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# 物理代写|统计物理代写Statistical Physics of Matter代考|PHYC40650 Walks

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## 物理代写|统计物理代写Statistical Physics of Matter代考|Random Walks

Random walks are used to model many phenomena, including the motion of particles in liquids (Brownian motion, named after Robert Brown (see Figure 2.34)) [58, 59],

stock price fluctuations $[60,61]$,foraging behavior of animals [62], and

polymer chains at the theta point $[63,64]$.

Moreover, random walks are also important in search algorithms such as the famous PageRank algorithm [65]. We could continue this list with many more examples from various disciplines [66]. For the interested reader, we refer to Refs. [67, 68].

Mathematically, a random walk is a stochastic process

$$X_{t}=X_{0}+\sum_{j=1}^{t} Z_{j},$$

where $Z_{1}, Z_{2}, \ldots, Z_{t}$ denote $t$ independent random variables that can be scalars or vectors. We set $X_{0}=0$ and note that $X_{t}$ can be defined either on a lattice $\left(\mathbb{Z}^{d}\right)$ or on a continuous space $\mathbb{R}^{d}$. In one dimension, one possibility is to consider binary random variables $Z_{j}$. That is, $Z_{j}=1$ with probability $p$ and $Z_{j}=-1$ with probability $q=1-p$. Out of the $t$ total steps, the walker moves $l$ steps to the left $\left(Z_{j}=-1\right)$ and $m$ steps to the right $\left(Z_{j}=1\right)$. Thus, after $t$ steps, the walker is at position $X_{t}=m-l=2 m-t$. The probability that the walker is at position $X_{t}=2 m-t$ is distributed according to a binomial distribution [68]

$$P\left(X_{t}=2 m-t\right)=\left(\begin{array}{c} t \ m \end{array}\right) p^{m} q^{t-m} .$$

In Figure 2.35, we illustrate the binomial distribution of eq. (2.41) and the corresponding Gaussian approximation (see eq. (2.47)). We observe a data collapse when plotting $P(r, t) t^{1 / 2}$ against $r / t^{1 / 2}$.

## 物理代写|统计物理代写Statistical Physics of Matter代考|Self-Avoiding Walks

An important extension of a random walk is the concept of walks that cannot intersect their own trajectory (i.e., fulfill the condition of “excluded volume”). The most prominent example is the so-called self-avoiding walk (SAW). In one dimension, the walker would just move in one direction, and for dimensions $d \geq 4$, the intersection probability is vanishingly small, so we observe a regular random-walk behavior with fractal dimension $d_{\mathrm{f}}=2$ [68]. In two and three dimensions, the fractal dimensions are $d_{\mathrm{f}}=4 / 3$ and $d_{\mathrm{f}} \approx 5 / 3$, respectively [75-77]. The SAW was first introduced by Paul Flory (see Figure 2.39) to describe polymers in a good solvent $[75,78]$.

For the SAW, all configurations of same chain length $N$ have the same statistical weight [79]:
$$\Omega_{N}=\mu^{N} N^{\theta},$$

where $\mu$ denotes a chemical potential and $\theta=11 / 32$ for all two-dimensional lattices $[76,80]$. The generating function is equivalent to a grand canonical partition function (see Section 3.1.4)
$$Z(x)=\sum_{N} \Omega_{N} x^{N}=\sum_{N}(\mu x)^{N} N^{\theta},$$
where $x$ is the fugacity which corresponds to the statistical weight of adding one element to the chain. The radius of convergence of $Z(x)$ defines the critical fugacity $x_{c}=1 / \mu$. The average chain length is
$$\langle N\rangle=\left.\frac{\partial \ln (Z)}{\partial x}\right|{x=1}=\frac{\sum{N} N \Omega_{N}}{\sum_{N} \Omega_{N}}= \begin{cases}\text { finite, } & \text { if } x_{c}>1, \ \text { critical, } & \text { if } x_{c}=1, \ \text { infinite, } & \text { if } x_{c}<1\end{cases}$$

## 物理代写|统计物理代写Statistical Physics of Matter代考|Random Walks

$\theta$ 点处的聚合物链 $[63,64]$.

$$X_{t}=X_{0}+\sum_{j=1}^{t} Z_{j},$$

$$P\left(X_{t}=2 m-t\right)=(t m) p^{m} q^{t-m} .$$

## 物理代写|统计物理代写Statistical Physics of Matter代考|Self-Avoiding Walks

$$\Omega_{N}=\mu^{N} N^{\theta},$$

$$Z(x)=\sum_{N} \Omega_{N} x^{N}=\sum_{N}(\mu x)^{N} N^{\theta},$$
$$\langle N\rangle=\frac{\partial \ln (Z)}{\partial x} \mid x=1=\frac{\sum N N \Omega_{N}}{\sum_{N} \Omega_{N}}=\left{\text { finite, } \quad \text { if } x_{c}>1, \text { critical, } \quad \text { if } x_{c}=1, \text { infinite }, \quad \text { if } x_{c}<1\right.$$

## MATLAB代写

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