Showing papers by "Jean-Pierre Eckmann published in 2006"

Nonequilibrium Energy Profiles for a Class of 1-D Models

Abstract: As a paradigm for heat conduction in 1 dimension, we propose a class of models represented by chains of identical cells, each one of which contains an energy storage device called a ``tank''. Energy exchange among tanks is mediated by tracer particles, which are injected at characteristic temperatures and rates from heat baths at the two ends of the chain. For stochastic and Hamiltonian models of this type, we develop a theory that allows one to derive rigorously – under physically natural assumptions – macroscopic equations for quantities related to heat transport, including mean energy profiles and tracer densities. Concrete examples are treated for illustration, and the validity of the Fourier Law in the present context is discussed.

Hierarchical structures induce long-range dynamical correlations in written texts

Abstract: Thoughts and ideas are multidimensional and often concurrent, yet they can be expressed surprisingly well sequentially by the translation into language. This reduction of dimensions occurs naturally but requires memory and necessitates the existence of correlations, e.g., in written text. However, correlations in word appearance decay quickly, while previous observations of long-range correlations using random walk approaches yield little insight on memory or on semantic context. Instead, we study combinations of words that a reader is exposed to within a “window of attention,” spanning about 100 words. We define a vector space of such word combinations by looking at words that co-occur within the window of attention, and analyze its structure. Singular value decomposition of the co-occurrence matrix identifies a basis whose vectors correspond to specific topics, or “concepts” that are relevant to the text. As the reader follows a text, the “vector of attention” traces out a trajectory of directions in this “concept space.” We find that memory of the direction is retained over long times, forming power-law correlations. The appearance of power laws hints at the existence of an underlying hierarchical network. Indeed, imposing a hierarchy similar to that defined by volumes, chapters, paragraphs, etc. succeeds in creating correlations in a surrogate random text that are identical to those of the original text. We conclude that hierarchical structures in text serve to create long-range correlations, and use the reader’s memory in reenacting some of the multidimensionality of the thoughts being expressed.

Concepts and Results in Chaotic Dynamics: A Short Course

Abstract: The study of dynamical systems is a well established field Having given graduate-level courses on the subject for many years, the authors have now written this book to provide a panorama of the aspects that are of interest to mathematicians and physicists alike Avoiding belaboured proofs, the exposition concentrates instead on abundant illustrations and examples, while still retaining sufficient mathematical precision Besides the standard topics of the field, questions of physical measurement and stochastic properties of chaotic dynamical systems are given much attention

Thermal rectification in billiardlike systems.

Abstract: We study the thermal rectification phenomenon in billiard systems with interacting particles. This interaction induces a local dynamical response of the billiard to an external thermodynamic gradient. To explain this dynamical effect we study the steady state of an asymmetric billiard in terms of the particle and energy reflection coefficients. This allows us to obtain expressions for the region in parameter space where large thermal rectifications are expected. Our results are confirmed by extensive numerical simulations.

Memory Effects in Nonequilibrium Transport for Deterministic Hamiltonian Systems

Abstract: We consider nonequilibrium transport in a simple chain of identical mechanical cells in which particles move around. In each cell, there is a rotating disc, with which these particles interact, and this is the only interaction in the model. It was shown in Ref. 1 that when the cells are weakly coupled, to a good approximation, the jump rates of particles and the energy-exchange rates from cell to cell follow linear profiles. Here, we refine that study by analyzing higher-order effects which are induced by the presence of external gradients for situations in which memory effects, typical of Hamiltonian dynamics, cannot be neglected. For the steady state we propose a set of balance equations for the particle number and energy in terms of the reflection probabilities of the cell and solve it phenomenologically. Using this approximate theory we explain how these asymmetries affect various aspects of heat and particle transport in systems of the general type described above and obtain in the infinite volume limit the deviation from the theory in Ref. 1 to first-order. We verify our assumptions with extensive numerical simulations.