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

Isostatic phase transition and instability in stiff granular materials

In this letter, structural rigidity concepts are used to understand the origin of instabilities in granular aggregates. It is shown that: a) The contact network of a noncohesive granular aggregate becomes exactly isostatic in the limit of large stiffness-to-load ratio. b) Isostaticity is responsible for the anomalously large susceptibility to perturbation of these systems, and c) The load-stress response function of granular materials is critical (power-law distributed) in the isostatic limit. Thus there is a phase transition in the limit of intinitely large stiffness, and the resulting isostatic phase is characterized by huge instability to perturbation.Comment: RevTeX, 4 pages w/eps figures [psfig]. To appear in Phys. Rev. Let

Strain versus stress in a model granular material: a Devil's staircase

The series of equilibrium states reached by disordered packings of rigid, frictionless discs in two dimensions, under gradually varying stress, are studied by numerical simulations. Statistical properties of trajectories in configuration space are found to be independent of specific assumptions ruling granular dynamics, and determined by geometry only. A monotonic increase in some macroscopic loading parameter causes a discrete sequence of rearrangements. For a biaxial compression, we show that, due to the statistical importance of such events of large magnitudes, the dependence of the resulting strain on stress direction is a Levy flight in the thermodynamic limit.Comment: REVTeX, 4 pages, 5 included PostScript figures. New version altered throughout text, very close to published pape

A Ball in a Groove

We study the static equilibrium of an elastic sphere held in a rigid groove by gravity and frictional contacts, as determined by contact mechanics. As a function of the opening angle of the groove and the tilt of the groove with respect to the vertical, we identify two regimes of static equilibrium for the ball. In the first of these, at large opening angle or low tilt, the ball rolls at both contacts as it is loaded. This is an analog of the "elastic" regime in the mechanics of granular media. At smaller opening angles or larger tilts, the ball rolls at one contact and slides at the other as it is loaded, analogously with the "plastic" regime in the mechanics of granular media. In the elastic regime, the stress indeterminacy is resolved by the underlying kinetics of the ball response to loading.Comment: RevTeX 3.0, 4 pages, 2 eps figures included with eps

Exactly solvable analogy of small-world networks

We present an exact description of a crossover between two different regimes of simple analogies of small-world networks. Each of the sites chosen with a probability $p$ from $n$ sites of an ordered system defined on a circle is connected to all other sites selected in such a way. Every link is of a unit length. Thus, while $p$ changes from 0 to 1, an averaged shortest distance between a pair of sites changes from $\bar{\ell} \sim n$ to $\bar{\ell} = 1$. We find the distribution of the shortest distances $P(\ell)$ and obtain a scaling form of $\bar{\ell}(p,n)$. In spite of the simplicity of the models under consideration, the results appear to be surprisingly close to those obtained numerically for usual small-world networks.Comment: 4 pages with 3 postscript figure

Internal states of model isotropic granular packings. I. Assembling process, geometry and contact networks

This is the first paper of a series of three, reporting on numerical simulation studies of geometric and mechanical properties of static assemblies of spherical beads under an isotropic pressure. Frictionless systems assemble in the unique random close packing (RCP) state in the low pressure limit if the compression process is fast enough, slower processes inducing traces of crystallization, and exhibit specific properties directly related to isostaticity of the force-carrying structure. The different structures of frictional packings assembled by various methods cannot be classified by the sole density. While lubricated systems approach RCP densities and coordination number z^*~=6 on the backbone in the rigid limit, an idealized "vibration" procedure results in equally dense configurations with z^*~=4.5. Near neighbor correlations on various scales are computed and compared to available laboratory data, although z^* values remain experimentally inaccessible. Low coordination packings have many rattlers (more than 10% of the grains carry no force), which should be accounted for on studying position correlations, and a small proportion of harmless "floppy modes" associated with divalent grains. Frictional packings, however slowly assembled under low pressure, retain a finite level of force indeterminacy, except in the limit of infinite friction.Comment: 29 pages. Published in Physical Review

Numerical Study of the Stress Response of Two-Dimensional Dense Granular Packings

We investigate the Green function of two-dimensional dense random packings of grains in order to discriminate between the different theories of stress transmission in granular materials. Our computer simulations allow for a detailed quantitative investigation of the dynamics which is difficult to obtain experimentally. We show that both hyperbolic and parabolic models of stress transmission fail to predict the correct stress distribution in the studied region of the parameters space. We demonstrate that the compressional and shear components of the stress compare very well with the predictions of isotropic elasticity for a wide range of pressures and porosities and for both frictional and frictionless packings. However, the states used in this study do not include the critical isostatic point for frictional particles, so that our results do not preclude the fact that corrections to elasticity may appear at the critical point of jamming, or for other sample preparation protocols, as discussed in the main text. We show that the agreement holds in the bulk of the packings as well as at the boundaries and we validate the linear dependence of the stress profile width with depth.Comment: 7 pages, 5 figure

Hopping Conductivity of a Nearly-1d Fractal: a Model for Conducting Polymers

We suggest treating a conducting network of oriented polymer chains as an anisotropic fractal whose dimensionality D=1+\epsilon is close to one. Percolation on such a fractal is studied within the real space renormalization group of Migdal and Kadanoff. We find that the threshold value and all the critical exponents are strongly nonanalytic functions of \epsilon as \epsilon tends to zero, e.g., the critical exponent of conductivity is \epsilon^{-2}\exp (-1-1/\epsilon). The distribution function for conductivity of finite samples at the percolation threshold is established. It is shown that the central body of the distribution is given by a universal scaling function and only the low-conductivity tail of distribution remains $\epsilon$-dependent. Variable range hopping conductivity in the polymer network is studied: both DC conductivity and AC conductivity in the multiple hopping regime are found to obey a quasi-1d Mott law. The present results are consistent with electrical properties of poorly conducting polymers.Comment: 27 pages, RevTeX, epsf, 5 .eps figures, to be published in Phys. Rev.

Complex temperatures zeroes of partition function in spin-glass models

An approximate method is proposed for investigating complex-temperature properties of real-dimensional spin-glass models. The method uses the complex-temperature data of the ferromagnetic model on the same lattice. The universality line in the complex-temperature space is obtained.Comment: latex, corrected some misprint

Stress in frictionless granular material: Adaptive Network Simulations

We present a minimalistic approach to simulations of force transmission through granular systems. We start from a configuration containing cohesive (tensile) contact forces and use an adaptive procedure to find the stable configuration with no tensile contact forces. The procedure works by sequentially removing and adding individual contacts between adjacent beads, while the bead positions are not modified. In a series of two-dimensional realizations, the resulting force networks are shown to satisfy a linear constraint among the three components of average stress, as anticipated by recent theories. The coefficients in the linear constraint remain nearly constant for a range of shear loadings up to about .6 of the normal loading. The spatial distribution of contact forces shows strong concentration along ``force chains". The probability of contact forces of magnitude f shows an exponential falloff with f. The response to a local perturbing force is concentrated along two characteristic rays directed downward and laterally.Comment: 8 pages, 8 figure