Stefano Ruffo

Chaotic behavior in nonlinear Hamiltonian systems and equilibrium statistical mechanics

The relation between chaotic dynamics of nonlinear Hamiltonian systems and equilibrium statistical mechanics in its canonical ensemble formulation has been investigated for two different nonlinear Hamiltonian systems. We have compared time averages obtained by means of numerical simulations of molecular dynamics type with analytically computed ensemble averages. The numerical simulation of the dynamic counterpart of the canonical ensemble is obtained by considering the behavior of a small part of a given system, described by a microcanonical ensemble, in order to have fluctuations of the energy of the subsystem. The results for the Fermi-Pasta-Ulam model (i.e., a one-dimensional anharmonic solid) show a substantial agreement between time and ensemble averages independent of the degree of stochasticity of the dynamics. On the other hand, a very different behavior is observed for a chain of weakly coupled rotators, where linear exchange effects are absent. In the high-temperature limit (weak coupling) we have a strong disagreement between time and ensemble averages for the specific heat even if the dynamics is chaotic. This behavior is related to the presence of spatially localized chaos, which prevents the complete filling of the accessible phase space of the system. Localized chaos is detected by the distribution of all the characteristic Liapunov exponents.

The spectrum of a one-dimensional hierarchical model

The spectrum of a discrete Schrödinger operator with a hierarchically distributed potential is studied both by a renormalization group technique and by numerical analysis. A suitable choice of the potential makes it possible to reduce the original problem to a two-dimensional map. Scaling laws for the band-edge energyEbe and for the integrated density of states η are predicted together with the global properties of the spectrum. Different scaling regimes are obtained depending on a hierarchy positive parameterR: for R<1/2 the usual scaling laws for the periodic case are obtained, while forR>1/2 the scaling behavior depends explicitly onR.

Kolmogorov Pathways from Integrability to Chaos and Beyond

Two limits of Newtonian mechanics were worked out by Kolmogorov. On one side it was shown that in a generic integrable Hamiltonian system, regular quasi-periodic motion persists when a small perturbation is applied. This result, known as Kolmogorov-Arnold-Moser (KAM) theorem, gives mathematical bounds for integrability and perturbations. On the other side it was proven that almost all numbers on the interval between zero and one are uncomputable, have positive Kolmogorov complexity and, therefore, can be considered as random. In the case of nonlinear dynamics with exponential (i.e. Lyapunov) instability this randomnesss, hidden in the initial conditions, rapidly explodes with time, leading to unpredictable chaotic dynamics in a perfectly deterministic system. Fundamental mathematical theorems were obtained in these two limits, but the generic situation corresponds to the intermediate regime between them. This intermediate regime, which still lacks a rigorous description, has been mainly investigated by physicists with the help of theoretical estimates and numerical simulations. In this contribution we outline the main achievements in this area with reference to speci.c examples of both lowdimensional and high-dimensional dynamical systems. We shall also discuss the successes and limitations of numerical methods and the modern trends in physical applications, including quantum computations.

Apparent Fractal Dimensions in the HMF Model

Abstract We show that recent observations of fractal dimensions in the μ‐space of N‐body Hamiltonian systems with long‐range interactions are due to finite N and finite resolution effects. We provide strong numerical evidence that, in the continuum (Vlasov) limit, a set which initially is not a fractal (e.g., a line in 2D) remains such for all finite times. We perform this analysis for the Hamiltonian mean field (HMF) model, which describes the motion of a system of N fully coupled rotors. The analysis can be indirectly confirmed by studying the evolution of a large set of initial points for the Chirikov standard map.

Phase Transitions in Convection Experiments

We discuss a phase transition observed in a convection experiment in an annular geometry in terms of Probabilistic Cellular Automata (PCA) rules. We introduce a simple toy model which reproduces some of the behaviors observed in the experiment. We investigate for this model the effect of “noise” superposed on a second order phase transition.

Periodic Orbits in a Coupled Map Lattice Model

The main feature of the coupled map lattice model that we investigate is the presence of periodic solutions. These can be studied analytically for homogeneous initial conditions and there is evidence that this dynamics provides some information for generic initial conditions. Due to such periodic properties this model could be helpful for the study of turbulent behaviour in reaction-diffusion processes.

An analysis of intermediate transverse momentum pion and photon production inpp collisions through a QCD resummation mechanism

We propose a resummation of QCD radiative corrections in large transverse momentum hadronic processes, based on the introduction of an infrared cut-off, linked to the confinement radius. An analysis of the inclusive reactionspp → π0X andpp → γX and of γ/π0 ratio is presented.

Lyapunov Spectra in Hamiltonian Dynamics

Let us consider a system of N classical interacting particles. The corresponding Hamilton equations of motion are: n n

BETHE - LIKE APPROXIMATION FOR SELF-AVOIDING RANDOM WALKS AND SURFACES (AND FRUSTRATIONS)

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