Showing papers by "Jean-Pierre Eckmann published in 2011"

Noise and Topology in Driven Systems - an Application to Interface Dynamics

Abstract: Motivated by a stochastic differential equation describing the dynamics of interfaces, we study the bifurcation behavior of a more general class of such equations. These equations are characterized by a 2-dimensional phase space (describing the position of the interface and an internal degree of freedom). The noise accounts for thermal fluctuations of such systems. The models considered show a saddle-node bifurcation and have furthermore homoclinic orbits, i.e., orbits leaving an unstable fixed point and returning to it. Such systems display intermittent behavior. The presence of noise combined with the topology of the phase space leads to unexpected behavior as a function of the bifurcation parameter, i.e., of the driving force of the system. We explain this behavior using saddle point methods and considering global topological aspects of the problem. This then explains the non-monotonous force-velocity dependence of certain driven interfaces.

Rattling and freezing in a 1D transport model

Abstract: We consider a heat conduction model introduced by Collet and Eckmann (2009 Commun. Math. Phys. 287 1015–38). This is an open system in which particles exchange momentum with a row of (fixed) scatterers. We assume simplified bath conditions throughout, and give a qualitative description of the dynamics extrapolating from the case of a single particle for which we have a fairly clear understanding. The main phenomenon discussed is freezing, or the slowing down of particles with time. As particle number is conserved, this means fewer collisions per unit time, and less contact with the baths; in other words, the conductor becomes less effective. Careful numerical documentation of freezing is provided, and a theoretical explanation is proposed. Freezing being an extremely slow process; however, the system behaves as though it is in a steady state for long durations. Quantities such as energy and fluxes are studied, and are found to have curious relationships with particle density.

Decay of Correlations in a Topological Glass

Abstract: In this paper we continue the study of a topological glassy system. The state space of the model is given by all triangulations of a sphere with $N$ nodes, half of which are red and half are blue. Red nodes want to have 5 neighbors while blue ones want 7. Energies of nodes with other numbers of neighbors are supposed to be positive. The dynamics is that of flipping the diagonal between two adjacent triangles, with a temperature dependent probability. We consider the system at very low temperatures. We concentrate on several new aspects of this model: Starting from a detailed description of the stationary state, we conclude that pairs of defects (nodes with the "wrong" degree) move with very high mobility along 1-dimensional paths. As they wander around, they encounter single defects, which they then move "sideways" with a geometrically defined probability. This induces a diffusive motion of the single defects. If they meet, they annihilate, lowering the energy of the system. We both estimate the decay of energy to equilibrium, as well as the correlations. In particular, we find a decay like $t^{-0.4}$.

Self reference in word definitions

Abstract: Dictionaries are inherently circular in nature. A given word is linked to a set of alternative words (the definition) which in turn point to further descendants. Iterating through definitions in this way, one typically finds that definitions loop back upon themselves. The graph formed by such definitional relations is our object of study. By eliminating those links which are not in loops, we arrive at a core subgraph of highly connected nodes. We observe that definitional loops are conveniently classified by length, with longer loops usually emerging from semantic misinterpretation. By breaking the long loops in the graph of the dictionary, we arrive at a set of disconnected clusters. We find that the words in these clusters constitute semantic units, and moreover tend to have been introduced into the English language at similar times, suggesting a possible mechanism for language evolution.

RESEARCH ARTICLE Decay of Correlations in a Topological Glass

Abstract: Dedicated to David Sherrington, with admiration and best wishes In this paper we continue the study of a topological glassy system. The state space of the model is given by all triangulations of a sphere with N nodes, half of which are red and half are blue. Red nodes want to have 5 neighbors while blue ones want 7. Energies of nodes with other numbers of neighbors are supposed to be positive. The dynamics is that of flipping the diagonal betwe en two adjacent triangles, with a temperature dependent probability. We consider the system at very low temperatures. We concentrate on several new aspects of this model: Starting from a detailed description of the stationary state, we conclude that pairs of defects (nodes with the “wrong” degree) move with very high mobility along 1-dimensional paths. As they wander around, they encounter single defects, which they then move “sideways” with a geometrically defined probability. This i nduces a diffusive motion of the single defects. If they meet, they annihilate, lowering the energy of the system. We both estimate the decay of energy to equilibrium, as well as the correlations. In particular, we find a decay like t −0.4 .