122 research outputs found
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
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
First-order transition in small-world networks
The small-world transition is a first-order transition at zero density of
shortcuts, whereby the normalized shortest-path distance undergoes a
discontinuity in the thermodynamic limit. On finite systems the apparent
transition is shifted by . Equivalently a ``persistence
size'' can be defined in connection with finite-size
effects. Assuming , simple rescaling arguments imply that
. We confirm this result by extensive numerical simulation in one to
four dimensions, and argue that implies that this transition is
first-order.Comment: 4 pages, 3 figures, To appear in Europhysics Letter
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
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
A mean field description of jamming in non-cohesive frictionless particulate systems
A theory for kinetic arrest in isotropic systems of repulsive,
radially-interacting particles is presented that predicts exponents for the
scaling of various macroscopic quantities near the rigidity transition that are
in agreement with simulations, including the non-trivial shear exponent. Both
statics and dynamics are treated in a simplified, one-particle level
description, and coupled via the assumption that kinetic arrest occurs on the
boundary between mechanically stable and unstable regions of the static
parameter diagram. This suggests the arrested states observed in simulations
are at (or near) an elastic buckling transition. Some additional numerical
evidence to confirm the scaling of microscopic quantities is also provided.Comment: 9 pages, 3 figs; additional clarification of different elastic moduli
exponents, plus typo fix. To appear in PR
Rigidity percolation in a field
Rigidity Percolation with g degrees of freedom per site is analyzed on
randomly diluted Erdos-Renyi graphs with average connectivity gamma, in the
presence of a field h. In the (gamma,h) plane, the rigid and flexible phases
are separated by a line of first-order transitions whose location is determined
exactly. This line ends at a critical point with classical critical exponents.
Analytic expressions are given for the densities n_f of uncanceled degrees of
freedom and gamma_r of redundant bonds. Upon crossing the coexistence line, n_f
and gamma_r are continuous, although their first derivatives are discontinuous.
We extend, for the case of nonzero field, a recently proposed hypothesis,
namely that the density of uncanceled degrees of freedom is a ``free energy''
for Rigidity Percolation. Analytic expressions are obtained for the energy,
entropy, and specific heat. Some analogies with a liquid-vapor transition are
discussed. Particularizing to zero field, we find that the existence of a
(g+1)-core is a necessary condition for rigidity percolation with g degrees of
freedom. At the transition point gamma_c, Maxwell counting of degrees of
freedom is exact on the rigid cluster and on the (g+1)-rigid-core, i.e. the
average coordination of these subgraphs is exactly 2g, although gamma_r, the
average coordination of the whole system, is smaller than 2g. gamma_c is found
to converge to 2g for large g, i.e. in this limit Maxwell counting is exact
globally as well. This paper is dedicated to Dietrich Stauffer, on the occasion
of his 60th birthday.Comment: RevTeX4, psfig, 16 pages. Equation numbering corrected. Minor typos
correcte
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