$$S(\mathbf{q}):=\mathcal{FT}\lbrace\langle\rho(\mathbf{r})\rho(\mathbf{0})\rangle\rbrace(\mathbf{q})$$

is actually a circular convolution, where FFT can give a dramatic boost in contrast to calculate directly. The steps using FFT are:

- Calculate $\rho(\mathbf{r})$, which is a summation of Dirac-Delta function that can be estimated as a hisotgram;
- Calculate $\hat{\rho}(\mathbf{q})$ by FFT;
- $S(\mathbf{q})=|\hat{\rho}(\mathbf{q})|^2$;
- Calculate mean over modulus: $S(q)=\int S(\mathbf{q})\delta(|\mathbf{q}|-q)\mathrm{d}\mathbf{q}/\int \delta(|\mathbf{q}|-q)\mathrm{d}\mathbf{q}$

**Efficiency**

If the simulation box is divided into $N$ (in 3D systems, $N=N_xN_yN_z$ for example) bins, the FFT gives $O(N\log(N))$ complexity and step 4 is $O(N)$. Generally, in comparison with the direct method, one needs at least loop over number of particles and $N$ bins for $\mathbf{q}$, the number of particles is obviously, way larger than $\log(N)$. For data of most simulations are real numbers, rFFT designed for real inputs could further boost the program. Here is code for a 3D example, i, j, k represent $N_x$, $N_y$, $N_z$ respectively.

`fftn(a) == np.concatenate([rfftn(a), conj(rfftn(a))[-np.arange(i),-np.arange(j),np.arange(k-k//2-1,0,-1)]], axis=-1)`

**Accuracy**

- Binsize effect. The binsize in histogram should be smaller than half of the minimun of the pair distances in the system according to Nyquist-Shannon sampling theorem. For example, in Lennard-Jones systems, $0.4\sigma$ is a good choice. However, if values of only some small $q$ are concerned, i.e., in some self-assembly systems, there is no need to calculate in the "pair accuracy", the sampling distance smaller than half of the interested domain size is fine.
- Randomness in the bin Particles in bins are randomly distributed, especially for large binsize mentioned above. The summation $\sum_i \exp(-\mathbb{I}\mathbf{q}\cdot\mathbf{r}_i)$ can be decomposed into $$\sum_\mathrm{bin} \exp(-\mathbb{I}\mathbf{q}\cdot\mathbf{r}_\mathrm{bin})\left(\sum_i \exp( -\mathbb{I}\mathbf{q}\cdot\delta\mathbf{r}_i)\right)$$ The idea is straightforward: for particles inside a bin, i.e., for particle $i$, position $\mathbf{r}_i$ can be decomposed as $\mathbf{r}_i=\mathbf{r}_\mathrm{bin}+\delta\mathbf{r}_i$, and for particles in the bin, $\delta\mathbf{r}$ is assumed randomly distributed in the bin. The latter summation in the decomposition is thus represented as $n_\mathrm{bin}\overline{\exp(-\mathbb{I}\mathbf{q}\cdot\delta\mathbf{r})}$, the average is approximated according to the distribution of $\delta\mathbf{r}$. If we consider $\delta r$ (1D case, for example) uniformly distributed in $(-L/2, L/2)$ with $L$ be the binsize, the average is $\mathrm{sinc}(qL/2)$. Multiply the sinc function during step 4, the $S(q)$ will be corrected.

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