Pierre Gaspard

Chaos, Scattering and Statistical Mechanics

1. Dynamical systems and their linear stability 2. Topological chaos 3. Liouvillian dynamics 4. Probabalistic chaos 5. Chaotic scattering 6. Scattering theory of transport 7. Hydrodynamic modes of diffusion 8. Systems maintained out of equilibrium 9. Noises as microscopic chaos.

WAVEPACKET DANCING: ACHIEVING CHEMICAL SELECTIVITY BY SHAPING LIGHT PULSES

Abstract The Tannor-Rice pump-dump scheme for controlling the selectivity of product formation in a chemical reaction is improved by development of a method for optimizing the field of a particular product with respect to the shapes of the pump and dump pulses. Numerical studies of the optimization of product yield in a model system of the same type as studied by Tannor and Rice illustrate the enhancement possible with pulse shaping.

Spin relaxation in a complex environment

We report the study of a model of a two-level system interacting in a nondiagonal way with a complex environment described by Gaussian orthogonal random matrices (GORM). The effect of the interaction on the total spectrum and its consequences on the dynamics of the two-level system is analyzed. We show the existence of a critical value of the interaction, depending on the mean level spacing of the environment, above which the dynamics is self-averaging and closely obey a master equation for the time evolution of the observables of the two-level system. Analytic results are also obtained in the strong coupling regimes. We finally study the equilibrium values of the two-level system population and show under which condition it thermalizes to the environment temperature.

Scattering from a classically chaotic repellor

We report a study of the classical scattering of a point particle from three hard circular discs in a plane, which we propose as a model of an idealized unimolecular fragmentation. The system possesses a fractal and chaotic metastable classical state. On the basis of a coding of the system dynamics, we develop a method to construct the invariant probability measure and to calculate the particle escape rate, the Hausdorff dimension, the Kolmogorov–Sinai entropy per unit time and the mean largest Lyapunov exponent of the repellor. The relations between these characteristics of the system dynamics are discussed. In particular, we show that, in general, chaos inhibits escaping from the metastable state. The theory is compared with numerical simulations. We also introduce the classical tools necessary for the semiclassical quantization of the dynamics; the latter is discussed in the following paper.

Fluctuation theorem for nonequilibrium reactions

A fluctuation theorem is derived for stochastic nonequilibrium reactions ruled by the chemical master equation. The theorem is expressed in terms of the generating and large-deviation functions characterizing the fluctuations of a quantity which measures the loss of detailed balance out of thermodynamic equilibrium. The relationship to entropy production is established and discussed. The fluctuation theorem is verified in the Schlögl model of far-from-equilibrium bistability.

Noise, chaos, and (ε,τ)-entropy per unit time

Abstract The degree of dynamical randomness of different time processes is characterized in terms of the (e, τ)-entropy per unit time. The (e, τ)-entropy is the amount of information generated per unit time, at different scales τ of time and e of the observables. This quantity generalizes the Kolmogorov-Sinai entropy per unit time from deterministic chaotic processes, to stochastic processes such as fluctuations in mesoscopic physico-chemical phenomena or strong turbulence in macroscopic spacetime dynamics. The random processes that are characterized include chaotic systems, Bernoulli and Markov chains, Poisson and birth-and-death processes, Ornstein-Uhlenbeck and Yaglom noises, fractional Brownian motions, different regimes of hydrodynamical turbulence, and the Lorentz-Boltzmann process of nonequilibrium statistical mechanics. We also extend the (e, τ)-entropy to spacetime processes like cellular automata, Conways game of life, lattice gas automata, coupled maps, spacetime chaos in partial differential equations, as well as the ideal, the Lorentz, and the hard sphere gases. Through these examples it is demonstrated that the (e, τ)-entropy provides a unified quantitative measure of dynamical randomness to both chaos and noises, and a method to detect transitions between dynamical states of different degrees of randomness as a parameter of the system is varied.

What Can We Learn from Homoclinic Orbits in Chaotic Dynamics

State diagrams of two model systems involving three variables are constructed. The parameter dependence of different forms of complex nonperiodic behavior, and particularly of homoclinic orbits, is analyzed. It is shown that the onset of homoclinicity is reflected by deep changes in the qualitative behavior of the system.

Fluctuation theorem for currents and Schnakenberg network theory

A fluctuation theorem is proved for the macroscopic currents of a system in a nonequilibrium steady state, by using Schnakenberg network theory. The theorem can be applied, in particular, in reaction systems where the affinities or thermodynamic forces are defined globally in terms of the cycles of the graph associated with the stochastic process describing the time evolution.

On using shaped light pulses to control the selectivity of product formation in a chemical reaction: An application to a multiple level system

A previously reported method for shaping electromagnetic field pulses to achieve chemical selectivity is extended and applied to a simple multiple level model system. The pulse shaping approach is based on optimal control theory, where both the time‐dependent Schrodinger equation and the constant pulse energy are used as constraints on the variational scheme. A conjugate gradient direction method is used to direct the convergence of the iterative process used to calculate the optimum pulse shape. The method is applied to a five‐level system interacting with an optical (laser) field. Results demonstrating selectivity and stability are compared to those of other recent related investigations.

Experimental evidence for microscopic chaos

Many macroscopic dynamical phenomena, for example in hydrodynamics and oscillatory chemical reactions, have been observed to display erratic or random time evolution, in spite of the deterministic character of their dynamics—a phenomenon known as macroscopic chaos. On the other hand, it has been long supposed that the existence of chaotic behaviour in the microscopic motions of atoms and molecules in fluids or solids is responsible for their equilibrium and non-equilibrium properties. But this hypothesis of microscopic chaos has never been verified experimentally. Chaotic behaviour of a system is characterized by the existence of positive Lyapunov exponents, which determine the rate of exponential separation of very close trajectories in the phase space of the system. Positive Lyapunov exponents indicate that the microscopic dynamics of the system are very sensitive to its initial state, which, in turn, indicates that the dynamics are chaotic; a small change in initial conditions will lead to a large change in the microscopic motion. Here we report direct experimental evidence for microscopic chaos in fluid systems, obtained by the observation of brownian motion of a colloidal particle suspended in water. We find a positive lower bound on the sum of positive Lyapunov exponents of the system composed of the brownian particle and the surrounding fluid.