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# Spectral Statistics of Instantaneous Normal Modes in Liquids and Random Matrices

## Author

Listed:
• Srikanth Sastry
• Nivedita Deo
• Silvio Franz

## Abstract

We study the statistical properties of eigenvalues of the Hessian matrix $\cal H$ (matrix of second derivatives of the potential energy) for a classical atomic liquid, and compare these properties with predictions for random matrix models (RMM). The eigenvalue spectra (the Instantaneous Normal Mode or INM spectra) are evaluated numerically for configurations generated by molecular dynamics simulations. We find that distribution of spacings between nearest-neighbor eigenvalues, $s$, obeys quite well the Wigner prediction $s exp(-s^2)$, with the agreement being better for higher densities at fixed temperature. The deviations display a correlation with the number of localized eigenstates (normal modes) in the liquid; there are fewer localized states at higher densities which we quantify by calculating the participation ratios of the normal modes. We confirm this observation by calculating the spacing distribution for parts of the INM spectra with high participation ratios, obtaining greater conformity with the Wigner form. We also calculate the spectral rigidity and find a substantial dependence on the density of the liquid.

## Suggested Citation

• Srikanth Sastry & Nivedita Deo & Silvio Franz, 2000. "Spectral Statistics of Instantaneous Normal Modes in Liquids and Random Matrices," Working Papers 00-09-053, Santa Fe Institute.
• Handle: RePEc:wop:safiwp:00-09-053
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## References listed on IDEAS

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1. Gernot Grabher & Walter W. Powell (ed.), 2004. "Networks," Books, Edward Elgar Publishing, volume 0, number 2771.
2. Andreas Wagner & David Fell, 2000. "The Small World Inside Large Metabolic Networks," Working Papers 00-07-041, Santa Fe Institute.
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