infty)$ for a broad variety of observables by requiring that the FSS and infinite-system critical behaviors match with each other in the crossover critical regime with $t \to 0$ and $|x|\to\infty$. This crossover FSS theory predicts that when the criticality is approached at a slower speed as $|t|\sim L^{-\lambda}$ with $\lambda <1/\nu$, the FSS becomes $\lambda$-dependent and the exponent can be derived. As applications, explosive percolation and high-dimensional percolation are considered. For the former, it is shown that the widely observed anomalous phenomena at the infinite-system criticality $t=0$ can be attributed to the mixing effects of the standard FSS behaviors around the pseudocritical point in an event-based ensemble. For the latter, FSS exponents are found to be different at the infinite-system critical and the pseudocritical point if free boundary conditions are used, and they are related to each other by using the crossover FSS theory. From these observations, the FSS of percolation systems falls into three classifications. Extensive simulations are carried out to affirm these predictions.