Lyapunov Mode Dynamics in Hard-Disk Systems

The tangent dynamics of the Lyapunov modes and their dynamics as generated numerically - {\it the numerical dynamics} - is considered. We present a new phenomenological description of the numerical dynamical structure that accurately reproduces the experimental data for the quasi-one-dimensional hard-disk system, and shows that the Lyapunov mode numerical dynamics is linear and separate from the rest of the tangent space. Moreover, we propose a new, detailed structure for the Lyapunov mode tangent dynamics, which implies that the Lyapunov modes have well-defined (in)stability in either direction of time. We test this tangent dynamics and its derivative properties numerically with partial success. The phenomenological description involves a time-modal linear combination of all other Lyapunov modes on the same polarization branch and our proposed Lyapunov mode tangent dynamics is based upon the form of the tangent dynamics for the zero modes

Method of constructing exactly solvable chaos

We present a new systematic method of constructing rational mappings as ergordic transformations with nonuniform invariant measures on the unit interval [0,1]. As a result, we obtain a two-parameter family of rational mappings that have a special property in that their invariant measures can be explicitly written in terms of algebraic functions of parameters and a dynamical variable. Furthermore, it is shown here that this family is the most generalized class of rational mappings possessing the property of exactly solvable chaos on the unit interval, including the Ulam=Neumann map y=4x(1-x). Based on the present method, we can produce a series of rational mappings resembling the asymmetric shape of the experimentally obtained first return maps of the Beloussof-Zhabotinski chemical reaction, and we can match some rational functions with other experimentally obtained first return maps in a systematic manner.Comment: 12 pages, 2 figures, REVTEX. Title was changed. Generalized Chebyshev maps including the precise form of two-parameter generalized cubic maps were added. Accepted for publication in Phys. Rev. E(1997

An algebraic Haag's theorem

Under natural conditions (such as split property and geometric modular action of wedge algebras) it is shown that the unitary equivalence class of the net of local (von Neumann) algebras in the vacuum sector associated to double cones with bases on a fixed space-like hyperplane completely determines an algebraic QFT model. More precisely, if for two models there is unitary connecting all of these algebras, then --- without assuming that this unitary also connects their respective vacuum states or spacetime symmetry representations --- it follows that the two models are equivalent. This result might be viewed as an algebraic version of the celebrated theorem of Rudolf Haag about problems regarding the so-called "interaction-picture" in QFT. Original motivation of the author for finding such an algebraic version came from conformal chiral QFT. Both the chiral case as well as a related conjecture about standard half-sided modular inclusions will be also discussed

Hot scatterers and tracers for the transfer of heat in collisional dynamics

We introduce stochastic models for the transport of heat in systems described by local collisional dynamics. The dynamics consists of tracer particles moving through an array of hot scatterers describing the effect of heat baths at fixed temperatures. Those models have the structure of Markov renewal processes. We study their ergodic properties in details and provide a useful formula for the cumulant generating function of the time integrated energy current. We observe that out of thermal equilibrium, the generating function is not analytic. When the set of temperatures of the scatterers is fixed by the condition that in average no energy is exchanged between the scatterers and the system, different behaviours may arise. When the tracer particles are allowed to travel freely through the whole array of scatterers, the temperature profile is linear. If the particles are locked in between scatterers, the temperature profile becomes nonlinear. In both cases, the thermal conductivity is interpreted as a frequency of collision between tracers and scatterers

Spatial Structure of Stationary Nonequilibrium States in the Thermostatted Periodic Lorentz Gas

We investigate analytically and numerically the spatial structure of the non-equilibrium stationary states (NESS) of a point particle moving in a two dimensional periodic Lorentz gas (Sinai Billiard). The particle is subject to a constant external electric field E as well as a Gaussian thermostat which keeps the speed |v| constant. We show that despite the singular nature of the SRB measure its projections on the space coordinates are absolutely continuous. We further show that these projections satisfy linear response laws for small E. Some of them are computed numerically. We compare these results with those obtained from simple models in which the collisions with the obstacles are replaced by random collisions.Similarities and differences are noted.Comment: 24 pages with 9 figure

On Bootstrap Percolation in Living Neural Networks

Recent experimental studies of living neural networks reveal that their global activation induced by electrical stimulation can be explained using the concept of bootstrap percolation on a directed random network. The experiment consists in activating externally an initial random fraction of the neurons and observe the process of firing until its equilibrium. The final portion of neurons that are active depends in a non linear way on the initial fraction. The main result of this paper is a theorem which enables us to find the asymptotic of final proportion of the fired neurons in the case of random directed graphs with given node degrees as the model for interacting network. This gives a rigorous mathematical proof of a phenomena observed by physicists in neural networks

Bursts in the Chaotic Trajectory Lifetimes Preceding the Controlled Periodic Motion

The average lifetime ($\tau(H)$) it takes for a randomly started trajectory to land in a small region ($H$) on a chaotic attractor is studied. $\tau(H)$ is an important issue for controlling chaos. We point out that if the region $H$ is visited by a short periodic orbit, the lifetime $\tau(H)$ strongly deviates from the inverse of the naturally invariant measure contained within that region ($\mu_N(H)^{-1}$). We introduce the formula that relates $\tau(H)/\mu_N(H)^{-1}$ to the expanding eigenvalue of the short periodic orbit visiting $H$.Comment: Accepted for publication in Phys. Rev. E, 3 PS figure

Molecular dynamics approach: from chaotic to statistical properties of compound nuclei

Statistical aspects of the dynamics of chaotic scattering in the classical model of $\alpha$-cluster nuclei are studied. It is found that the dynamics governed by hyperbolic instabilities which results in an exponential decay of the survival probability evolves to a limiting energy distribution whose density develops the Boltzmann form. The angular distribution of the corresponding decay products shows symmetry with respect to $\pi/2$ angle. Time estimated for the compound nucleus formation ranges within the order of $10^{-21}$s.Comment: 11 pages, LaTeX, non

Normal transport properties for a classical particle coupled to a non-Ohmic bath

We study the Hamiltonian motion of an ensemble of unconfined classical particles driven by an external field F through a translationally-invariant, thermal array of monochromatic Einstein oscillators. The system does not sustain a stationary state, because the oscillators cannot effectively absorb the energy of high speed particles. We nonetheless show that the system has at all positive temperatures a well-defined low-field mobility over macroscopic time scales of order exp(-c/F). The mobility is independent of F at low fields, and related to the zero-field diffusion constant D through the Einstein relation. The system therefore exhibits normal transport even though the bath obviously has a discrete frequency spectrum (it is simply monochromatic) and is therefore highly non-Ohmic. Such features are usually associated with anomalous transport properties

Information-Geometric Indicators of Chaos in Gaussian Models on Statistical Manifolds of Negative Ricci Curvature

A new information-geometric approach to chaotic dynamics on curved statistical manifolds based on Entropic Dynamics (ED) is proposed. It is shown that the hyperbolicity of a non-maximally symmetric 6N-dimensional statistical manifold M_{s} underlying an ED Gaussian model describing an arbitrary system of 3N degrees of freedom leads to linear information-geometric entropy growth and to exponential divergence of the Jacobi vector field intensity, quantum and classical features of chaos respectively.Comment: 8 pages, final version accepted for publicatio