Kinetic Theory Estimates for the Kolmogorov-Sinai Entropy and the Largest Lyapunov Exponents for Dilute, Hard-Ball Gases and for Dilute, Random Lorentz Gases

The kinetic theory of gases provides methods for calculating Lyapunov exponents and other quantities, such as Kolmogorov-Sinai entropies, that characterize the chaotic behavior of hard-ball gases. Here we illustrate the use of these methods for calculating the Kolmogorov-Sinai entropy, and the largest positive Lyapunov exponent, for dilute hard-ball gases in equilibrium. The calculation of the largest Lyapunov exponent makes interesting connections with the theory of propagation of hydrodynamic fronts. Calculations are also presented for the Lyapunov spectrum of dilute, random Lorentz gases in two and three dimensions, which are considerably simpler than the corresponding calculations for hard-ball gases. The article concludes with a brief discussion of some interesting open problems.Comment: 41 pages (REVTEX); 7 figs., 4 of which are included in LaTeX source. (Fig.7 doesn't print well on some printers) This revised paper will appear in "Hard Ball Systems and the Lorentz Gas", D. Szasz ed., Encyclopaedia of Mathematical Sciences, Springe

Shuttle/spacelab MMAP/electromagnetic environment experiment phase B definition study

Progress made during the first five months of the Phase B definition study for the MMAP/Electromagnetic Environment Experiment (EEE) was described. An antenna/receiver assembly has been defined and sized for stowing in a three pallet bay area in the shuttle. Six scanning modes for the assembly are analyzed and footprints for various antenna sizes are plotted. Mission profiles have been outlined for a 400 km height, 57 deg inclination angle, circular orbit. Viewing time over 7 geographical areas are listed. Shuttle interfaces have been studied to determine what configuration the antenna assembly must have to be shared with other experiments of the Microwave Multi-Applications Payload (MMAP) and to be stowed in the shuttle bay. Other results reported include a frequency plan, a proposed antenna subsystem design, a proposed receiver design, preliminary outlines of the experiment controls and an analysis of on-board and ground data processing schemes

Chaotic Properties of Dilute Two and Three Dimensional Random Lorentz Gases II: Open Systems

We calculate the spectrum of Lyapunov exponents for a point particle moving in a random array of fixed hard disk or hard sphere scatterers, i.e. the disordered Lorentz gas, in a generic nonequilibrium situation. In a large system which is finite in at least some directions, and with absorbing boundary conditions, the moving particle escapes the system with probability one. However, there is a set of zero Lebesgue measure of initial phase points for the moving particle, such that escape never occurs. Typically, this set of points forms a fractal repeller, and the Lyapunov spectrum is calculated here for trajectories on this repeller. For this calculation, we need the solution of the recently introduced extended Boltzmann equation for the nonequilibrium distribution of the radius of curvature matrix and the solution of the standard Boltzmann equation. The escape-rate formalism then gives an explicit result for the Kolmogorov Sinai entropy on the repeller.Comment: submitted to Phys Rev

The Kolmogorov-Sinai Entropy for Dilute Gases in Equilibrium

We use the kinetic theory of gases to compute the Kolmogorov-Sinai entropy per particle for a dilute gas in equilibrium. For an equilibrium system, the KS entropy, h_KS is the sum of all of the positive Lyapunov exponents characterizing the chaotic behavior of the gas. We compute h_KS/N, where N is the number of particles in the gas. This quantity has a density expansion of the form h_KS/N = a\nu[-\ln{\tilde{n}} + b + O(\tilde{n})], where \nu is the single-particle collision frequency and \tilde{n} is the reduced number density of the gas. The theoretical values for the coefficients a and b are compared with the results of computer simulations, with excellent agreement for a, and less than satisfactory agreement for b. Possible reasons for this difference in b are discussed.Comment: 15 pages, 2 figures, submitted to Phys. Rev.

Time-oscillating Lyapunov modes and auto-correlation functions for quasi-one-dimensional systems

The time-dependent structure of the Lyapunov vectors corresponding to the steps of Lyapunov spectra and their basis set representation are discussed for a quasi-one-dimensional many-hard-disk systems. Time-oscillating behavior is observed in two types of Lyapunov modes, one associated with the time translational invariance and another with the spatial translational invariance, and their phase relation is specified. It is shown that the longest period of the Lyapunov modes is twice as long as the period of the longitudinal momentum auto-correlation function. A simple explanation for this relation is proposed. This result gives the first quantitative connection between the Lyapunov modes and an experimentally accessible quantity.Comment: 4 pages, 3 figure

Wave packet autocorrelation functions for quantum hard-disk and hard-sphere billiards in the high-energy, diffraction regime

We consider the time evolution of a wave packet representing a quantum particle moving in a geometrically open billiard that consists of a number of fixed hard-disk or hard-sphere scatterers. Using the technique of multiple collision expansions we provide a first-principle analytical calculation of the time-dependent autocorrelation function for the wave packet in the high-energy diffraction regime, in which the particle's de Broglie wave length, while being small compared to the size of the scatterers, is large enough to prevent the formation of geometric shadow over distances of the order of the particle's free flight path. The hard-disk or hard-sphere scattering system must be sufficiently dilute in order for this high-energy diffraction regime to be achievable. Apart from the overall exponential decay, the autocorrelation function exhibits a generally complicated sequence of relatively strong peaks corresponding to partial revivals of the wave packet. Both the exponential decay (or escape) rate and the revival peak structure are predominantly determined by the underlying classical dynamics. A relation between the escape rate, and the Lyapunov exponents and Kolmogorov-Sinai entropy of the counterpart classical system, previously known for hard-disk billiards, is strengthened by generalization to three spatial dimensions. The results of the quantum mechanical calculation of the time-dependent autocorrelation function agree with predictions of the semiclassical periodic orbit theory.Comment: 24 pages, 13 figure

Leading Pollicott-Ruelle Resonances and Transport in Area-Preserving Maps

The leading Pollicott-Ruelle resonance is calculated analytically for a general class of two-dimensional area-preserving maps. Its wave number dependence determines the normal transport coefficients. In particular, a general exact formula for the diffusion coefficient D is derived without any high stochasticity approximation and a new effect emerges: The angular evolution can induce fast or slow modes of diffusion even in the high stochasticity regime. The behavior of D is examined for three particular cases: (i) the standard map, (ii) a sawtooth map, and (iii) a Harper map as an example of a map with nonlinear rotation number. Numerical simulations support this formula.Comment: 5 pages, 1 figur

From Lyapunov modes to the exponents for hard disk systems

We demonstrate the preservation of the Lyapunov modes by the underlying tangent space dynamics of hard disks. This result is exact for the zero modes and correct to order $\epsilon$ for the transverse and LP modes where $\epsilon$ is linear in the mode number. For sufficiently large mode numbers the dynamics no longer preserves the mode structure. We propose a Gram-Schmidt procedure based on orthogonality with respect to the centre space that determines the values of the Lyapunov exponents for the modes. This assumes a detailed knowledge of the modes, but from that predicts the values of the exponents from the modes. Thus the modes and the exponents contain the same information

Rational Approximate Symmetries of KdV Equation

We construct one-parameter deformation of the Dorfman Hamiltonian operator for the Riemann hierarchy using the quasi-Miura transformation from topological field theory. In this way, one can get the approximately rational symmetries of KdV equation and then investigate its bi-Hamiltonian structure.Comment: 14 pages, no figure