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# 物理代写|统计物理代写Statistical Physics of Matter代考|PHYS328 Fractal Dimension: Mathematical Definition

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## 物理代写|统计物理代写Statistical Physics of Matter代考|Fractal Dimension: Mathematical Definition

Keeping these examples in mind, we will now give the precise mathematical definition of fractal dimension. To determine the fractal dimension of an object, we can use the following (theoretical) procedure.

Consider all possible coverings of an object with $d$-dimensional spheres of radius $r_i \leq \epsilon$, where $\epsilon$ is the resolution. Let $N_\epsilon(C)$ be the number of spheres used in the covering $C$. Then, the volume of the covering is A covering of an object $X \subseteq \mathbb{R}^d$ is a collection of sets whose union contains $X$ as a subset. If we denote the collection of sets by $C=\left{A_i \mid i \in I\right}$, we call $C$ a covering of $X$ if
$$X \subseteq \bigcup_{i \in I} A_i$$

## 物理代写|统计物理代写Statistical Physics of Matter代考|The Box Counting Method

The box counting method is a method to numerically determine the fractal dimension of an object. It is conceptually easy because the underlying idea is close to the mathematical definition.

In the box counting method, we superimpose a grid with grid spacing $\epsilon$ on the fractal object. We define the number of boxes in the grid that are not empty contain a part of the fractal as $N(\epsilon)$. We do this for a large range of $\epsilon$ and plot $N(\epsilon)$ as a function of $\epsilon$ in a log-log plot. We show a typical result in Figure 2.29.

We recognize a region where the slope is constant in the plot. In this region, the object is self-similar and the slope equals the fractal dimension. Outside this self-similar regime, the finite resolution and finite size of the object destroy the selfsimilarity. To convince ourselves that this indeed reproduces the fractal dimension that we defined before, recall that $\epsilon$ defines a characteristic length scale, while $N(\epsilon)$ is proportional to the volume of the fractal object. Numerically, the box counting method has rather pronounced finite-size effects, because at the border of an object very sparsely occupied cells can have an important weight.

## 物理代写|统计物理代写物质统计物理学代考|分形维数:数学定义

$$X \subseteq \bigcup_{i \in I} A_i$$

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