206 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
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
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.
Combinatorial models of rigidity and renormalization
We first introduce the percolation problems associated with the graph
theoretical concepts of -sparsity, and make contact with the physical
concepts of ordinary and rigidity percolation. We then devise a renormalization
transformation for -percolation problems, and investigate its domain of
validity. In particular, we show that it allows an exact solution of
-percolation problems on hierarchical graphs, for . 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 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
Spreading and shortest paths in systems with sparse long-range connections
Spreading according to simple rules (e.g. of fire or diseases), and
shortest-path distances are studied on d-dimensional systems with a small
density p per site of long-range connections (``Small-World'' lattices). The
volume V(t) covered by the spreading quantity on an infinite system is exactly
calculated in all dimensions. We find that V(t) grows initially as t^d/d for
t>t^*$,
generalizing a previous result in one dimension. Using the properties of V(t),
the average shortest-path distance \ell(r) can be calculated as a function of
Euclidean distance r. It is found that
\ell(r) = r for r<r_c=(2p \Gamma_d (d-1)!)^{-1/d} log(2p \Gamma_d L^d), and
\ell(r) = r_c for r>r_c.
The characteristic length r_c, which governs the behavior of shortest-path
lengths, diverges with system size for all p>0. Therefore the mean separation s
\sim p^{-1/d} between shortcut-ends is not a relevant internal length-scale for
shortest-path lengths. We notice however that the globally averaged
shortest-path length, divided by L, is a function of L/s only.Comment: 4 pages, 1 eps fig. Uses psfi
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
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
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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