Random Matrix Theory of Rigidity in Soft Matter

We study the rigidity or softness of soft matter using the characteristic scale of coupling formation developed in random matrix theory. The eigensystems of the timescale-dependent cross-correlation matrix, which are obtained from the time series data of the atomic coordinates of a protein produced by the all-atom molecular dynamics of the solvent, are analyzed. As an example, we present a result for a protein lysozyme, PDBID: 1AKI. We find that there are at least three different time scales involved in the coupling formation of correlated sectors of atoms and at least two different time scales for the size of the correlated sectors. These five time scales coexist simultaneously. We compare the results with those of the normal mode analysis and find a crossover of the distribution of the dominant vibrational components.