Chaos and a Resonance Mechanism for Structure Formation in Inflationary Models

We exhibit a resonance mechanism of amplification of density perturbations in inflationary mo-dels, using a minimal set of ingredients (an effective cosmological constant, a scalar field minimally coupled to the gravitational field and matter), common to most models in the literature of inflation. This mechanism is based on the structure of homoclinic cylinders, emanating from an unstable periodic orbit in the neighborhood of a saddle-center critical point, present in the phase space of the model. The cylindrical structure induces oscillatory motions of the scales of the universe whenever the orbit visits the neighborhood of the saddle-center, before the universe enters a period of exponential expansion. The oscillations of the scale functions produce, by a resonance mechanism, the amplification of a selected wave number spectrum of density perturbations, and can explain the hierarchy of scales observed in the actual universe. The transversal crossings of the homoclinic cylinders induce chaos in the dynamics of the model, a fact intimately connected to the resonance mechanism occuring immediately before the exit to inflation.Comment: 4 pages. This essay received an Honorable Mention from the Gravity Research Foundation, 1998-Ed. To appear in Mod. Phys. Lett.

Large cities are less green

We study how urban quality evolves as a result of carbon dioxide emissions as urban agglomerations grow. We employ a bottom-up approach combining two unprecedented microscopic data on population and carbon dioxide emissions in the continental US. We first aggregate settlements that are close to each other into cities using the City Clustering Algorithm (CCA) defining cities beyond the administrative boundaries. Then, we use data on $\rm{CO}_2$ emissions at a fine geographic scale to determine the total emissions of each city. We find a superlinear scaling behavior, expressed by a power-law, between $\rm{CO}_2$ emissions and city population with average allometric exponent $\beta = 1.46$ across all cities in the US. This result suggests that the high productivity of large cities is done at the expense of a proportionally larger amount of emissions compared to small cities. Furthermore, our results are substantially different from those obtained by the standard administrative definition of cities, i.e. Metropolitan Statistical Area (MSA). Specifically, MSAs display isometric scaling emissions and we argue that this discrepancy is due to the overestimation of MSA areas. The results suggest that allometric studies based on administrative boundaries to define cities may suffer from endogeneity bias

Numerical study of a model for non-equilibrium wetting

We revisit the scaling properties of a model for non-equilibrium wetting [Phys. Rev. Lett. 79, 2710 (1997)], correcting previous estimates of the critical exponents and providing a complete scaling scheme. Moreover, we investigate a special point in the phase diagram, where the model exhibits a roughening transition related to directed percolation. We argue that in the vicinity of this point evaporation from the middle of plateaus can be interpreted as an external field in the language of directed percolation. This analogy allows us to compute the crossover exponent and to predict the form of the phase transition line close to its terminal point.Comment: 8 pages, 8 figure

Fracturing the optimal paths

Optimal paths play a fundamental role in numerous physical applications ranging from random polymers to brittle fracture, from the flow through porous media to information propagation. Here for the first time we explore the path that is activated once this optimal path fails and what happens when this new path also fails and so on, until the system is completely disconnected. In fact numerous applications can be found for this novel fracture problem. In the limit of strong disorder, our results show that all the cracks are located on a single self-similar connected line of fractal dimension $D_{b} \approx 1.22$. For weak disorder, the number of cracks spreads all over the entire network before global connectivity is lost. Strikingly, the disconnecting path (backbone) is, however, completely independent on the disorder.Comment: 4 pages,4 figure

Equivalence between different classical treatments of the O(N) nonlinear sigma model and their functional Schrodinger equations

In this work we derive the Hamiltonian formalism of the O(N) non-linear sigma model in its original version as a second-class constrained field theory and then as a first-class constrained field theory. We treat the model as a second-class constrained field theory by two different methods: the unconstrained and the Dirac second-class formalisms. We show that the Hamiltonians for all these versions of the model are equivalent. Then, for a particular factor-ordering choice, we write the functional Schrodinger equation for each derived Hamiltonian. We show that they are all identical which justifies our factor-ordering choice and opens the way for a future quantization of the model via the functional Schrodinger representation.Comment: Revtex version, 17 pages, substantial change