Elastic collapse in disordered isostatic networks

Isostatic networks are minimally rigid and therefore have, generically, nonzero elastic moduli. Regular isostatic networks have finite moduli in the limit of large sizes. However, numerical simulations show that all elastic moduli of geometrically disordered isostatic networks go to zero with system size. This holds true for positional as well as for topological disorder. In most cases, elastic moduli decrease as inverse power-laws of system size. On directed isostatic networks, however, of which the square and cubic lattices are particular cases, the decrease of the moduli is exponential with size. For these, the observed elastic weakening can be quantitatively described in terms of the multiplicative growth of stresses with system size, giving rise to bulk and shear moduli of order exp{-bL}. The case of sphere packings, which only accept compressive contact forces, is considered separately. It is argued that these have a finite bulk modulus because of specific correlations in contact disorder, introduced by the constraint of compressivity. We discuss why their shear modulus, nevertheless, is again zero for large sizes. A quantitative model is proposed that describes the numerically measured shear modulus, both as a function of the loading angle and system size. In all cases, if a density p>0 of overconstraints is present, as when a packing is deformed by compression, or when a glass is outside its isostatic composition window, all asymptotic moduli become finite. For square networks with periodic boundary conditions, these are of order sqrt{p}. For directed networks, elastic moduli are of order exp{-c/p}, indicating the existence of an "isostatic length scale" of order 1/p.Comment: 6 pages, 6 figues, to appear in Europhysics Letter

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.Comment: 1 page revTe

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.Comment: 8 pages, 6 .eps figures. Uses epsfig and ijmpc.sty (included). To appear in Int. J. Mod. Phys.

Isostaticity in two dimensional pile of rigid disks

We study the static structure of piles made of polydisperse disks in the rigid limit with and without friction using molecular dynamic simulations for various elasticities of the disks and pile preparation procedures. The coordination numbers are calculated to examine the isostaticity of the pile structure. For the frictionless pile, it is demonstrated that the coordination number converges to 4 in the rigid limit, which implies that the structure of rigid disk pile is isostatic. On the other hand, for the frictional case with the infinite friction constant, the coordination number depends on the preparation procedure of the pile, but we find that the structure becomes very close to isostatic with the coordination number close to 3 in the rigid limit when the pile is formed through the process that tends to make a pile of random configuration.Comment: 3 pages, 3 figures, Submitted to J. Phys. Soc. Jp

Isostaticity and Mechanical Response of Two-Dimensional Granular Piles

We numerically study the static structure and the mechanical response of two-dimensional granular piles. The piles consist of polydisperse disks with and without friction. Special attention is paid for the rigid grain limit by examining the systems with various disk elasticities. It is shown that the static pile structure of frictionless disks becomes isostatic in the rigid limit, while the isostaticity of frictional pile depends on the pile forming procedure, but in the case of the infinite friction is effective, the structure becomes very close to isostatic in the rigid limit. The mechanical response of the piles are studied by infinitesimally displacing one of the disks at the bottom. It is shown that the total amount of the displacement in the pile caused by the perturbation diverges in the case of frictionless pile as it becomes isostatic, while the response remains finite for the frictional pile. In the frictionless isostatic pile, the displacement response in each sample behaves rather complicated way, but its average shows wave like propagation.Comment: 23 pages, 10 figure

Combinatorial models of rigidity and renormalization

We first introduce the percolation problems associated with the graph theoretical concepts of $(k,l)$-sparsity, and make contact with the physical concepts of ordinary and rigidity percolation. We then devise a renormalization transformation for $(k,l)$-percolation problems, and investigate its domain of validity. In particular, we show that it allows an exact solution of $(k,l)$-percolation problems on hierarchical graphs, for $k\leq l<2k$. We introduce and solve by renormalization such a model, which has the interesting feature of showing both ordinary percolation and rigidity percolation phase transitions, depending on the values of the parameters.Comment: 22 pages, 6 figure

First-order transition in small-world networks

The small-world transition is a first-order transition at zero density $p$ of shortcuts, whereby the normalized shortest-path distance undergoes a discontinuity in the thermodynamic limit. On finite systems the apparent transition is shifted by $\Delta p \sim L^{-d}$. Equivalently a ``persistence size'' $L^* \sim p^{-1/d}$ can be defined in connection with finite-size effects. Assuming $L^* \sim p^{-\tau}$, simple rescaling arguments imply that $\tau=1/d$. We confirm this result by extensive numerical simulation in one to four dimensions, and argue that $\tau=1/d$ implies that this transition is first-order.Comment: 4 pages, 3 figures, To appear in Europhysics Letter

Statistical Laws and Mechanics of Voronoi Random Lattices

We investigate random lattices where the connectivities are determined by the Voronoi construction, while the location of the points are the dynamic degrees of freedom. The Voronoi random lattices with an associated energy are immersed in a heat bath and investigated using a Monte Carlo simulation algorithm. In thermodynamic equilibrium we measure coordination number distributions and test the Aboav-Weaire and Lewis laws.Comment: 14 pages (figures not included), LaTeX, HLRZ-26/9

Floppy modes and the free energy: Rigidity and connectivity percolation on Bethe Lattices

We show that negative of the number of floppy modes behaves as a free energy for both connectivity and rigidity percolation, and we illustrate this result using Bethe lattices. The rigidity transition on Bethe lattices is found to be first order at a bond concentration close to that predicted by Maxwell constraint counting. We calculate the probability of a bond being on the infinite cluster and also on the overconstrained part of the infinite cluster, and show how a specific heat can be defined as the second derivative of the free energy. We demonstrate that the Bethe lattice solution is equivalent to that of the random bond model, where points are joined randomly (with equal probability at all length scales) to have a given coordination, and then subsequently bonds are randomly removed.Comment: RevTeX 11 pages + epsfig embedded figures. Submitted to Phys. Rev.

Yard-Sale exchange on networks: Wealth sharing and wealth appropriation

Yard-Sale (YS) is a stochastic multiplicative wealth-exchange model with two phases: a stable one where wealth is shared, and an unstable one where wealth condenses onto one agent. YS is here studied numerically on 1d rings, 2d square lattices, and random graphs with variable average coordination, comparing its properties with those in mean field (MF). Equilibrium properties in the stable phase are almost unaffected by the introduction of a network. Measurement of decorrelation times in the stable phase allow us to determine the critical interface with very good precision, and it turns out to be the same, for all networks analyzed, as the one that can be analytically derived in MF. In the unstable phase, on the other hand, dynamical as well as asymptotic properties are strongly network-dependent. Wealth no longer condenses on a single agent, as in MF, but onto an extensive set of agents, the properties of which depend on the network. Connections with previous studies of coalescence of immobile reactants are discussed, and their analytic predictions are successfully compared with our numerical results.Comment: 10 pages, 7 figures. Submitted to JSTA