910 research outputs found
Retrieving information from a noisy "knowledge network"
We address the problem of retrieving information from a noisy version of the
``knowledge networks'' introduced by Maslov and Zhang. We map this problem onto
a disordered statistical mechanics model, which opens the door to many
analytical and numerical approaches. We give the replica symmetric solution,
compare with numerical simulations, and finally discuss an application to real
datas from the United States Senate.Comment: 10 pages, 4 figures. Writing of the last section improved; version
accepted in JSTA
Hierarchical models of rigidity percolation
We introduce models of generic rigidity percolation in two dimensions on
hierarchical networks, and solve them exactly by means of a renormalization
transformation. We then study how the possibility for the network to self
organize in order to avoid stressed bonds may change the phase diagram. In
contrast to what happens on random graphs and in some recent numerical studies
at zero temperature, we do not find a true intermediate phase separating the
usual rigid and floppy ones.Comment: 20 pages, 8 figures. Figures improved, references added, small
modifications. Accepted in Phys. Rev.
Microcanonical Analysis of Exactness of the Mean-Field Theory in Long-Range Interacting Systems
Classical spin systems with nonadditive long-range interactions are studied
in the microcanonical ensemble. It is expected that the entropy of such a
system is identical to that of the corresponding mean-field model, which is
called "exactness of the mean-field theory". It is found out that this
expectation is not necessarily true if the microcanonical ensemble is not
equivalent to the canonical ensemble in the mean-field model. Moreover,
necessary and sufficient conditions for exactness of the mean-field theory are
obtained. These conditions are investigated for two concrete models, the
\alpha-Potts model with annealed vacancies and the \alpha-Potts model with
invisible states.Comment: 23 pages, to appear in J. Stat. Phy
Oscillating elastic defects: competition and frustration
We consider a dynamical generalization of the Eshelby problem: the strain
profile due to an inclusion or "defect" in an isotropic elastic medium. We show
that the higher the oscillation frequency of the defect, the more localized is
the strain field around the defect. We then demonstrate that the qualitative
nature of the interaction between two defects is strongly dependent on
separation, frequency and direction, changing from "ferromagnetic" to
"antiferromagnetic" like behavior. We generalize to a finite density of defects
and show that the interactions in assemblies of defects can be mapped to XY
spin-like models, and describe implications for frustration and
frequency-driven pattern transitions.Comment: 4 pages, 5 figure
Ensemble Inequivalence in Mean-field Models of Magnetism
Mean-field models, while they can be cast into an {\it extensive}
thermodynamic formalism, are inherently {\it non additive}. This is the basic
feature which leads to {\it ensemble inequivalence} in these models. In this
paper we study the global phase diagram of the infinite range
Blume-Emery-Griffiths model both in the {\it canonical} and in the {\it
microcanonical} ensembles. The microcanonical solution is obtained both by
direct state counting and by the application of large deviation theory. The
canonical phase diagram has first order and continuous transition lines
separated by a tricritical point. We find that below the tricritical point,
when the canonical transition is first order, the phase diagrams of the two
ensembles disagree. In this region the microcanonical ensemble exhibits energy
ranges with negative specific heat and temperature jumps at transition
energies. These two features are discussed in a general context and the
appropriate Maxwell constructions are introduced. Some preliminary extensions
of these results to weakly decaying nonintegrable interactions are presented.Comment: Chapter of the forthcoming "Lecture Notes in Physics" volume:
``Dynamics and Thermodynamics of Systems with Long Range Interactions'', T.
Dauxois, S. Ruffo, E. Arimondo, M. Wilkens Eds., Lecture Notes in Physics
Vol. 602, Springer (2002). (see http://link.springer.de/series/lnpp/
Exactly solvable models of adaptive networks
A satisfiability (SAT-UNSAT) transition takes place for many optimization
problems when the number of constraints, graphically represented by links
between variables nodes, is brought above some threshold. If the network of
constraints is allowed to adapt by redistributing its links, the SAT-UNSAT
transition may be delayed and preceded by an intermediate phase where the
structure self-organizes to satisfy the constraints. We present an analytic
approach, based on the recently introduced cavity method for large deviations,
which exactly describes the two phase transitions delimiting this adaptive
intermediate phase. We give explicit results for random bond models subject to
the connectivity or rigidity percolation transitions, and compare them with
numerical simulations.Comment: 4 pages, 4 figure
Lyapunov exponents as a dynamical indicator of a phase transition
We study analytically the behavior of the largest Lyapunov exponent
for a one-dimensional chain of coupled nonlinear oscillators, by
combining the transfer integral method and a Riemannian geometry approach. We
apply the results to a simple model, proposed for the DNA denaturation, which
emphasizes a first order-like or second order phase transition depending on the
ratio of two length scales: this is an excellent model to characterize
as a dynamical indicator close to a phase transition.Comment: 8 Pages, 3 Figure
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