# Dynamic structure factor of vibrating fractals. Academic Article

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### abstract

• Motivated by novel experimental work and the lack of an adequate theory, we study the dynamic structure factor $S(k,t)$ of large vibrating fractal networks at large wave numbers $k$. We show that the decay of $S(k,t)$ is dominated by the spatially averaged mean square displacement of a network node, which evolves subdiffusively in time, $⟨\mathbf{(}{\stackrel{\ensuremath{\rightarrow}}{u}}_{i}(t)\ensuremath{-}{\stackrel{\ensuremath{\rightarrow}}{u}}_{i}(0){\mathbf{)}}^{2}⟩\ensuremath{\sim}{t}^{\ensuremath{\nu}}$, where $\ensuremath{\nu}$ depends on the spectral dimension ${d}_{s}$ and fractal dimension ${d}_{f}$. As a result, $S(k,t)$ decays as a stretched exponential $S(k,t)\ensuremath{\approx}S(k){e}^{\ensuremath{-}({\ensuremath{\Gamma}}_{k}t{)}^{\ensuremath{\nu}}}$ with ${\ensuremath{\Gamma}}_{k}\ensuremath{\sim}{k}^{2/\ensuremath{\nu}}$. Applications to a variety of fractal-like systems are elucidated.

### publication date

• January 1, 2012