## Fractal dimensions Fractals are scale-invariant systems, or "self-similar". For polymers, porous, colloidal systems, etc., fractals are widely exist. To accurately define fractals and fractal dimension ([Hausdorff dimension](https://en.wikipedia.org/wiki/Inductive_dimension)) requires complicated mathematical descriptions. Here we just show a descriptive definition for our material system. Consider a self-similar structure, if $n(R)$ is the number of particles inside a sphere of radius $R$, and satisfy $n(R)\propto R^{D}$ $D$ is called the fractal dimension. ### Radial distribution function and fractal dimension The definition of radial distribution function is given in [[Real space correlations, radial distribution function]]. In short, it describes the probability density of finding a particle at radius $r$. Then $n(R)$ can be written as $\begin{aligned} n(R) = \langle n \rangle \int \! \mathrm{d}^3 r \, g(r) &= cR^D \\ 4\pi \langle n \rangle \int_0^{R} \! \mathrm{d}r \, r^2 g(r) &= cR^D \\ 4\pi \langle n \rangle R^2 g(R) &= c D R^{D-1} \\ g(R) &= \frac{c D}{4 \pi \langle n \rangle} R^{D-3} \end{aligned}$ Here $c$ is a constant (so $\frac{1}{3}$ gets into it), $\langle n \rangle$ is the average particle number, or say particle number density, defined as $\langle n \rangle = \frac{1}{V} \int_V n(R) \, \mathrm{d}V=\frac{N}{V}$ where $V$ is the volume over which the average is taken, and $n(R)$ is the number of particles within a radius $R$. This $\langle n \rangle$ is not a function of $R$, just an average value. Above formula gives the $g(R)$ for certain fractal structure. >[!Notice] >A normal question is, we all know that radial distribution function has some peaks for small $r$, due to stereo repulsion. So it is definitely not a power function. How can above expression being valid? >This is because for our actual mesoscopic system, we are limited by the particle, or the minimum structure size $r_{0}$. Above relation does not really works for $r$ close to $r_{0}$. >![[Drawing 2024-05-21 01.12.04.excalidraw.svg]] >This is the property of fractal systems! The probability of having $\langle n\rangle$, or say $\rho_{0}$, average particle density, is not 1 but decreases with $r$. For actual systems, we have both the minimum limits $r_{0}$, and the maximum limits, $\xi$, cause they do not have infinite self-similar. So we should add a limiting factor to ensure $g(r)$ decay correctly and meet the integral average. $g(r) = \frac{cD}{4\pi \langle n \rangle} r^{D-3} e^{-\frac{r}{\xi}}$ ![[Drawing 2024-05-21 01.18.35.excalidraw.svg]] This $e^{-r/\xi}$ is the cut-off function describing the perimeter of the fractal system. It should stay ~ 1 when $\frac{r}{\xi} \leq 1$, but for large $\frac{r}{\xi}$ it falls off faster than any power law. It is worth to mention that the constant $c\propto r_{0}^{D}$ and its unit compensates the $R^{D-3}$. This ensures $g(r)$ is dimensionless. >[!Info] >See the paper ***Size distribution effect on the power law regime of the structure factor of fractal aggregates*** https://journals.aps.org/pre/pdf/10.1103/PhysRevE.60.7143. Local file [[Size distribution effect on the power law regime of the structure factor of fractal aggregates]] >