= Finite-volume random scattering spectrum
{title2=$\langle\sigma_d\rangle\simeq\mu^2k^4|V|S_W(\mathbf k_s)/(4\pi^2)$}
In the <Born approximation for scalar wave scattering>, with $n=1+\mu W$ and zero-mean stationary unit-variance $W$, define $S_W(\mathbf q)=\int C(\mathbf s)e^{-i\mathbf q\cdot\mathbf s}d^3s$. The leading <expectation> of the <differential cross-section for scalar wave scattering> is
$$
\langle\sigma_d\rangle=\frac{\mu^2k^4}{4\pi^2}\int C(\mathbf s)|V\cap(V-\mathbf s)|e^{-i\mathbf k_s\cdot\mathbf s}\,d^3s+O(\mu^3),
\quad\mathbf k_s=k(\widehat{\mathbf r}-\widehat{\mathbf r}_0).
$$
For a regular volume much larger than the covariance range, the overlap is approximately $|V|$, giving $\mu^2k^4|V|S_W(\mathbf k_s)/(4\pi^2)$. <Statistical isotropy>, beyond <statistical homogeneity>, permits the scalar argument $|\mathbf k_s|=2k\sin(\theta/2)$. If the spectrum is defined for the index fluctuation rather than $W$, it already contains $\mu^2$.
Back to article page
