Ronnie Mainieri

It is shown that a spin system with long range interactions can be converted into a chaotic dynamical system that is differentiable and low-dimensional. The thermodynamic limit of the spin system is then equivalent to studying the long term behavior of the dynamical system. Cycle expansions of chaotic systems (expansion of the Fredholm determinant)...

Spiral-defect populations in low-Prandtl number Rayleigh-Bénard convection with slow rotation about a vertical axis were measured in carbon dioxide at high pressure. The results indicate that spirals act like "thermally excited" defects and that the winding direction of a spiral is analogous to a magnetic spin. Rotation about a vertical axis, the s...

A cycle expansion for the Lyapunov exponent of a product of random matrices is derived. The formula is non-perturbative and numerically effective, which allows the Lyapunov exponent to be computed to high accuracy. In particular, the free energy and the heat capacity are computed for the one-dimensional Ising model with quenched disorder. The formu...

In this series of four 90 minutes lectures given at the University of Porto, June 14-17 2010, we present some techniques introduced in the past 5 years to study the spectrum of transfer operators associated to differentiable hyperbolic dynamical systems, in particular through dynamical zeta functions or dynamical determinants (which are power serie...

ented by a single point in an abstract space called state space or phase space J. As the system changes, so does the representative point in phase space. We refer to the evolution of such points as dynamics, and the function ft which specifies where the representative point is at time t as the evolution rule. If there is a definite rule f that tell...

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Cycle expansions applied to classical and quantum physical systems. This particular chapter is devoted to explaining diffractive effects in quantum systems using cycle expansion. http://chaosbook.org/version15/chapters/whelan-2p.pdf

The trace formula for the evolution operator associated with nonlinear stochastic flows with weak additive noise is cast in the path integral formalism. We integrate over the neighbourhood of a given saddlepoint exactly by means of a smooth conjugacy, a locally analytic nonlinear change of field variables. The perturbative corrections are transferr...

In partially linear systems, such as the Lorenz model, chaotic synchronization is possible in only some of the variables. We show that, for the nonsynchronizing variable, synchronization up to a scale factor is possible. We explain the mechanism for this projective form of chaotic synchronization in three-dimensional systems. Projective synchroniza...

We study the a.c. transport through a two-dimensional quantum point contact (QPC) using a Boltzmann-like kinetic equation derived for the partial Wigner distribution function. An integral equation for a potential inside a QPC is solved numerically. It is shown that the electric field inside a QPC is an inhomogeneous function of the spatial coordina...

Periodic orbit theory is all effective tool for the analysis of classical and quantum chaotic systems. In this paper we extend this approach to stochastic systems, in particular to mappings with additive noise. The theory is cast in the standard field-theoretic formalism and weak noise perturbation theory written in terms of Feynman diagrams. The r...

This is the final report of a three-year, Laboratory Directed Research and Development (LDRD) project at the Los Alamos National Laboratory (LANL). The authors calculated observables in strongly chaotic systems. This is difficult to do because of a lack of a workable orbit classification for such systems. This is due to global geometrical informati...

We present the results of molecular dynamics modeling on the structural properties of grain boundaries (GB) in thin polycrystalline films. The transition from crystalline boundaries with low mismatch angle to amorphous boundaries is investigated. It is shown that the structures of the GBs satisfy a thermodynamical criterion. The potential energy of...

We review the convergence of chaotic integrals computed by Monte Carlo simulation, the trace method, dynamical zeta function, and Fredholm determinant on a simple one-dimensional example: the parabola repeller. There is a dramatic difference in convergence between these approaches. The convergence of the Monte Carlo method follows an inverse power...

Using molecular dynamics simulations, the grain boundaries in thin polycrystalline silicon films (considered as promising material for future nanoelectronic devices) are investigated. It is shown that in polysilicon film with randomly oriented grains the majority of grain boundaries are disordered. However, some grains with small mutual orientation...

Complex systems are difficult to characterize and to simulate. By considering a series of explicit systems, through experiments and analysis, this project has shown that dynamical systems can be used to model complex systems. A complex dynamical system requires an exponential amount of computer work to simulate accurately. Direct methods are not pr...

The idea of building autonomous robots that can carry out complex and nonrepetitive tasks is an old one, so far unrealized in any meaningful hardware. Tilden has shown recently that there are simple, processor-free solutions to building autonomous mobile machines that continuously adapt to unknown and hostile environments, are designed primarily to...

