Eur. Phys. J. E 22, 235–240 (2007) DOI: 10.1140/epje/e2007-00033-x T HE EUROPEAN P HYSICAL JOURNAL E Correlations and aggregate statistics in granular packs T. Aste a and T. Di Matteo Department of Applied Mathematics, Research School of Physical Sciences and Engineering, The Australian National University, 0200 Canberra, Australia Received 4 January 2007 Published online: 11 April 2007 – c  EDP Sciences, Societ`a Italiana di Fisica, Springer-Verlag 2007 Abstract. We study how the aggregate statistical properties for density fluctuations in granular aggregates scale with the sample size and how such a scaling is associated with the correlations between grains. Correlations are studied both between grain positions and between Vorono¨ ı cell volumes, showing distinct behaviors and properties. A non-linear scaling in the aggregate volume fluctuations as function of the sample size is discovered and the connection between such anomalous scaling and correlations is explained. It emerges that volume fluctuations might be described by means of a single universal equation for all samples at all cluster sizes. PACS. 45.70.-n Granular systems – 45.70.Cc Static sandpiles; Granular compaction – 81.05.Rm Porous materials; Granular materials The study of volume fluctuations in large granular ag- gregates could be the way to validate some statisti- cal mechanics theoretical descriptions of granular matter [1–11], and therefore studies in this domain have recently attracted a large interest from the scientific community [12,9,13]. In a recent paper [13] we have shown that, at local level, the fluctuations of the Vorono¨ ı cell volumes can be predicted with a remarkable precision by using a distribution function – with no adjustable parameters – which reveals a universal dependence on the packing fraction of the sample. On the other hand, in most ex- periments the only measurable parameters are the global volume fluctuations on the whole sample. The general- ization to large aggregates of the local theoretical distri- bution for the Vorono¨ ı cells depends on the presence of correlations. More generally, to understand correlations is essential in any statistical mechanics approach of any given system because the correlation length establishes the level of detail at which the system description must be tuned. In this paper we study correlations and their effects on the scaling laws of the volume fluctuations. The structure of granular materials is disordered but not random. These systems present a spontaneous or- ganization which can be measured both at local and at global level. Such an organization is the consequence of several different mechanisms which are both physical (e.g. mechanical stability) and geometrical (e.g. close packing configurations). One of the key-questions, which we ad- dress in this paper, is to identify the length scale at which structural organization is present. In other words, we want to identify the length-scale above which correlations are a e-mail: tomaso.aste@anu.edu.au smearing out and average quantities become the relevant control parameters. In this paper we tackle this question by looking at changes in the statistical properties as function of the size of the packing-aggregates. Specifically, we compute two different kinds of correlations: (1) correlations between grain positions; (2) correlations between local Vorono¨ ı cells constructed from the grain centers. In particular we calculate the first kind of correlation by dividing the sam- ple in cubic grids of different sizes and comparing the occu- pation numbers in adjacent grid-units. The second kind of correlation is computed in two different ways: (a) a direct method is used for adjacent Vorono¨ ı cells; (b) an indirect method is used at larger scales where the average corre- lation between couples in clusters is computed from the dependence of the aggregated distribution of the Vorono¨ ı volumes on the cluster sizes. Such analysis concerns six experimental samples made of monosized acrylic beads prepared in a cylindrical con- tainer and having packing fractions ranging between 0.58 to 0.64 [14–17]. Such a dataset was acquired by means of X-ray Computed Tomography and it records the positions of more than 385 000 sphere centers. The precision on the coordinates is better than 0.1 % of the sphere-diameters and the sphere polydispersity is within 2 %. In this pa- per we refer to these samples with labels A, B, C, D, E and F which are the same used to respectively identify the samples in the previous papers [14–17] where other kinds of structural analysis were performed. The present investigations are performed over an internal region (G) at 4 sphere-diameters away from the sample boundaries. (Spheres outside G are considered when computing the