Random multiplicativeprocesses and the response functions of granular packings

Abstract

It has recently been suggested that the property of isostaticity ofthe contact network of a frictionless polydisperse granular packingin the limit of low applied pressure is responsible for some of theanomalous static behaviour of packings. In this paper we discuss thefact that, on disordered isostatic networks,displacement-displacement and stress-stress static Green functionsare described by coupled random multiplicative processes and thushave a truncated power-law distribution, with a cut-off that growsexponentially with distance. The expectation values of Greenfunctions on these systems differ from observed averages by anexponentially large factor unless the number of samples over whichaverages are taken is exponentially large. Thus predicted averageswill seldom be observed in experiments. If the external pressure isincreased sufficiently, excess contacts are created, the packingbecomes hyperstatic, and the above-mentioned anomalous propertiesdisappear because Green functions now have a bounded distribution.Thus the low-pressure, isostatic, limit is a critical pointwhere the Green function distribution becomes scale-free. Thiscriticality is induced by multiplicative noise.