Coexisting crystal and liquid-like properties in a 2D long-range self-consistent model.

A two-dimensional class of mean-field models serving as a minimal frame to study long-range interaction in two space dimensions is considered. In the case of an anisotropic mixed attractive-repulsive interaction, an initially spatially homogeneous cold fluid is dynamically unstable and evolves towards a quasi-stationary state in which the less energetic particles get trapped into clusters forming a Bravais-like lattice, mimicking a crystalline state. Superimposed to this, one observes in symplectic numerical simulations a flux of slightly more energetic particles channeling through this crystalline background. The resultant system combines the rigidity features of a solid, as particles from a displaced core are shown to snap back into place after a transient, and the dynamical diffusive features of a liquid for the fraction of channeling and free particles. The combination of solid and liquid properties is numerically observed here within the classical context. The quantum transposition of the model may be experimentally reached using the latest ultracold atoms techniques to generate long-range interactions.

Truncated Lévy flights and weak ergodicity breaking in the Hamiltonian mean-field model.

The dynamics of the Hamiltonian mean-field model is studied in the context of continuous-time random walks. We show that the sojourn times in cells in the momentum space are well described by a one-sided truncated Lévy distribution. Consequently, the system is nonergodic for long observation times that diverge with the number of particles. Ergodicity is attained only after very long times both at thermodynamic equilibrium and at quasistationary out-of-equilibrium states.

Scaling of the dynamics of homogeneous states of one-dimensional long-range interacting systems.

Quasistationary states of long-range interacting systems have been studied at length over the last 15 years. It is known that the collisional terms of the Balescu-Lenard and Landau equations vanish for one-dimensional systems in homogeneous states, thus requiring a new kinetic equation with a proper dependence on the number of particles. Here we show that the scalings discussed in the literature are mainly due either to small size effects or the use of unsuitable variables to describe the dynamics. The scaling obtained from both simulations and theoretical considerations is proportional to the square of the number of particles, and a general form for the kinetic equation valid for the homogeneous regime is obtained. Numerical evidence is given for the Hamiltonian mean field and ring models, and a kinetic equation valid for the homogeneous state is obtained for the former system.

Dynamics and physical interpretation of quasistationary states in systems with long-range interactions.

The time evolution of the one-particle distribution function of an N-particle classical Hamiltonian system with long-range interactions satisfies the Vlasov equation in the limit of infinite N. In this paper we present a new derivation of this result using a different approach allowing a discussion of the role of interparticle correlations on the system dynamics. Otherwise for finite N collisional corrections must be introduced. This has allowed a quite comprehensive study of the quasistationary states (QSSs) though many aspects of the physical interpretations of these states still remain unclear. In this paper a proper definition of time scale for long time evolution is discussed, and several numerical results are presented for different values of N. Previous reports indicate that the lifetimes of the QSS scale as N1.7 or even the system properties scale with exp(N). However, preliminary results presented here indicates that time scale goes as N2 for a different type of initial condition. We also discuss how the form of the interparticle potential determines the convergence of the N-particle dynamics to the Vlasov equation. The results are obtained in the context of the following models: the Hamiltonian mean field, the Self-gravitating ring model, and one- and two-dimensional systems of gravitating particles. We have also provided information of the validity of the Vlasov equation for finite N.

Nonequilibrium phase transitions and violent relaxation in the Hamiltonian mean-field model.

We discuss the nature of nonequilibrium phase transitions in the Hamiltonian mean-field model using detailed numerical simulations of the Vlasov equation and molecular dynamics. Starting from fixed magnetization water bag initial distributions and varying the energy, the states obtained after a violent relaxation undergo a phase transition from magnetized to nonmagnetized states when going from lower to higher energies. The phase transitions are either first order or are composed of a cascade of phase reentrances. This result is at variance with most previous results in the literature mainly based on the Lynden-Bell theory of violent relaxation. The latter is a rough approximation and, consequently, is not suited for an accurate description of nonequilibrium phase transition in long-range interacting systems.

Phase transitions in simplified models with long-range interactions.

We study the origin of phase transitions in several simplified models with long-range interactions. For the self-gravitating ring model, we are unable to observe a possible phase transition predicted by Nardini and Casetti [Phys. Rev. E 80, 060103R (2009).] from an energy landscape analysis. Instead we observe a sharp, although without any nonanalyticity, change from a core-halo to a core-only configuration in the spatial distribution functions for low energies. By introducing a different class of solvable simplified models without any critical points in the potential energy we show that a behavior similar to the thermodynamics of the ring model is obtained, with a first-order phase transition from an almost homogeneous high-energy phase to a clustered phase and the same core-halo to core configuration transition at lower energies. We discuss the origin of these features for the simplified models and show that the first-order phase transition comes from the maximization of the entropy of the system as a function of energy and an order parameter, as previously discussed by Hahn and Kastner [Phys. Rev. E 72, 056134 (2005); Eur. Phys. J. B 50, 311 (2006)], which seems to be the main mechanism causing phase transitions in long-range interacting systems.

Entropy of classical systems with long-range interactions.

We discuss the form of the entropy for classical Hamiltonian systems with long-range interaction using the Vlasov equation which describes the dynamics of a N particle in the limit N-->infinity. The stationary states of the Hamiltonian system are subject to infinite conserved quantities due to the Vlasov dynamics. We show that the stationary states correspond to an extremum of the Boltzmann-Gibbs entropy, and their stability is obtained from the condition that this extremum is a maximum. As a consequence, the entropy is a function of an infinite set of Lagrange multipliers that depend on the initial condition. We also discuss in this context the meaning of ensemble inequivalence and the temperature.

Hedgehog signaling in vertebrate eye development: a growing puzzle.

The vertebrate retina has been widely used as a model to study the development of the central nervous system. Its accessibility and relatively simple organization allow analysis of basic mechanisms such as cell proliferation, differentiation and death. For this reason, it could represent an ideal place to solve the puzzle of Hh signaling during neural development. However, the extensive wealth of data, sometimes apparently discordant, has made the retina one of the most complicated models for studying the role of the Hh cascade. Given the complexity of the field, a deep analysis of the data arising from different animal models is essential. In this review, we will compare and discuss all reported roles of Hh signaling in eye development to shed light on its multiple functions.

When similarity is a liability: effects of sex-based preferential selection on reactions to like-sex and different-sex others.

In 2 laboratory studies, 145 male and female undergraduates were selected for the role of manager either on the basis of merit or preferentially on the basis of their sex. Results of the first study indicated that when female subjects had been selected preferentially as compared with on a merit basis, they reacted more negatively to female (but not to male) applicants for an entry-level position in terms of personnel evaluations and competence ratings and they recommended female applicants for hire less frequently and less enthusiastically. No differences in personnel evaluations were found as a result of preferential selection when subjects were male (Study 1) or when subjects were provided with favorable information about their ability (Study 2). Implications for implementation of affirmative action programs are discussed.