567 research outputs found
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
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
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/
Algebraic damping in the one-dimensional Vlasov equation
We investigate the asymptotic behavior of a perturbation around a spatially
non homogeneous stable stationary state of a one-dimensional Vlasov equation.
Under general hypotheses, after transient exponential Landau damping, a
perturbation evolving according to the linearized Vlasov equation decays
algebraically with the exponent -2 and a well defined frequency. The
theoretical results are successfully tested against numerical -body
simulations, corresponding to the full Vlasov dynamics in the large limit,
in the case of the Hamiltonian mean-field model. For this purpose, we use a
weighted particles code, which allows us to reduce finite size fluctuations and
to observe the asymptotic decay in the -body simulations.Comment: 26 pages, 8 figures; text slightly modified, references added, typos
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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
Linear theory and violent relaxation in long-range systems: a test case
In this article, several aspects of the dynamics of a toy model for longrange
Hamiltonian systems are tackled focusing on linearly unstable unmagnetized
(i.e. force-free) cold equilibria states of the Hamiltonian Mean Field (HMF).
For special cases, exact finite-N linear growth rates have been exhibited,
including, in some spatially inhomogeneous case, finite-N corrections. A random
matrix approach is then proposed to estimate the finite-N growth rate for some
random initial states. Within the continuous, , approach,
the growth rates are finally derived without restricting to spatially
homogeneous cases. All the numerical simulations show a very good agreement
with the different theoretical predictions. Then, these linear results are used
to discuss the large-time nonlinear evolution. A simple criterion is proposed
to measure the ability of the system to undergo a violent relaxation that
transports it in the vicinity of the equilibrium state within some linear
e-folding times
Inhomogeneous Quasi-stationary States in a Mean-field Model with Repulsive Cosine Interactions
The system of N particles moving on a circle and interacting via a global
repulsive cosine interaction is well known to display spatially inhomogeneous
structures of extraordinary stability starting from certain low energy initial
conditions. The object of this paper is to show in a detailed manner how these
structures arise and to explain their stability. By a convenient canonical
transformation we rewrite the Hamiltonian in such a way that fast and slow
variables are singled out and the canonical coordinates of a collective mode
are naturally introduced. If, initially, enough energy is put in this mode, its
decay can be extremely slow. However, both analytical arguments and numerical
simulations suggest that these structures eventually decay to the spatially
uniform equilibrium state, although this can happen on impressively long time
scales. Finally, we heuristically introduce a one-particle time dependent
Hamiltonian that well reproduces most of the observed phenomenology.Comment: to be published in J. Phys.
The cavity method for large deviations
A method is introduced for studying large deviations in the context of
statistical physics of disordered systems. The approach, based on an extension
of the cavity method to atypical realizations of the quenched disorder, allows
us to compute exponentially small probabilities (rate functions) over different
classes of random graphs. It is illustrated with two combinatorial optimization
problems, the vertex-cover and coloring problems, for which the presence of
replica symmetry breaking phases is taken into account. Applications include
the analysis of models on adaptive graph structures.Comment: 18 pages, 7 figure
Lyapunov exponent of many-particle systems: testing the stochastic approach
The stochastic approach to the determination of the largest Lyapunov exponent
of a many-particle system is tested in the so-called mean-field
XY-Hamiltonians. In weakly chaotic regimes, the stochastic approach relates the
Lyapunov exponent to a few statistical properties of the Hessian matrix of the
interaction, which can be calculated as suitable thermal averages. We have
verified that there is a satisfactory quantitative agreement between theory and
simulations in the disordered phases of the XY models, either with attractive
or repulsive interactions. Part of the success of the theory is due to the
possibility of predicting the shape of the required correlation functions,
because this permits the calculation of correlation times as thermal averages.Comment: 11 pages including 6 figure
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