46 research outputs found
Strength Reduction in Electrical and Elastic Networks
Particular aspects of problems ranging from dielectric breakdown to metal
insu- lator transition can be studied using electrical o elastic networks. We
present an expression for the mean breakdown strength of such networks.First,
we intro- duce a method to evaluate the redistribution of current due to the
removal of a finite number of elements from a hyper-cubic network of
conducatances.It is used to determine the reduction of breakdown strength due
to a fracture of size .Numerical analysis is used to show that the
analogous reduction due to random removal of elements from electrical and
elastic networks follow a similar form.One possible application, namely the use
of bone density as a diagnostic tools for osteorosporosis,is discussed.Comment: one compressed file includes: 9 PostScrpt figures and a text fil
Structural Properties of Self-Attracting Walks
Self-attracting walks (SATW) with attractive interaction u > 0 display a
swelling-collapse transition at a critical u_{\mathrm{c}} for dimensions d >=
2, analogous to the \Theta transition of polymers. We are interested in the
structure of the clusters generated by SATW below u_{\mathrm{c}} (swollen
walk), above u_{\mathrm{c}} (collapsed walk), and at u_{\mathrm{c}}, which can
be characterized by the fractal dimensions of the clusters d_{\mathrm{f}} and
their interface d_{\mathrm{I}}. Using scaling arguments and Monte Carlo
simulations, we find that for u<u_{\mathrm{c}}, the structures are in the
universality class of clusters generated by simple random walks. For
u>u_{\mathrm{c}}, the clusters are compact, i.e. d_{\mathrm{f}}=d and
d_{\mathrm{I}}=d-1. At u_{\mathrm{c}}, the SATW is in a new universality class.
The clusters are compact in both d=2 and d=3, but their interface is fractal:
d_{\mathrm{I}}=1.50\pm0.01 and 2.73\pm0.03 in d=2 and d=3, respectively. In
d=1, where the walk is collapsed for all u and no swelling-collapse transition
exists, we derive analytical expressions for the average number of visited
sites and the mean time to visit S sites.Comment: 15 pages, 8 postscript 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
Using Nonlinear Response to Estimate the Strength of an Elastic Network
Disordered networks of fragile elastic elements have been proposed as a model
of inner porous regions of large bones [Gunaratne et.al., cond-mat/0009221,
http://xyz.lanl.gov]. It is shown that the ratio of responses of such
a network to static and periodic strain can be used to estimate its ultimate
(or breaking) stress. Since bone fracture in older adults results from the
weakening of porous bone, we discuss the possibility of using as a
non-invasive diagnostic of osteoporotic bone.Comment: 4 pages, 4 figure
On the study of jamming percolation
We investigate kinetically constrained models of glassy transitions, and
determine which model characteristics are crucial in allowing a rigorous proof
that such models have discontinuous transitions with faster than power law
diverging length and time scales. The models we investigate have constraints
similar to that of the knights model, introduced by Toninelli, Biroli, and
Fisher (TBF), but differing neighbor relations. We find that such knights-like
models, otherwise known as models of jamming percolation, need a ``No Parallel
Crossing'' rule for the TBF proof of a glassy transition to be valid.
