3,429 research outputs found

    A Model of Heat Conduction

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    We define a deterministic ``scattering'' model for heat conduction which is continuous in space, and which has a Boltzmann type flavor, obtained by a closure based on memory loss between collisions. We prove that this model has, for stochastic driving forces at the boundary, close to Maxwellians, a unique non-equilibrium steady state

    Spectral Properties of Hypoelliptic Operators

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    We study hypoelliptic operators with polynomially bounded coefficients that are of the form K = sum_{i=1}^m X_i^T X_i + X_0 + f, where the X_j denote first order differential operators, f is a function with at most polynomial growth, and X_i^T denotes the formal adjoint of X_i in L^2. For any e > 0 we show that an inequality of the form |u|_{delta,delta} <= C(|u|_{0,eps} + |(K+iy)u|_{0,0}) holds for suitable delta and C which are independent of y in R, in weighted Sobolev spaces (the first index is the derivative, and the second the growth). We apply this result to the Fokker-Planck operator for an anharmonic chain of oscillators coupled to two heat baths. Using a method of Herau and Nier [HN02], we conclude that its spectrum lies in a cusp {x+iy|x >= |y|^tau-c, tau in (0,1], c in R}.Comment: 3 figure

    Curvature of Co-Links Uncovers Hidden Thematic Layers in the World Wide Web

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    Beyond the information stored in pages of the World Wide Web, novel types of ``meta-information'' are created when they connect to each other. This information is a collective effect of independent users writing and linking pages, hidden from the casual user. Accessing it and understanding the inter-relation of connectivity and content in the WWW is a challenging problem. We demonstrate here how thematic relationships can be located precisely by looking only at the graph of hyperlinks, gleaning content and context from the Web without having to read what is in the pages. We begin by noting that reciprocal links (co-links) between pages signal a mutual recognition of authors, and then focus on triangles containing such links, since triangles indicate a transitive relation. The importance of triangles is quantified by the clustering coefficient (Watts) which we interpret as a curvature (Gromov,Bridson-Haefliger). This defines a Web-landscape whose connected regions of high curvature characterize a common topic. We show experimentally that reciprocity and curvature, when combined, accurately capture this meta-information for a wide variety of topics. As an example of future directions we analyze the neural network of C. elegans (White, Wood), using the same methods.Comment: 8 pages, 5 figures, expanded version of earlier submission with more example

    Non-Linear Stability Analysis of Higher Order Dissipative Partial Differential Equations

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    We extend the invariant manifold method for analyzing the asymptotics of dissipative partial differential equations on unbounded spatial domains to treat equations in which the linear part has order greater than two. One important example of this type of equation which we analyze in some detail is the Cahn-Hilliard equation. We analyze the marginally stable solutions of this equation in some detail. A second context in which such equations arise is in the Ginzburg-Landau equation, or other pattern forming equations, near a codimension-two bifurcation

    Scattering Phases and Density of States for Exterior Domain

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    For a bounded open domain Ω∈ℜ2\Omega\in \real^2 with connected complement and piecewise smooth boundary, we consider the Dirichlet Laplacian -\DO on Ω\Omega and the S-matrix on the complement Ωc\Omega^c. Using the restriction AEA_E of (−Δ−E)−1(-\Delta-E)^{-1} to the boundary of Ω\Omega , we establish that AE0−1/2AEAE0−1/2−1A_{E_0}^{-1/2}A_EA_{E_0}^{-1/2}-1 is trace class when E0E_0 is negative and give bounds on the energy dependence of this difference. This allows for precise bounds on the total scattering phase, the definition of a ζ\zeta-function, and a Krein spectral formula, which improve similar results found in the literature.Comment: 15 pages, Postscript, A

    Transport Properties of the Lorentz Gas in Terms of Periodic Orbits

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    We establish a formula relating global diffusion in a space periodic dynamical system to cycles in the elementary cell which tiles the space under translations.Comment: 8 pages, Postscript, A

    Temperature Profiles in Hamiltonian Heat Conduction

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    We study heat transport in the context of Hamiltonian and related stochastic models with nearest-neighbor coupling, and derive a universal law for the temperature profiles of a large class of such models. This law contains a parameter α\alpha, and is linear only when α=1\alpha=1. The value of α\alpha depends on energy-exchange mechanisms, including the range of motion of tracer particles and their times of flight.Comment: Revised text, same results Second revisio

    Geometric Stability Analysis for Periodic Solutions of the Swift-Hohenberg Equation

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    In this paper we describe invariant geometrical ~structures in the phase space of the Swift-Hohenberg equation in a neighborhood of its periodic stationary states. We show that in spite of the fact that these states are only marginally stable (i.e., the linearized problem about these states has continuous spectrum extending all the way up to zero), there exist finite dimensional invariant manifolds in the phase space of this equation which determine the long-time behavior of solutions near these stationary solutions. In particular, using this point of view, we obtain a new demonstration of Schneider's recent proof that these states are nonlinearly stable.Comment: 44 pages, plain tex, 0 figure
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