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Dynamical fluctuations close to Jamming versus Vibrational Modes : a pedagogical discussion illustrated on hard spheres simulations, colloidal and granular experiments

Studying 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 system. Only recently however, these modes have been accessed experimentally in colloidal and granular media, computing the eigenmodes of the covariance matrix of the particle positions. At the same time new conceptual and methodological questions have appeared, regarding the interpretation of these results. In the present paper, we first give an overview of the theoretical framework which is appropriate to discuss the interpretation of 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 which are necessary to extract reliable results from experimental data. Concentrating on the case of hard sphere simulations, colloidal and granular experiments, we discuss how to test the existence of a metastable state, the statistical independence of the sampling, the effect of the experimental resolution, and the harmonic hypothesis underlying the approach, highlighting both its promises and limitations.