Furthermore, most knight-like models fail a ``No Perpendicular Crossing''
requirement, and thus need modification to be made rigorous. We also show how
the ``No Parallel Crossing'' requirement can be used to evaluate the provable
glassiness of other correlated percolation models, by looking at models with
more stable directions than the knights model. Finally, we show that the TBF
proof does not generalize in any straightforward fashion for three-dimensional
versions of the knights-like models.Comment: 13 pages, 18 figures; Spiral model does satisfy property
Phase diagram of an Ising model with long-range frustrating interactions: a theoretical analysis
We present a theoretical study of the phase diagram of a frustrated Ising
model with nearest-neighbor ferromagnetic interactions and long-range
(Coulombic) antiferromagnetic interactions. For nonzero frustration, long-range
ferromagnetic order is forbidden, and the ground-state of the system consists
of phases characterized by periodically modulated structures. At finite
temperatures, the phase diagram is calculated within the mean-field
approximation. Below the transition line that separates the disordered and the
ordered phases, the frustration-temperature phase diagram displays an infinite
number of ``flowers'', each flower being made by an infinite number of
modulated phases generated by structure combination branching processes. The
specificities introduced by the long-range nature of the frustrating
interaction and the limitation of the mean-field approach are finally
discussed.Comment: 32 pages, 7 figure
Percolation in three-dimensional random field Ising magnets
The structure of the three-dimensional random field Ising magnet is studied
by ground state calculations. We investigate the percolation of the minority
spin orientation in the paramagnetic phase above the bulk phase transition,
located at [Delta/J]_c ~= 2.27, where Delta is the standard deviation of the
Gaussian random fields (J=1). With an external field H there is a disorder
strength dependent critical field +/- H_c(Delta) for the down (or up) spin
spanning. The percolation transition is in the standard percolation
universality class. H_c ~ (Delta - Delta_p)^{delta}, where Delta_p = 2.43 +/-
0.01 and delta = 1.31 +/- 0.03, implying a critical line for Delta_c < Delta <=
Delta_p. When, with zero external field, Delta is decreased from a large value
there is a transition from the simultaneous up and down spin spanning, with
probability Pi_{uparrow downarrow} = 1.00 to Pi_{uparrow downarrow} = 0. This
is located at Delta = 2.32 +/- 0.01, i.e., above Delta_c. The spanning cluster
has the fractal dimension of standard percolation D_f = 2.53 at H = H_c(Delta).
We provide evidence that this is asymptotically true even at H=0 for Delta_c <
Delta <= Delta_p beyond a crossover scale that diverges as Delta_c is
approached from above. Percolation implies extra finite size effects in the
ground states of the 3D RFIM.Comment: replaced with version to appear in Physical Review
On the freezing of variables in random constraint satisfaction problems
The set of solutions of random constraint satisfaction problems (zero energy
groundstates of mean-field diluted spin glasses) undergoes several structural
phase transitions as the amount of constraints is increased. This set first
breaks down into a large number of well separated clusters. At the freezing
transition, which is in general distinct from the clustering one, some
variables (spins) take the same value in all solutions of a given cluster. In
this paper we study the critical behavior around the freezing transition, which
appears in the unfrozen phase as the divergence of the sizes of the
rearrangements induced in response to the modification of a variable. The
formalism is developed on generic constraint satisfaction problems and applied
in particular to the random satisfiability of boolean formulas and to the
coloring of random graphs. The computation is first performed in random tree
ensembles, for which we underline a connection with percolation models and with
the reconstruction problem of information theory. The validity of these results
for the original random ensembles is then discussed in the framework of the
cavity method.Comment: 32 pages, 7 figure
Effects of Pore Walls and Randomness on Phase Transitions in Porous Media
We study spin models within the mean field approximation to elucidate the
topology of the phase diagrams of systems modeling the liquid-vapor transition
and the separation of He--He mixtures in periodic porous media. These
topologies are found to be identical to those of the corresponding random field
and random anisotropy spin systems with a bimodal distribution of the
randomness. Our results suggest that the presence of walls (periodic or
otherwise) are a key factor determining the nature of the phase diagram in
porous media.Comment: REVTeX, 11 eps figures, to appear in Phys. Rev.
Rings and rigidity transitions in network glasses
Three elastic phases of covalent networks, (I) floppy, (II) isostatically
rigid and (III) stressed-rigid have now been identified in glasses at specific
degrees of cross-linking (or chemical composition) both in theory and
experiments. Here we use size-increasing cluster combinatorics and constraint
counting algorithms to study analytically possible consequences of
self-organization. In the presence of small rings that can be locally I, II or
III, we obtain two transitions instead of the previously reported single
percolative transition at the mean coordination number , one from a
floppy to an isostatic rigid phase, and a second one from an isostatic to a
stressed rigid phase. The width of the intermediate phase and the
order of the phase transitions depend on the nature of medium range order
(relative ring fractions). We compare the results to the Group IV
chalcogenides, such as Ge-Se and Si-Se, for which evidence of an intermediate
phase has been obtained, and for which estimates of ring fractions can be made
from structures of high T crystalline phases.Comment: 29 pages, revtex, 7 eps figure