arXiv subjects

Cristian F. Moukarzel

Publications and source records attributed to Cristian F. Moukarzel.

27 records · Page 2Linked to original sources

arXiv1998

Granular matter instability: A structural rigidity point of view

Stress paths in granular matter often suffer sudden large-scale rearrangements when the system is slightly perturbed, i.e. granular systems are unstable. We show in this paper that the observed instability is due to the minimally rigid, or isostatic, character of the system's contact network. It is first demonstrated that the contact network of a granular packing becomes isostatic (minimally rigid) in any dimension in the limit of large stiffness-to-load ratio. We next show that, in this isostatic limit, the load-stress response function becomes power-law distributed and takes exponentially large (growing as exp(H) where H is the system's height) positive and negative values. Large negative values of the load-stress response function imply instability, since only positive (compressive) stresses are allowed in non-cohesive granular packings. Thus there is an isostatic phase transition in the limit of large stiffness (or small load), and the resulting isostatic phase has an anomalously large susceptibility to perturbation.

cond-mat.soft↗

arXiv1998

Reply to the comment by Jacobs and Thorpe

Reply to a comment on "Infinite-Cluster geometry in central-force networks", PRL 78 (1997), 1480. A discussion about the order of the rigidity percolation transition.

cond-mat.stat-mech↗

arXiv1998

A fast algorithm for backbones

A matching algorithm for the identification of backbones in percolation problems is introduced. Using this procedure, percolation backbones are studied in two- to five-dimensional systems containing 1.7x10^7 sites, two orders of magnitude larger than was previously possible using burning algorithms.

cond-mat.stat-mech↗

arXiv1997

Complex-Field Zeros of the random-energy model

The complex-field zeros of the Random Energy Model are analytically determined. For T T_c a region is found which is free of zeros and is associated to the paramagnetic phase of the system. This region is separated from the former by a line of zeros. The real axis is found to be enclosed in a cloud of zeros in the whole frozen phase of the system.

cond-mat.dis-nn↗

arXiv1997

First-order rigidity transition on Bethe Lattices

Tree models for rigidity percolation are introduced and solved. A probability vector describes the propagation of rigidity outward from a rigid border. All components of this ``vector order parameter'' are singular at the same rigidity threshold, $p_c$. The infinite-cluster probability $P_{\infty}$ is usually first-order at $p_c$, but often behaves as $P_{\infty} \sim ΔP_{\infty} + (p-p_c)^{1/2}$, indicating critical fluctuations superimposed on a first order jump. Our tree models for rigidity are in qualitative disagreement with ``constraint counting'' mean field theories. In an important sub-class of tree models ``Bootstrap'' percolation and rigidity percolation are equivalent.

cond-mat.stat-mech↗

arXiv1997

Geometrical consequences of foam equilibrium

The equilibrium conditions impose nontrivial geometrical constraints on the configurations that a two-dimensional foam can attain. In the first place, the three centers of the films that converge to a vertex have to be on a line, i.e. all vertices are aligned. Moreover an equilibrated foam must admit a reciprocal figure. This means that it must be possible to find a set of points P_i on the plane, one per bubble, such that the segments (P_i P_j) are normal to the corresponding foam films. It is furthermore shown that these constraints are equivalent to the requirement that the foam be a Sectional Multiplicative Voronoi Partition (SMVP). A SMVP is a cut with a two-dimensional plane, of a three-dimensional Multiplicative Voronoi Partition. Thus given an arbitrary equilibrated foam, we can always find point-like sources (one per bubble) in three dimensions that reproduce this foam as a generalized Voronoi partition. These sources are the only degrees of freedom that we need in oder to fully describe the foam.

cond-mat.soft↗

arXiv1996

Stressed backbone and elasticity of random central-force systems

We use a new algorithm to find the stress-carrying backbone of ``generic'' site-diluted triangular lattices of up to 10^6 sites. Generic lattices can be made by randomly displacing the sites of a regular lattice. The percolation threshold is Pc=0.6975 +/- 0.0003, the correlation length exponent ν=1.16 +/- 0.03 and the fractal dimension of the backbone Db=1.78 +/- 0.02. The number of ``critical bonds'' (if you remove them rigidity is lost) on the backbone scales as L^{x}, with x=0.85 +/- 0.05. The Young's modulus is also calculated.

cond-mat.stat-mech↗

arXiv1996

Infinite-cluster geometry in central-force networks

We show that the infinite percolating cluster (with density P_inf) of central-force networks is composed of: a fractal stress-bearing backbone (Pb) and; rigid but unstressed ``dangling ends'' which occupy a finite volume-fraction of the lattice (Pd). Near the rigidity threshold pc, there is then a first-order transition in P_inf = Pd + Pb, while Pb is second-order with exponent Beta'. A new mean field theory shows Beta'(mf)=1/2, while simulations of triangular lattices give Beta'_tr = 0.255 +/- 0.03.

cond-mat.stat-mech↗

arXiv1996

An efficient algorithm for two-dimensional central-force rigidity

Given a structure made up of n sites connected by b bars, the problem of recognizing which subsets of sites form rigid units is not a trivial one, because of the non-local character of rigidity in central-force systems. Even though this is a very old problem of statics, no simple algorithms are available for it so the most usual approach has been to solve the elastic equations, which is very time-consuming for large systems. Recently an integer algorithm was proposed for this problem in two dimensions, using matching methods from graph theory and Laman's theorem for two-dimensional graphs. The method is relatively simple, but its time complexity grows as $n^2$ in the worst case, and almost as fast on practical cases, so that an improvement is highly desirable. I describe here a further elaboration of that procedure, which relies upon the description of the system as a collection of rigid bodies connected by bars, instead of sites connected by bars. Sets of rigidly connected objects are replaced by a unique body, and this is done recursively as more rigid connections between bodies are discovered at larger scales. As a consequence of this ``rescaling transformation'', our algorithm has much improved average behavior, even when its worst-case complexity remains $n^2$. The time complexity of the body-bar algorithm is found to scale as $n^{1.12}$ for the randomly diluted triangular lattice, while the original site-bar version scales as $n^{1.9}$ on the same problem.

physics.comp-ph↗