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## 物理代写|统计物理代写Statistical Physics of Matter代考|PHYS730 Finite Size Scaling

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## 物理代写|统计物理代写STATISTICAL PHYSICS OF MATTER代考|Finite Size Scaling

In the neighborhood of the percolation threshold $p_{c}$, we find that the dependence of the second moment of the cluster-size distribution $\chi$ (see eq. (2.4)) on $p$ and $L$ can be reduced to a one variable function. This is a consequence of a general scaling law that exists around all critical points.

By plotting $\chi$ around $p_{c}$ against the occupation probability $p$ for several values of $L$, we obtain curves that differ strongly around the critical point. The most important difference is the height of the peaks which is found to grow as a power law with the linear system size $L$. Using this observation and the fact that the critical region shrinks like $L^{-1 / v}$ as shown in eq. (2.13), we can rescale the horizontal and vertical axis of the left panel of Figure $2.18$ to get a data collapse as shown in the corresponding right panel. Such a data collapse can be also observed in related models such as kinetic gelation (see Figure 2.19). ${ }^{5}$

Let us try to comprehend what this data collapse means. We have a function $\chi$ that was originally a function of two parameters (the occupation probability $p$ and the system size $L$ ) and that now behaves as though it was a one-parameter function. Mathematically, the data collapse of $\chi$ as shown in Figures $2.18$ and $2.19$ can be written as

For site percolation on a square lattice with side length $L$, we show (left) the system-size dependence of the second moment of the cluster-size distribution (eq. (2.4)) and (right) the corresponding finite-size scaling (eq. (2.15)). The number of samples is $10^{4}$. We normalized $\chi(p, L)$ by $\max {p}[\chi(p, L=1024)]$ in the left panel. $$\chi(p, L)=L^{\frac{\gamma}{v}} \mathbf{N}{\chi}\left[\left(p-p_{c}\right) L^{\frac{1}{v}}\right]$$
where $\boldsymbol{s}{\chi}$ is called a scaling function of $\chi .$ This is an example of the finite size scaling first proposed in Ref. [41]. When we approach the critical occupation probability $p{c}$, the scaling function $\boldsymbol{N}{\chi}$ approaches a constant and we find that the peak $\chi{\max }$ depends on the system size, so
$$\chi_{\max }(L) \propto L^{\frac{\gamma}{v}}$$

## 物理代写|统计物理代写STATISTICAL PHYSICS OF MATTER代考|Size Dependence of the Order Parameter

We are now going to consider the order parameter $P(p)$ at the critical occupation probability $p_{c}$. We denote the size of the largest cluster by $s_{\infty}$ and the side length of the lattice by $L$. When we plot $s_{\infty}$ against $L$, we notice that there is a power law at work,
$$s_{\infty} \propto L^{d_{\mathrm{f}}},$$
where the exponent $d_{\mathrm{f}}$ depends on the dimension of the system. We illustrate this behavior in Figure 2.20. For two-dimensional lattices, we find $d_{\mathrm{f}}=91 / 48$ and for three dimensions we find $d_{\mathrm{f}}=2.5226(1)$ [42]. The exponent $d_{\mathrm{f}}$ is called the fractal dimension, which will be explained in more detail in Section 2.4.

The size of the largest cluster is proportional to the product of system size $L^{d}$ and $P(p)$ (i.e., the probability of a lattice site to belong to the largest cluster). That is,
$$s_{\infty} \propto L^{d} P(p) .$$
We now combine Eqs. (2.19) and (2.10) and obtain
$$s_{\infty} \propto L^{d} P(p) \propto L^{d}\left|p-p_{c}\right|^{\beta} \propto L^{d} L^{-\beta / v}$$
where we used that the correlation length is of the order of $L$ for a finite system in the vicinity of $p_{c}$. Based on Eqs. (2.18) and (2.20), we can establish the following connection:
$$d_{\mathrm{f}}=d-\frac{\beta}{v}$$

## 物理代写㳘计物理代写STATISTICAL PHYSICS OF MATTER代考|Finite Size Scaling

(2.13)，我们可以重新缩放图左面板的横纵轴 $2.18$ 获得数据折峝，如相应的右侧面板所示。在动力学懝胶等相 关模型中也可以观察到这种数据崩溃（见图 2.19）。 5

$L)$ 现在的行为就好像它是一个单参数函数。在数学上，数据崩溃 $\chi$ 如图所示 $2.18$ 和 $2.19$ 可以写成

(2.4) ) 和 (右) 相应的有限尺寸缩放 (方程 (2.15)) 。样本数为 $10^{4}$. 我们标准化 $\chi(p, L)$ 经过 $\max p[\chi(p, L=1024)]$ 在左侧面板中。
$$\chi(p, L)=L^{\frac{\gamma}{v}} \mathbf{N} \chi\left[\left(p-p_{c}\right) L^{\frac{1}{v}}\right]$$

$$\chi_{\max }(L) \propto L^{\frac{\gamma}{v}}$$

## 物理代写|统计物理代写STATISTICAL PHYSICS OF MATTER代考|Size Dependence of the Order Parameter

$$s_{\infty} \propto L^{d_{\mathrm{f}}}$$

$$s_{\infty} \propto L^{d} P(p)$$

$$s_{\infty} \propto L^{d} P(p) \propto L^{d}\left|p-p_{c}\right|^{\beta} \propto L^{d} L^{-\beta / v}$$

$$d_{\mathrm{f}}=d-\frac{\beta}{v}$$

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