A Model of Heat Conduction

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

Liapunov Multipliers and Decay of Correlations in Dynamical Systems

The essential decorrelation rate of a hyperbolic dynamical system is the decay rate of time-correlations one expects to see stably for typical observables once resonances are projected out. We define and illustrate these notions and study the conjecture that for observables in $C^1$, the essential decorrelation rate is never faster than what is dictated by the {\em smallest} unstable Liapunov multiplier

Extensive Properties of the Complex Ginzburg-Landau Equation

We study the set of solutions of the complex Ginzburg-Landau equation in $\real^d, d<3$. We consider the global attracting set (i.e., the forward map of the set of bounded initial data), and restrict it to a cube $Q_L$ of side $L$. We cover this set by a (minimal) number $N_{Q_L}(\epsilon)$ of balls of radius $\epsilon$ in \Linfty(Q_L). We show that the Kolmogorov $\epsilon$-entropy per unit length, $H_\epsilon =\lim_{L\to\infty} L^{-d} \log N_{Q_L}(\epsilon)$ exists. In particular, we bound $H_\epsilon$ by \OO(\log(1/\epsilon), which shows that the attracting set is smaller than the set of bounded analytic functions in a strip. We finally give a positive lower bound: H_\epsilon>\OO(\log(1/\epsilon))Comment: 24 page

Non-Equilibrium Statistical Mechanics of Strongly Anharmonic Chains of Oscillators

We study the model of a strongly non-linear chain of particles coupled to two heat baths at different temperatures. Our main result is the existence and uniqueness of a stationary state at all temperatures. This result extends those of Eckmann, Pillet, Rey-Bellet to potentials with essentially arbitrary growth at infinity. This extension is possible by introducing a stronger version of H\"ormander's theorem for Kolmogorov equations to vector fields with polynomially bounded coefficients on unbounded domains.Comment: ~60 pages, 3 figure

Strange Heat Flux in (An)Harmonic Networks

We study the heat transport in systems of coupled oscillators driven out of equilibrium by Gaussian heat baths. We illustrate with a few examples that such systems can exhibit ``strange'' transport phenomena. In particular, {\em circulation} of heat flux may appear in the steady state of a system of three oscillators only. This indicates that the direction of the heat fluxes can in general not be "guessed" from the temperatures of the heat baths. Although we primarily consider harmonic couplings between the oscillators, we explain why this strange behavior persists under weak anharmonic perturbations

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

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