Showing papers by "Jean-Pierre Eckmann published in 2003"

Spectral Properties of Hypoelliptic Operators

Abstract: We study hypoelliptic operators with polynomially bounded coefficients that are of the form K=∑ 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 ɛ>0 we show that an inequality of the form ||u||δ,δ≤C(||u||0,ɛ+||(K+iy)u||0,0) holds for suitable δ and C which are independent of y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|τ−c,τ(0,1],cR}.

Liapunov Multipliers and Decay of Correlations in Dynamical Systems

Abstract: 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.

Strange Heat Flux in (An)Harmonic Networks

Abstract: 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-Equilibrium Steady States

Abstract: The mathematical physics of mechanical systems in thermal equilibrium is a well studied, and relatively easy, subject, because the Gibbs distribution is in general an adequate guess for the equilibrium state. On the other hand, the mathematical physics of non-equilibrium systems, such as that of a chain of masses connected with springs to two (infinite) heat reservoirs is more difficult, precisely because no sucha priori guess exists. Recent work has, however, revealed that under quite general conditions,

Dialog in e-Mail Traffic

Abstract: Connectivity and topology are known to yield information about networks, whose origin is self-organized, but the impact of {\it temporal dynamics} in a network is still mostly unexplored. Using an information theoretic approach to e-mail exchange, we show that an e-mail network allows for a separation of static and dynamic structures within it. The static structures are related to organizational units such as departments. The temporally linked structures turn out to be more goal-oriented, functional units such as committees and user groups.

On the Fractal Dimension of the Visible Universe

Abstract: Estimates of the fractal dimension $D$ of the set of galaxies in the universe, based on ever improving data sets, tend to settle on $D\approx 2$. This result raised a raging debate due to its glaring contradiction with astrophysical models that expect a homogeneous universe. A recent mathematical result indicates that there is no contradiction, since measurements of the dimension of the {\em visible} subset of galaxies is bounded from above by D=2 even if the true dimension is anything between D=2 and D=3. We demonstrate this result in the context of a simple fractal model, and explain how to proceed in order to find a better estimate of the true dimension of the set of galaxies.

Non-Vanishing Profiles for the Kuramoto-Sivashinsky Equation on the Infinite Line

Abstract: We study the Kuramoto-Sivashinsky equation on the infinite line with initial conditions having arbitrarily large limits $\pm Y$ at $x=\pm\infty$. We show that the solutions have the same limits for all positive times. This implies that an attractor for this equation cannot be defined in $L^\infty$. To prove this, we consider profiles with limits at $x=\pm\infty$, and show that initial conditions $L^2$-close to such profiles lead to solutions which remain $L^2$-close to the profile for all times. Furthermore, the difference between these solutions and the initial profile tends to 0 as $x\to\pm\infty$, for any fixed time $t>0$. Analogous results hold for $L^2$-neighborhoods of periodic stationary solutions. This implies that profiles and periodic stationary solutions partition the phase space into mutually unattainable regions.