7 research outputs found
Tail universalities in rank distributions as an algebraic problem: the beta-like function
Although power laws of the Zipf type have been used by many workers to fit
rank distributions in different fields like in economy, geophysics, genetics,
soft-matter, networks etc., these fits usually fail at the tails. Some
distributions have been proposed to solve the problem, but unfortunately they
do not fit at the same time both ending tails. We show that many different data
in rank laws, like in granular materials, codons, author impact in scientific
journal, etc. are very well fitted by a beta-like function. Then we propose
that such universality is due to the fact that a system made from many
subsystems or choices, imply stretched exponential frequency-rank functions
which qualitatively and quantitatively can be fitted with the proposed
beta-like function distribution in the limit of many random variables. We prove
this by transforming the problem into an algebraic one: finding the rank of
successive products of a given set of numbers
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
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