or

will take place at

(Italy) on friday 7- saturday 8 may, 2004.

Gianni Arioli: U. Milano Politecnico

New methods for multi-mode periodic solutions for the Fermi-Pasta-Ulam model

Abstract: We introduce two novel methods for studying periodic solutions of the FPU beta-model, both numerically and rigorously. One is a variational approach, based on the dual formulation of the problem, and the other involves computer-assisted proofs. These methods are used e.g. to construct a new type of solutions, whose energy is spread among several modes, associated with closely spaced resonances.

Roberto Artuso: U. dell'Insubria (cioe' Como)

"Weak chaos and anomalous transport"

Abstract: The presence of a mixed phase space, or even the appearance of a single marginal fixed point in a chaotic system may greatly influence transport properties. In particular Gaussian character is lost, diffusion is typically anomalous and the whole spectrum of transport exponents (ruling the asymptotic behavior of arbitrary q-order moments) may be highly non-trivial. We have provided a way to compute this spectrum from periodic orbits, via cycle expansions, and applied the theory to chains of 1d maps and Lorentz gas, where we explicitly show how multiscaling in the spectrum typically arise.

Dario Bambusi: U. Milano 1

Some Local Normal Form Results for FPU Chains in the Thermodynamic Limit

Abstract: I will present some local normal form results for FPU chains. Explicit estimates of the remainder valid for a class of states with finite specific energy will be provided. Some dynamical consequences will also be obtained. A method to obtain lower bounds for the equipartition time will also be discussed. Such a method is still heuristic, but it gives results in good agreement with numerical computations.

Giancarlo Benettin: U. Padova

"Preliminary results on a two-dimensional version of the FPU model"

Abstract: some recent results on FPU appear to profit in a possibly important way of the one-dimensional character of the FPU model. It looks then interesting to repeat some crucial numerical experiments in higher dimension. Some recent highly preliminary results on a two-dimensional model with triangular lattice will be presented.

Gero Friesecke: University of Warwick / TU Munich

"Long-time stability of FPU solitary waves and recurrence of Fourier spectrum"

Abstract: We will begin with a brief history of the discovery of coherent modes in approximations to (Kruskal/Zabusky), special cases of (Toda) and general (Friesecke/Wattis, Friesecke/Pego, Iooss, G.James) FPU lattices.

We will then explain the broad ideas behind the recent rigorous result (joint with Robert Pego, Maryland) that low-energy FPU solitary waves are stable globally in time, in the sense of start close, stay close, and its corollary that on each low-energy surface an open set of initial data has perpetually recurrent Fourier spectrum. (Nonlinearity 17 2004 229-252)

Various interesting features of FPU not limited to coherent modes or Fourier recurrence will come to light along the way, including the existence of a natural symplectic form for FPU, or rigorous relationships to certain nonlinear partial differential equations in appropriate scaling regimes.

Luigi Galgani: U. Milano 1

The fpu problem and the classical theory of specific heats

Abstract. The thesis is maintained that, according to all the available results, the FPU paradox (absence of equipartition) survives in the thermodynamic limit. This should however be intended in the following weak sense: there exists a critical specific energy below which a metastable glassy--like state is formed, which would later evolve towards equipartition on a much longer time scale. A general review of the available results is given, with emphasis on some rather old results of Fucito et al. and of Parisi, and to some very recent results. A possible thermodynamic interpretation is discussed.

Antonio Giorgilli: U. Milano 2 (Bicocca)

"Metastable states in the Fermi-Pasta-Ulam system"

The phenomenon of "packets of modes" in the FPU system is illustrated. If the total energy is initially given to a few low--frequency modes then one observes a fast relaxation to a state with energy distributed on what we call "natural packets of modes", involving all modes up to a certain characteristic frequency depending only on the initial specific energy given to the system. Such a packet persists for a long time that increases exponentially with the inverse of the specific energy, so that it becomes rapidly unobservable. The numerical calculations indicate that this behaviour persists in the thermodynamic limit.

Roberto Livi: U. Firenze

Heat transport and the FPU model

Abstract: The FPU model was introduced for tackling the classical problem of heat conduction raised by Pierre Debye in 1914: the nonlinearity inherent real solids was conjectured to be the basic ingredient for producing finite heat conductivity. As Peierls showed, this is correct at least in 3d. Conversely, in 1d and 2d conservation laws impose stronger constraints to statistical fluctutations and anomalous transport is found. Beyond pure academic interest, a divergent heat conductivity in the thermodynamic limit seems to characterize several low--dimensional systems of physical interest, e.g. polymers, biomolecules, carbon nanotubes, thin films. In this seminar we survey some of the most recent numerical and analytical results about the FPU model.

Karsten Matthies: Freie U., Berlin

Atomic-scale localization of high-energy solitary waves for Lennard-Jones type interactions

Abstract: The FPU system on an infinite lattice carries localized travelling waves for generic nonlinear potentials. In this talk we discus the behavior of these waves in the high-energy limit for Lennard-Jones type interactions. The limiting profile is a highly discrete wave, concentrated on a single atomic spacing. Hence dispersionless energy transport is not restricted to the long-wave regime. The behavior will be related to a hard-sphere model.

Stefano Ruffo: U. Firenze

Forced-damped FPU lattices

Abstract: We discuss numerical results obtained when forcing and damping in various ways a FPU lattice. A variety of standing and traveling localized excitations are obtained, which bear some connection, in the low amplitude limit, with exact solutions of the continuum mKdV equation, or, in the large amplitude case, with approximate solutions obtained by truncating exact solutions of the form of nonlinear waves.

