Clusters anomalous subdiffusion

Anomalous subdiffusion occurs on percolation clusters or on objects that in a statistical sense can be described as fractal, by which we mean that selfsimilarity describes simply the scaling of mass with length. Connections between v, the fractal dimension of the cluster, D, and the spectral dimension, d, have been established, relations that were originally derived by Alexander and Orbach [35], who developed a theory of vibrational excitations on fractal objects which they called fractons. An elegant scaling argument by Rammal and Toulouse [140] also leads to these relations, and we summarize their results. 

Figure 10. Plot of log ((R2(t)) - R2(0))) versus log (t) for a cluster of 735 water molecules (circles), cytochrome c (triangles), and cytochrome c hydrated by 400 water molecules (squares). Plotted fit to water data from 0.1 ps to 0.9 ps has a slope of 0.95 0.05, indicating normal diffusion. The slope of the plotted fit to both cytochrome c results from 0.1 ps to 3.0ps is 0.52 0.01, indicating anomalous subdiffusion with exponent v = 0.26.

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