Extracting vibrational modes from fluctuations: A pedagogical discussion

The study of the jamming transition of granular and colloidal systems, has lead to a proliferation of theoretical and numerical results formulated in the language of the eigenspectrum of the dynamical matrix for these disordered systems. Only recently however, have these modes been accessed experimentally in colloidal and granular media, by computing the eigenmodes of the covariance matrix of the particle positions. At the same time, new conceptual and methodological questions regarding the interpretation of these results have appeared. In the present paper, we first give an overview of the theoretical framework which is appropriate to interpret the eigenmodes and eigenvalues of the correlation matrix in terms of the vibrational properties of these systems. We then illustrate several aspects of the statistical and data analysis techniques necessary to extract reliable results from experimental data. Concentrating on the cases of hard sphere simulations, colloidal and granular experiments, we discuss how to test, in turn, for the existence of a metastable state and the statistical independence of the sampling, the effect of experimental resolution, and the harmonic hypothesis underlying the approach; highlighting both the promises and limitations of this approach.