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# 物理代写|统计物理代写Statistical Physics of Matter代考|PHY625 Correlation Length

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## 物理代写|统计物理代写STATISTICAL PHYSICS OF MATTER代考|Correlation Length

To further analyze percolation clusters, we now consider the correlation function $c(r)$ that describes connectivity correlations as a function of the radial distance $r$. For one cluster (e.g., the largest cluster), we can use density correlations since, by definition, all sites in a cluster are connected. Otherwise, if we want to study correlation effects between multiple clusters, we should focus on connectivity correlations. Large values of the correlation function $c(r)$ mean that two quantities strongly influence each other, whereas zero means that they are uncorrelated.

Numerically, we can obtain the correlation function $c(r)$ by placing concentric circles or spheres around a site in the center region of the cluster. The correlation function is defined as

$$c(r)=\frac{\Gamma\left(\frac{d}{2}\right)}{2 \pi^{d / 2} r^{d-1} \Delta r}[M(r+\Delta r)-M(r)],$$
where $M(r)$ (mass of the cluster) denotes the number of occupied sites in a $d$ dimensional hypersphere of radius $r$. The prefactor in eq. (2.8) is the inverse of the volume of a $d$-dimensional annulus between $r$ and $r+\Delta r$, and $\Gamma(\cdot)$ is the gamma function. We see that $c(r)$ is the number of occupied sites within an annulus of thickness $\Delta r$ at a distance $r$ from the center and normalized by the volume of the annulus. We apply this method to the largest percolation cluster for different occupation probabilities $p$. When we compute $c(r)$ for a given cluster (see Figure 2.15), we find that the correlation function usually decreases exponentially with distance $r$, eventually with an offset $C$. Mathematically, this means
$$c(r) \propto C+\exp \left(-\frac{r}{\xi}\right),$$
where the constant $C$ is equal to the order parameter of percolation, $P(p)$, and thus vanishes for $p<p_{c}$. The newly introduced quantity $\xi$ is the correlation length. It describes the typical length scale over which the correlation function decays. Below $p_{c}$, the correlation length $\xi$ is proportional to the radius of a typical cluster.

When we analyze the dependence of the correlation length $\xi$ on the occupation probability $p$ (see Figure 2.16), we find that it diverges at the critical occupation probability $p_{c}[39]$
$$\xi(p) \propto\left|p-p_{c}\right|^{-v} \quad \text { where } \quad v= \begin{cases}\frac{4}{3}, & \text { in } 2 \mathrm{D}, \ 0.8751(11), & \text { in 3D } .\end{cases}$$

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

We encounter problems when the system size $L$ is smaller than the correlation length $\xi$. Instead of the singularity that was mentioned in eq. (2.10), the correlation length $\xi$ only takes on finite values (see Figure 2.17). We consider the values $p_{1}$ and $p_{2}$ that are defined by correlation lengths $\xi\left(p_{1}\right)$ and $\xi\left(p_{2}\right)$ that are of the order of the linear system size $L$. The region between $p_{1}$ and $p_{2}$ is called the critical region. We cannot trust any quantity obtained numerically within this critical region. Based on

$$L=\xi\left(p_{1}\right) \propto\left(p_{1}-p_{c}\right)^{-v}$$
and
$$p_{1}-p_{2} \approx 2\left(p_{1}-p_{c}\right),$$
we find that the critical region shrinks with system size like
$$p_{1}-p_{2} \propto L^{-\frac{1}{v}} .$$
In the limit $L \rightarrow \infty$, the critical region will vanish. We can obviously not realize such a limit on a computer. If we identify our numerically obtained effective critical occupation probability $p_{\mathrm{eff}}(L)$ with $p_{1}\left(\xi\left(p_{1}\right) \approx L\right)$, we will always obtain $p_{\mathrm{eff}}(L)0$ is a constant. The best we can do at this point is using the data acquired for finite system sizes and extrapolate to the values of the infinite system. To do so, we can use eq. (2.14), plot $p_{\text {eff }}$ as a function of $L^{-1 / v}$, and extrapolate the data to the point at which the vertical axis is crossed (corresponding to the limit $L^{-1 / v} \rightarrow 0$ ). This method enables us to find the critical occupation probability $p_{c}$.

## 物理代写|统计物理代写STATISTICAL PHYSICS OF MATTER代考|Correlation Length

(例如，最大的集群)，我们可以使用密度相关性，因为根据定义，集群中的所有站点都是连接的。否则，如

$$c(r)=\frac{\Gamma\left(\frac{d}{2}\right)}{2 \pi^{d / 2} r^{d-1} \Delta r}[M(r+\Delta r)-M(r)],$$

$$c(r) \propto C+\exp \left(-\frac{r}{\xi}\right)$$

$$\xi(p) \propto\left|p-p_{c}\right|^{-v} \quad \text { where } \quad v=\left{\frac{4}{3}, \quad \text { in } 2 \mathrm{D}, 0.8751(11), \quad \text { in } 3 \mathrm{D} .\right.$$

## 物理代写|统计物理代写STATISTICAL PHYSICS OF MATTER代 考|Finite Size Effects

2.17）。我们考虑价值观 $p_{1}$ 和 $p_{2}$ 由相关长度定义 $\xi\left(p_{1}\right)$ 和 $\xi\left(p_{2}\right)$ 是线性系统大小的数量级 $L$. 之间的区域 $p_{1}$ 和 $p_{2}$

$$L=\xi\left(p_{1}\right) \propto\left(p_{1}-p_{c}\right)^{-v}$$

$$p_{1}-p_{2} \approx 2\left(p_{1}-p_{c}\right)$$

$$p_{1}-p_{2} \propto L^{-\frac{1}{v}}$$

$p_{\text {eff }}$ 作为一个函数 $L^{-1 / v}$ ，并将数据外推到垂直轴相交的点 (对应于极限 $L^{-1 / v} \rightarrow 0$ )。这种方法使我们能够

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