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.
We would like to thank J.-P. Bouchaud and G. Biroli for inspiring the correlation matrix approach and pointing to the Marcenko–Pastur theorem and acknowledge the soft matter groups at U. Penn. for collaborating and sharing their data on
the NIPA system. None of the present work would have been done without the numerous discussions, we have had at various level with those involved in the analysis in terms of vibrational modes of experimental and numerical data. In particular we thank Wouter Ellenbroek, Andrea Liu and Corey O’Hern for helpful discussions on the interpretation of modes. A special thought to Wim van Sarloos and Martin van Hecke who hosted us in Leiden and always encouraged us to persevere in this delicate game, the rules and traps of which we have tried to expose in this paper.