This is the final report of a three-year, Laboratory-Directed Research and Development (LDRD) project at the Los Alamos National Laboratory (LANL). Research was performed in analytic and computational techniques for dealing with hard chaos, especially the powerful tool of cycle expansions. This work has direct application to the understanding of el...

We review the convergence of chaotic integrals computed by Monte Carlo simulation, the trace method, dynamical zeta function, and Fredholm determinant on a simple one-dimensional example: the parabola repeller. There is a dramatic difference in convergence between these approaches. The convergence of the Monte Carlo method follows an inverse power...

Cycle expansions are an efficient scheme for computing the properties of chaotic systems. When enumerating the orbits for a cycle expansion not all orbits that one would expect at first are present --- some are pruned. This pruning leads to convergence difficulties when computing properties of chaotic systems. In numerical schemes, I show that prun...

We obtain a five-step approximation to the quasiperiodic dynamic scaling function for experimental Rayleigh-Be'nard convection data. When errors are taken into account in the experiment, the f(alpha) spectrum of scalings is equivalent to just two of these five scales. To overcome this limitation, we develop a robust technique for extracting the sca...

Nanometer scale electronics present a challenge for the computer architect. These quantum devices have small gain and are difficult to interconnect. I have analyzed current device capabilities and explored two general design requirements for the design of computers: error correction and long range connections. These two principles follow when Turin...

We repeat the numerical experiments for diffusion limited aggregation (DLA) and show that there is a potentially infinite set of conserved quantities for the long time asymptotics. We connect these observations with the exact integrability of the continuum limit of the DLA (quasistatic Stefan problem). The conserved quantities of the Stefan problem...

Trajectory scaling functions are the basic element in the study of chaotic dynamical systems from which any long-time average can be computed. They have never been extracted from an experimental time series, the reason being their sensitivity to noise. It is shown, by numerical simulations, that the scaling function is more sensitive to drift in th...

By viewing the covers of a fractal as a statistical mechanical system, the exact capacity of a multifractal is computed. The procedure can be extended to any multifractal described by a scaling function to show why the capacity and Hausdorff dimension are expected to be equal.

Trajectory scaling functions are the basic element in the study of chaotic dynamical systems, from which any long time average can be computed. It has never been extracted from an experimental time series the reason being its sensitivity to noise. It is shown, by numerical simulations, that the sensitivity of the scaling function is to drift in the...

For one dimensional maps the trajectory scaling functions is invariant under coordinate transformations and can be used to compute any ergodic average. It is the most stringent test between theory and experiment, but so far it has proven difficult to extract from experimental data. It is shown that the main difficulty is a dephasing of the experime...

The use of cycle expansions for spin systems with long-range interactions is explored numerically. It is found that the thermodynamic {zeta} function is an effective tool, both in practice and in principle, for the study of Ising models with power-law interactions. To deal with phase transitions the cycle expansion is factorized, and accurate phase...

Using cycle expansion for the thermodynamic zeta function, a formula is derived for the Lyapunov exponent of a product of random hyperbolic matrices chosen from a discrete set. This allows for an accurate numerical solution of the Ising model in one dimension with quenched disorder. The formula is compared with weak disorder expansions and with the...

We study universal scaling properties of quasiperiodic Rayleigh-Benard convection in a ³He--superfluid-⁴He mixture. The critical line is located in a parameter space of Rayleigh and Prandtl numbers using a transient-Poincare-section technique to identify transitions from nodal periodic points to spiral periodic points within resonance horns. We...

We study the frequency lockings of two intrinsic hydrodynamic modes of a convecting ³He-superfluid-⁴He mixture by independently varying the Rayleigh and Prandtl numbers. We establish points on the critical line in this parameter space using a transient technique to locate the spiral-node transition in the interior of three resonance horns. Unive...

Hopf bifurcations in two-dimensional maps give rise to closed invariant curves and circle maps induced on these curves. It is not obvious whether the induced maps will exhibit the full array of scaling phenomena familiar from the study of one-dimensional maps. In the present work we numerically study a variable Jacobian map (the coupled logistic ma...

spinsystem with long-range interactions can be converted into a chaotic dynamical system that is differentiable and low-dimensional. The thermodynamic limit quantities of the spin system are then equivalent to lo ng time averages of the dynamical system. In this way the spin system averages can be recast as the cycle expansions. If the resulting dy...