Showing papers by "Jean-Pierre Eckmann published in 1999"

Non-Equilibrium Statistical Mechanics of Anharmonic Chains Coupled to Two Heat Baths at Different Temperatures

Abstract: We study the statistical mechanics of a finite-dimensional non-linear Hamiltonian system (a chain of anharmonic oscillators) coupled to two heat baths (described by wave equations). Assuming that the initial conditions of the heat baths are distributed according to the Gibbs measures at two different temperatures we study the dynamics of the oscillators. Under suitable assumptions on the potential and on the coupling between the chain and the heat baths, we prove the existence of an invariant measure for any temperature difference, i.e., we prove the existence of steady states. Furthermore, if the temperature difference is sufficiently small, we prove that the invariant measure is unique and mixing. In particular, we develop new techniques for proving the existence of invariant measures for random processes on a non-compact phase space. These techniques are based on an extension of the commutator method of Hormander used in the study of hypoelliptic differential operators.

Entropy Production in Nonlinear, Thermally Driven Hamiltonian Systems

Abstract: We consider a finite chain of nonlinear oscillators coupled at its ends to two infinite heat baths which are at different temperatures. Using our earlier results about the existence of a stationary state, we show rigorously that for arbitrary temperature differences and arbitrary couplings, such a system has a unique stationary state. (This extends our earlier results for small temperature differences.) In all these cases, any initial state will converge (at an unknown rate) to the stationary state. We show that this stationary state continually produces entropy. The rate of entropy production is strictly negative when the temperatures are unequal and is proportional to the mean energy flux through the system

Extensive Properties of the Complex Ginzburg–Landau Equation

Abstract: We study the set of solutions of the complex Ginzburg–Landau equation in R 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 (ɛ) of balls of radius ɛ in $L infin(Q L ). We show that the Kolmogorov ɛ-entropy per unit length, \(\) exists. In particular, we bound \(\) by \(\), 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: \(\).

Hydrodynamic Lyapunov Modes in Translation Invariant Systems

Abstract: We study the implications of translation invariance on the tangent dynamics of extended dynamical systems, within a random matrix approximation. In a model system, we show the existence of hydrodynamic modes in the slowly growing part of the Lyapunov spectrum, which are analogous to the hydrodynamic modes discovered numerically by [Dellago, Ch., Posch, H.A., Hoover, W.G., Phys. Rev. E 53, 1485 (1996)]. The hydrodynamic Lyapunov vectors loose the typical random structure and exhibit instead the structure of weakly perturbed coherent long wavelength waves. We show further that the amplitude of the perturbations vanishes in the thermodynamic limit, and that the associated Lyapunov exponents are universal.

The definition and measurement of the topological entropy per unit volume in parabolic PDEs

Abstract: We define the topological entropy per unit volume in parabolic PDEs such as the complex Ginzburg-Landau equation, and show that it exists and is bounded by the upper Hausdorff dimension times the maximal expansion rate. We then give a constructive implementation of a bound on the inertial range of such equations. Using this bound, we are able to propose a finite sampling algorithm which allows (in principle) to measure this entropy from experimental data. AMS classification number: 35B37

Topological Entropy and epsilon-Entropy for Damped Hyperbolic Equations

Abstract: We study damped hyperbolic equations on the infinite line. We show that on the global attracting set $G$ the $\epsilon$-entropy (per unit length) exists in the topology of $W^{1,\infty}$. We also show that the topological entropy per unit length of $G$ exists. These results are shown using two main techniques: Bounds in bounded domains in position space and for large momenta, and a novel submultiplicativity argument in $W^{1,\infty}$.

Non-Equilibrium Statistical Mechanics of Strongly Anharmonic Chains of Oscillators

Abstract: 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.

Porosities and dimensions of measures satisfying the doubling condition

Abstract: This paper explains the implications of a mathematical theory (by the same authors) of holes in fractals and their relation to dimension for measurements The novelty of our approach is to consider the fractal measure on a set rather than just the support of that measure This should take into account in a more precise way the distribution of data points in measured sets, such as the distribution of galaxies

Porosities and dimensions of measures

Abstract: We introduce a concept of porosity for measures and study relations between dimensions and porosities for two classes of measures: measures on $R^n$ which satisfy the doubling condition and strongly porous measures on $R$.