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.

Ensemble inequivalence: A formal approach

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 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: 4 pages, no figures, given at the NEXT2001 conference on non-extensive thermodynamic

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 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

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/

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 $0\leq\alpha<1$

Phase transitions of quasistationary states in the Hamiltonian Mean Field model

The out-of-equilibrium dynamics of the Hamiltonian Mean Field (HMF) model is studied in presence of an externally imposed magnetic field h. Lynden-Bell's theory of violent relaxation is revisited and shown to adequately capture the system dynamics, as revealed by direct Vlasov based numerical simulations in the limit of vanishing field. This includes the existence of an out-of-equilibrium phase transition separating magnetized and non magnetized phases. We also monitor the fluctuations in time of the magnetization, which allows us to elaborate on the choice of the correct order parameter when challenging the performance of Lynden-Bell's theory. The presence of the field h removes the phase transition, as it happens at equilibrium. Moreover, regions with negative susceptibility are numerically found to occur, in agreement with the predictions of the theory.Comment: 6 pages, 7 figure