Showing papers by "Jean-Pierre Eckmann published in 2009"

Remarks on bootstrap percolation in metric networks

Abstract: We examine bootstrap percolation in d-dimensional, directed metric graphs in the context of recent measurements of firing dynamics in 2D neuronal cultures. There are two regimes depending on the graph size N. Large metric graphs are ignited by the occurrence of critical nuclei, which initially occupy an infinitesimal fraction, f∗ → 0, of the graph and then explode throughout a finite fraction. Smaller metric graphs are effectively random in the sense that their ignition requires the initial ignition of a finite, unlocalized fraction of the graph, f∗ > 0. The crossover between the two regimes is at a size N∗ which scales exponentially with the connectivity range λ like N∗ ∼ exp λ d .T he neuronal cultures are finite metric graphs of size N � 10 5 − 10 6 , which, for

Remarks on Bootstrap Percolation in Metric Networks

Abstract: We examine bootstrap percolation in d-dimensional, directed metric graphs in the context of recent measurements of firing dynamics in 2D neuronal cultures. There are two regimes, depending on the graph size N. Large metric graphs are ignited by the occurrence of critical nuclei, which initially occupy an infinitesimal fraction, f_* -> 0, of the graph and then explode throughout a finite fraction. Smaller metric graphs are effectively random in the sense that their ignition requires the initial ignition of a finite, unlocalized fraction of the graph, f_* >0. The crossover between the two regimes is at a size N_* which scales exponentially with the connectivity range \lambda like_* \sim \exp\lambda^d. The neuronal cultures are finite metric graphs of size N \simeq 10^5-10^6, which, for the parameters of the experiment, is effectively random since N<< N_*. This explains the seeming contradiction in the observed finite f_* in these cultures. Finally, we discuss the dynamics of the firing front.

Depinning of domain walls with an internal degree of freedom

Abstract: Taking into account the coupling between the position of the wall and an internal degree of freedom, namely, its phase $\ensuremath{\varphi}$, we examine, in the rigid-wall approximation, the dynamics of a magnetic domain wall subject to a weak pinning potential. We determine the corresponding force-velocity characteristics, which display several unusual features when compared to the standard depinning behavior. At zero temperature, there exists a bistable regime for low forces, with a logarithmic behavior close to the transition. For weak pinning, there occurs a succession of bistable transitions corresponding to different topological modes of the phase evolution. At finite temperature, the force-velocity characteristics become nonmonotonous. We compare our results to recent experiments on permalloy nanowires.

A Model of Heat Conduction

Abstract: In this paper, we first define a deterministic particle model for heat conduction. It consists of a chain of N identical subsystems, each of which contains a scatterer and with particles moving among these scatterers. Based on this model, we then derive heuristically, in the limit of N → ∞ and decreasing scattering cross-section, a Boltzmann equation for this limiting system. This derivation is obtained by a closure argument based on memory loss between collisions. We then prove that the Boltzmann equation has, for stochastic driving forces at the boundary, close to Maxwellians, a unique non-equilibrium steady state.

Superdiffusive Heat Transport in a Class of Deterministic One-dimensional Many-Particle Lorentz Gases

Abstract: We study heat transport in a one-dimensional chain of a finite number N of identical cells, coupled at its boundaries to stochastic particle reservoirs. At the center of each cell, tracer particles collide with fixed scatterers, exchanging momentum. In a recent paper (Collet and Eckmann in Commun. Math. Phys. 287:1015, 2009), a spatially continuous version of this model was derived in a scaling regime where the scattering probability of the tracers is γ∼1/N, corresponding to the Grad limit. A Boltzmann-like equation describing the transport of heat was obtained. In this paper, we show numerically that the Boltzmann description obtained in Collet and Eckmann (Commun. Math. Phys. 287:1015, 2009) is indeed a bona fide limit of the particle model. Furthermore, we study the heat transport of the model when the scattering probability is 1, corresponding to deterministic dynamics. Thought as a lattice model in which particles jump between different scatterers the motion is persistent, with a persistence probability determined by the mass ratio among particles and scatterers, and a waiting time probability distribution with algebraic tails. We find that the heat and particle currents scale slower than 1/N, implying that this model exhibits anomalous heat and particle transport.

Typical Behavior for One Map

Abstract: Before we study parametrized families of maps, we want to analyze individual maps. We are interested in the possible behavior of the successive images of an initial point x0 on the interval [-1,1] for a fixed map f. For this we first outline a graphical method for determining the iterates \({\rm x}_{{\rm n}} = {\rm f}^{{\rm n}}({\rm x}_0)\). Here, we define \({\rm f}_{{\rm n}}({\rm x}_0) = {\rm f}({\rm f}^{{\rm n-1}}({\rm x}_0))\). The following figure 1.5 shows how this is done through the rule: Go from x0 to the graph of the function, from the graph to the diagaonal, from the diagonal to the graph,….

Higher Dimensional Systems

Abstract: As we have seen in section I.1, there are many situations in which a map \(\mathbb{R}^{{\rm n}} \rightarrow \mathbb{R}^{{\rm n}}\) seems more appropriate than a map \(\mathbb{R}^1 \rightarrow \mathbb{R}^1\). The purpose of this section is to illustrate how the features of one-dimensional maps described in I.6 generalize to higher dimensions. We have already seen in the previous section that the numerical results seem to indicate a persistence of the universal behavior for higher dimensional systems. While we refer for the theorems to Section III.4, we want to illustrate here the salient features of their hypotheses in two dimensions.