arXiv subjects

Jean-Pierre Eckmann

Publications and source records attributed to Jean-Pierre Eckmann.

At least 55 records · Page 3Linked to original sources

arXiv2008

Ergodicity and Slowing Down in Glass-Forming Systems with Soft Potentials: No Finite-Temperature Singularities

The aim of this paper is to discuss some basic notions regarding generic glass forming systems composed of particles interacting via soft potentials. Excluding explicitly hard-core interaction we discuss the so called `glass transition' in which super-cooled amorphous state is formed, accompanied with a spectacular slowing down of relaxation to equilibrium, when the temperature is changed over a relatively small interval. Using the classical example of a 50-50 binary liquid of N particles with different interaction length-scales we show that (i) the system remains ergodic at all temperatures. (ii) the number of topologically distinct configurations can be computed, is temperature independent, and is exponential in N. (iii) Any two configurations in phase space can be connected using elementary moves whose number is polynomially bounded in N, showing that the graph of configurations has the `small world' property. (iv) The entropy of the system can be estimated at any temperature (or energy), and there is no Kauzmann crisis at any positive temperature. (v) The mechanism for the super-Arrhenius temperature dependence of the relaxation time is explained, connecting it to an entropic squeeze at the glass transition. (vi) There is no Vogel-Fulcher crisis at any finite temperature T>0

cond-mat.dis-nn↗

arXiv2007

Leadership in 2D living neural networks

Eytan and Marom recently showed that the spontaneous burst activity of rat neuron cultures includes `first to fire' cells that consistently fire earlier than others. Here we analyze the behavior of these neurons in long term recordings of spontaneous activity of rat hippocampal and rat cortical neuron cultures from three different laboratories. We identify precursor events that may either subside (`small events') or can lead to a full-blown burst (`pre-bursts'). We find that the activation in the pre-burst typically has a first neuron (`leader'), followed by a localized response in its neighborhood. Locality is diminished in the bursts themselves. The long term dynamics of the leaders is relatively robust, evolving with a half-life of 23-34 hours. Stimulation of the culture can temporarily alter the leader distribution, but it returns to the previous distribution within about 1 hour. We show that the leaders carry information about the identity of the burst, as measured by the signature of the number of spikes per neuron in a burst. The number of spikes from leaders in the first few spikes of a precursor event is furthermore shown to be predictive with regard to the transition into a burst (pre-burst versus small event). We conclude that the leaders play a rôle in the development of the bursts, and conjecture that they are part of an underlying sub-network that is excited first and then act as nucleation centers for the burst.

q-bio.NC↗

arXiv2007

A Topological Glass

We propose and study a model with glassy behavior. 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 different numbers of neighbors are supposed to be positive. The dynamics is that of flipping the diagonal of two adjacent triangles, with a temperature dependent probability. We show that this system has an approach to a steady state which is exponentially slow, and show that the stationary state is unordered. We also study the local energy landscape and show that it has the hierarchical structure known from spin glasses. Finally, we show that the evolution can be described as that of a rarefied gas with spontaneous generation of particles and annihilating collisions.

cond-mat.stat-mech↗

arXiv2007

Controllability for chains of dynamical scatterers

In this paper, we consider a class of mechanical models which consists of a linear chain of identical chaotic cells, each of which has two small lateral holes and contains a rotating disk at its center. Particles are injected at characteristic temperatures and rates from stochastic heat baths located at both ends of the chain. Once in the system, the particles move freely within the cells and will experience elastic collisions with the outer boundary of the cells as well as with the disks. They do not interact with each other but can transfer energy from one to another through collisions with the disks. The state of the system is defined by the positions and velocities of the particles and by the angular positions and angular velocities of the disks. We show that each model in this class is controllable with respect to the baths, i.e. we prove that the action of the baths can drive the system from any state to any other state in a finite time. As a consequence, one obtains the existence of at most one regular invariant measure characterizing its states (out of equilibrium).

