929 research outputs found
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
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
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/
Breathing mode for systems of interacting particles
We study the breathing mode in systems of trapped interacting particles. Our
approach, based on a dynamical ansatz in the first equation of the
Bogolyubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy allows us to tackle at
once a wide range of power law interactions and interaction strengths, at
linear and non linear levels. This both puts in a common framework various
results scattered in the literature, and by widely generalizing these,
emphasizes universal characters of this breathing mode. Our findings are
supported by direct numerical simulations.Comment: 4 pages, 4 figure
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
Large deviation techniques applied to systems with long-range interactions
We discuss a method to solve models with long-range interactions in the
microcanonical and canonical ensemble. The method closely follows the one
introduced by Ellis, Physica D 133, 106 (1999), which uses large deviation
techniques. We show how it can be adapted to obtain the solution of a large
class of simple models, which can show ensemble inequivalence. The model
Hamiltonian can have both discrete (Ising, Potts) and continuous (HMF, Free
Electron Laser) state variables. This latter extension gives access to the
comparison with dynamics and to the study of non-equilibri um effects. We treat
both infinite range and slowly decreasing interactions and, in particular, we
present the solution of the alpha-Ising model in one-dimension with
Ensemble inequivalence in systems with long-range interactions
Ensemble inequivalence has been observed in several systems. In particular it
has been recently shown that negative specific heat can arise in the
microcanonical ensemble in the thermodynamic limit for systems with long-range
interactions. We display a connection between such behaviour and a mean-field
like structure of the partition function. Since short-range models cannot
display this kind of behaviour, this strongly suggests that such systems are
necessarily non-mean field in the sense indicated here. We illustrate our
results showing an application to the Blume-Emery-Griffiths model. We further
show that a broad class of systems with non-integrable interactions are indeed
of mean-field type in the sense specified, so that they are expected to display
ensemble inequivalence as well as the peculiar behaviour described above in the
microcanonical ensemble.Comment: 12 pages, no figure
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