Some recent theoretical results on amorphous packings of hard spheres

Abstract

The aim of this paper is to review and discuss qualitatively some results on the properties of amorphous packings of hard spheres that were recently obtained by means of the replica method. The theory gives predictions for the equation of state of the glass, the complexity of the metastable states, the scaling of the pressure close to jamming, the coordination of the packing and the pair correlation function in any space dimension d. The predictions compare very well with numerical simulations in d = 2, 3. The asymptotic predictions for dto ∞ are within the rigorous bounds. The theory can be extended to binary mixtures and to hard-core potentials with an attractive tail or square well.

Acknowledgments

I would like to thank the organizers of the X International Workshop on Disordered Systems for the opportunity to present these results. The paper is based on joint work with G. Parisi 26, 27, and I wish to thank him for a careful reading of the manuscript. I would also like to thank the authors of 10, 14, 41 for their kind permission to reproduce their numerical data.

This paper has been motivated by many discussions with G. Biroli, J.-P. Bouchaud, A. Cavagna, B. Coluzzi, A. Donev, S. Franz, A. Giuliani, C. O'Hern, J.L. Lebowitz, R. Monasson, W. Krauth, J. Kurchan, F. Sciortino, F.H. Stillinger, M. Tarzia, S. Torquato, P. Verrocchio, M. Wyart; I wish to thank all of them for their interest in our work and for their suggestions.

I benefited a lot from the stimulating atmosphere of the Department of Physics of Princeton University where part of this work was developed, and from the discussions at the end of my presentations in LPTHE Jussieu, ENS Paris and Princeton. I wish to thank the organizers for the invitation and all the participants for their comments.

This work has been supported by the Research Training Network STIPCO (HPRN-CT-2002-00319).