math-ph↗

arXiv2006

Memory Effects in Nonequilibrium Transport for Deterministic Hamiltonian Systems

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 \cite{eckmann-young} 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 \cite{eckmann-young} to first-order. We verify our assumptions with extensive numerical simulations.

cond-mat.stat-mech↗

arXiv2006

Thermal Rectification in Billiard-like Systems

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.

cond-mat.stat-mech↗

arXiv2005

Nonequilibrium Energy Profiles for a Class of 1-D Models

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 containing 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.

nlin.CD↗

arXiv2004

Dynamics of Triangulations

We study a few problems related to Markov processes of flipping triangulations of the sphere. We show that these processes are ergodic and mixing, but find a natural example which does not satisfy detailed balance. In this example, the expected distribution of the degrees of the nodes seems to follow the power law $d^{-4}$.

math-ph↗

arXiv2004

Lyapunov Modes in Hard-Disk Systems

We consider simulations of a 2-dimensional gas of hard disks in a rectangular container and study the Lyapunov spectrum near the vanishing Lyapunov exponents. To this spectrum are associated ``eigen-directions'', called Lyapunov modes. We carefully analyze these modes and show how they are naturally associated with vector fields over the container. We also show that the Lyapunov exponents, and the coupled dynamics of the modes (where it exists) follow linear laws, whose coefficients only depend on the density of the gas, but not on aspect ratio and very little on the boundary conditions.

nlin.CD↗

arXiv2004

Temperature Profiles in Hamiltonian Heat Conduction

We study heat transport in the context of Hamiltonian and related stochastic models with nearest-neighbor coupling, and derive a universal law for the temperature profiles of a large class of such models. This law contains a parameter $α$, and is linear only when $α=1$. The value of $α$ depends on energy-exchange mechanisms, including the range of motion of tracer particles and their times of flight.

cond-mat.stat-mech↗

arXiv2004

Using Curvature and Markov Clustering in Graphs for Lexical Acquisition and Word Sense Discrimination

We introduce two different approaches for clustering semantically similar words. We accommodate ambiguity by allowing a word to belong to several clusters. Both methods use a graph-theoretic representation of words and their paradigmatic relationships. The first approach is based on the concept of curvature and divides the word graph into classes of similar words by removing words of low curvature which connect several dispersed clusters. The second method, instead of clustering the nodes, clusters the links in our graph. These contain more specific contextual information than nodes representing just words. In so doing, we naturally accommodate ambiguity by allowing multiple class membership. Both methods are evaluated on a lexical acquisition task, using clustering to add nouns to the WordNet taxonomy. The most effective method is link clustering.

cond-mat.other↗

arXiv2003

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

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.

nlin.CD↗

arXiv2003

Strange Heat Flux in (An)Harmonic Networks

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.

nlin.CD↗

arXiv2003

Non-equilibrium steady states

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 {\em 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 such {\em a priori} guess exists. Recent work has, however, revealed that under quite general conditions, such states can not only be shown to exist, but are {\em unique}, using the Hörmander conditions and controllability. Furthermore, interesting properties, such as energy flux, exponentially fast convergence to the unique state, and fluctuations of that state have been successfully studied. Finally, the ideas used in these studies can be extended to certain stochastic PDE's using Malliavin calculus to prove regularity of the process.

math-ph↗

arXiv2003

Dialog in e-Mail Traffic

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.

cond-mat.dis-nn↗

arXiv2003

Liapunov Multipliers and Decay of Correlations in Dynamical Systems

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.

nlin.CD↗

arXiv2003

On the Fractal Dimension of the Visible Universe

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.

astro-ph↗

arXiv2002

The Number of Large Graphs with a Positive Density of Triangles

We give upper and lower bounds on the number of graphs of fixed degree which have a positive density of triangles. In particular, we show that there are very few such graphs, when compared to the number of graphs without this restriction. We also show that in this case the triangles seem to cluster even at low density.

math-ph↗