Angelo Vulpiani: U. Roma 1

Diffusion: from stochastic processes to chaos and beyond

Abstract: After one century from Einstein's works, Brownian Motion and diffusion are still central issues in physics. We start with an introduction to the subject and an analysis of stochastic and deterministic models for the phenomenon. Then we discuss about the possibility of determining the microscopic nature of diffusion by means of data analysis. Finally we focus on deterministic diffusion and, in particular, on what are the necessary conditions for having a (genuine) large scale diffusive motion.

Susanna Terracini, U. Milano Bicocca

"New analytic result about the periodic problem fro FPU-chains"

Abstract: In the recent work:

[AKT] G. Arioli, H. Koch, S. Terracini, preprint 2003,

the authors suggest the presence an original mechanism of secundary bifurcation, apparently connected with the concentration of the frequencies of primary bifurcation: the effectiveness of this mechanism is verified experimentally, both from the numerical point of view and with the use of computer-assisted proofs. We intend to give analytic proofs of the secondary bifurcation mechanism suggested in [AKT] and carried out in a recent study:

[STT] M. Tarallo, E. Serra, S. Terracini, preprint, 2004.

The approach exploits the of variational nature of the problem and requires a careful analysis of the jumps of Morse index along the primary bifurcation branches: detection of changes in the index will imply existence of secondary bifurcations. The achievement of the goal therefore depends on the understanding of the following items: (a) when the phenomena of concentration of frequencies observed in [AKT] take place; (b) how such phenomena are connected to possible changes of the Morse index. We treat the simplest case of concentration, namely the one that involves only one pair of frequencies. A more ambitious goal, will deal with more complex types of concentration (those involving more than two frequencies) and with the search for a deeper understanding of the properties of the solutions found, and in particular, the reasons of energy flow among the modes.

Stefano Olla: CEREMADE, Paris

Heat transport and non-equilibrium fluctuations for a chain of oscillators with random interaction.

Abstract: We consider a one-dimensional chain of N oscillators whose time evolution is given by a combination of Hamiltonian and stochastic dynamics. The Hamiltonian part is given by the usual linear chain of harmonic nearest neighbors oscillators. The stochastic part of the dynamics comes from a random exchange of kinetic energy between the nearest neighbors particles. The end-point particles are then coupled to two heath baths at different temperatures. We study the heath transport and the fluctuations in the corresponding stationary state, as the size of the system N tends to infinity. We prove a Fourier law, with conductivity decreasing like 1/N. Then we show that the temperature profile has gaussian fluctuations with long range correlations.

Antonio Ponno: U. Milano 1

"Scaling laws of the energy cascade in FPU models."

Abstract: Using canonical perturbation theory on a formal level, some scaling laws characterizing the energy transfer from large to small spatial scales in FPU models are justified. Fundamental quantities such as relaxation times and minimal wavelengths are shown to depend on the specific energy of the system.

Simone Paleari

"We consider the relaxation time to equipartition for FPU systems: we investigate, by means of numerical methods, how it scales with the specific energy of the chain, and the dependence of these behaviors with respect to the number of particles. Using the so called spectral entropy as a numerical indicator, we obtain strong indication towards exponentially, with respect to an inverse power of the specific energy, long relaxations times, also in the thermodynamic limit. We will also briefly comment on the use of the maximal Lyapunov exponent as an indicator of equipartition."

Yannick Sire: INSA-Toulouse

"Travelling breathers with exponentially small tails in a chain of nonlinear oscillators"

Abstract: We study the existence of travelling breathers in Klein-Gordon chains, which consist in one-dimensional networks of nonlinear oscillators in an anharmonic on-site potential, linearly coupled to their nearest neighbors. Travelling breathers are spatially localized solutions which appear time periodic in a referential in translation at constant velocity. Approximate solutions of this type have been constructed in the form of modulated plane waves, whose envelopes satisfy the nonlinear Schr\"odinger equation (M. Remoissenet, {\sl Phys. Rev. B} 33, n.4, 2386 (1986)). In the case of travelling waves (where the phase velocity of the plane wave equals the group velocity of the wave packet), the existence of nearby exact solutions has been proved by Iooss and Kirchg\"assner, who showed in addition that the exact travelling wave solutions possess a nonvanishing exponentially small oscillatory tail (G. Iooss, K. Kirchg\"assner, {\sl Commun. Math. Phys.} 211, 439-464 (2000)). However, a rigorous existence result has been lacking in the more general case when phase and group velocities are different. We present results dealing with the case when the breather period and the inverse of its velocity are commensurate. We show that the center manifold reduction method introduced by Iooss and Kirchg\"assner is still applicable when the problem is formulated in an appropriate way. This allows us to reduce the problem locally to a finite dimensional reversible system of ordinary differential equations, whose principal part admits homoclinic solutions to quasi-periodic orbits under general conditions on the potential. For an even potential, using the additional symmetry of the system, we obtain homoclinic orbits to small periodic ones for the full reduced system. For the oscillator chain, these orbits correspond to exact small amplitude travelling breather solutions superposed to an exponentially small oscillatory tail. Their principal part (excluding the tail) coincides at leading order with the nonlinear Schr\"odinger approximation.

Further short communcations titles and abstracts will be communicated as soon as possible.

Interested participants are invited to register: this is essential for those who wish to present communications or posters as well as for those who apply for support. Due to funds restrictions no secretarial help can be offered and participants will have to book hotels directly by fax, phone or internet. for a list of hotels often used by visitors click

e-mail: gallavotti@roma1.infn.it