OFFPRINT Localization behavior of vibrational modes in granular packings

We study the localization of vibrational modes of frictionless granular media. We introduce a new method, motivated by earlier work on non-Hermitian quantum problems, which works well both in the localized regime where the localization length ξ is much less than the linear size L and in the regime ξ L when modes are extended throughout our finite system. Our very lowest frequency modes show “quasi-localized” resonances away from the jamming point; the spatial extent of these regions increases as the jamming point is approached, as expected theoretically. Throughout the remaining frequency range, our data show no signature of the nearness of the jamming point and collapse well when properly rescaled with the system size. Using Random Matrix Theory, we derive the scaling relation ξ ∼L for the regime ξ≫L in d dimensions. Copyright c © EPLA, 2008 Introduction. – Over the past years, many questions concerning the behavior of disordered systems have been put in a new perspective by addressing them from the point of view of the more general jamming scenario [1]. Especially for granular systems it has turned out to be very fruitful to study the changes in the properties and the response of granular packings as one approaches the jamming point from the jammed side, where the packing gets close to an isostatic solid. An isostatic packing is indeed essentially a marginal solid which has just enough contacts to maintain a stable packing. From simple counting arguments, one finds that the average coordination number Z of a d-dimensional isostatic packing of frictionless spheres equals Ziso = 2d [2]. Upon approaching this marginal solid, many static and dynamic properties exhibit anomalous behavior, associated with the fact that the excess number of average bonds, ∆Z ≡Z −Ziso, goes to zero [3–6]. In fact, ∆Z itself scales anomalously, namely as the square root of the difference in density from the one at jamming [3]. Likewise, the ratio G/K of the shear modulus G over the compression modulus K is found to scale as ∆Z, and the density of states of the vibrational modes becomes flat at low frequencies above some crossover frequency ω ∼∆Z, due to the emergence of many low-frequency modes. Much of this behavior was explained by Wyart et al. [4–6] in terms of the existence (a)E-mail: [email protected] of an important cross-over length scale l ∼ 1/∆Z, the length up to which the response is close to that of an isostatic packing. This scale l diverges as the jamming point is approached, but is difficult to probe directly. Nevertheless, the length l has recently been uncovered as the important cross-over length to continuum behavior in the static response [7,8]. Although most of these results pertain explicitly to packings of frictionless spheres, there are several indications [9,10] that many of these observations and ideas can be generalized to frictional packings. It has been noted in several studies that both the response to a local or global deformation [7,11] and the behavior of the vibrational eigenmodes [4,6] of a packing become much more disordered as one approaches the jamming point: as the snapshots of two vibrational modes in fig. 1 illustrate, far above the jamming point the eigenmodes have a structure reminiscent of what one gets in a continuum theory of an elastic medium, but close to the jamming point one is immediately struck by the appearance of many disordered “swirls”. The arguments put forward by Wyart et al. [4–6] indicate that the excess low frequency modes cannot be localized on scales l since they are the vestiges of the global floppy modes that emerge at the isostatic point. Hence, if there are any low-frequency modes away from jamming and if indeed their localization length is l, we should see this as the jamming point is approached. The aim of this paper is to investigate whether this is indeed the case.