Energy Localisation and Transfer

The following sections are included:

• PREFACE

• PARTICIPANTS

• CONTENTS

This chapter provides a description of the main computational tools for the study of discrete breathers. It starts with the observation of breathers through simple numerical runs, their study using targeted initial conditions, and discrete breather impact on transient processes and thermal equilibrium. Next we describe a set of numerical methods to obtain breathers up to machine precision, including the Newton method. We explain the basic approaches of computing the linear stability properties of these excitations, and proceed to compute wave scattering by discrete breathers, and to briefly discuss computational aspects of studying dissipative breathers. In a final part of this chapter we present computational approaches of studying quantum discrete breathers.

These lecture-notes are meant to provide newcomers with an overview of the impact of vibrational spectroscopy in the field of nonlinear dynamics of atoms and molecules, in the perspective of energy localization. In the introduction, the terminology of nonlinear excitations and tentative experimental evidences are briefly recalled in a brief historical perspective. The basic principles of vibrational spectroscopy are presented in section 11 for infrared, Raman and inelastic neutron scattering. The potentialities for each technique to probing energy localization are discussed. In section 12, nonlinear dynamics in isolated molecules are treated within the framework of normal versus local mode representations. It is shown that these complementary representations are not necessarily distinctive of weak versus strong anharmonicity, in the context of chemical complexity. It is emphasized that local modes and energy localization are totally independent concepts. In section 4, examples of nonlinear dynamics in crystals are reviewed: multiphonon bound states, strong coupling between phonons and electrons probed with resonance Raman, local modes and quantum rotation in one-dimension probed with inelastic neutron scattering, strong coupling in hydrogen-bonded crystals and self-trapping probed with time-resolved vibrational-spectroscopy. The extended character of eigenstates in crystals free of impurities and disorder, the nature of the interaction of periodic lattices with plane waves, the Franck-Condon principle and the particle-wave duality in the quantum regime are key factors preventing observation of energy localization. It is shown that free spatially-localized nondissipative classical waves give rise to free pseudoparticles that behave as planar waves in the quantum regime. In conclusion, a clear demonstration that energy localization corresponds to eigenstates is eagerly expected for further evidencing these states with vibrational spectroscopy.

A key mathematical concept for the understanding of the dynamics of coherent structures in typical (as opposed to integrable) spatially extended systems is that of a slow manifold. The theory of slow manifolds is reviewed and extended here, with special emphasis on the Hamiltonian context. For a broad range of dynamical systems with a small separation of timescales parameter ε, given any integer r ≥ 1, existence of a slow manifold and slow vector field εX on it, containing all nearby equilibria and with error field of order (ε|X|)^r, is proved. In the Hamiltonian case, Hamiltonian slow dynamics is constructed and the theory is extended to slow manifolds with an internal oscillation. Applications are given to a variety of problems, including the interactions of localised excitations in several types of spatially extended system. In particular, the theory helps one to understand the interaction and propagation of discrete breathers.

Josephson-junction arrays are excellent experimental systems for the study of nonlinear phenomena in general and nonlinear localised excitations in particular. This chapter is an introduction to the physics of vortices, kinks and breathers in Josephson arrays. Special emphasis is placed in the description of discrete breather solutions.

The purpose of this review is to give an introduction into experiments with Josephson junction arrays. For many years this experimental field has been the test ground of nonlinear science. Josephson junctions and arrays are unique hardware to study solitons, synchronization, chaos, pattern formation, and localized excitations. After introducing the reader into fabrication and measurement techniques, I will discuss several experiments with Josephson arrays in both classical and quantum regimes.

As polymers, proteins have very unusual properties. The main one is that they are able to fold, that is, under physiological conditions, to adopt a very well defined three-dimensional structure. Such a folding process has been studied using simple lattice models that were shown to share many properties with natural proteins. Simple models, where a protein is represented as a network of harmonic springs, have also recently proved useful for studying the other functionally important motions of proteins, namely, their conformational changes. Both kinds of models could be used in order to test the hypothesis that non-linear energy localisation phenomena may occur in proteins and play a role in their functional dynamics.

We review the capability of nonlinear vibrational spectroscopy to study vibrational self-trapping in hydrogen-bonded molecular crystals. For that purpose, the two relevant coupling mechanisms, excitonic coupling and nonlinear exciton-phonon coupling, are first introduced separately using appropriately chosen molecular systems as examples. Both coupling mechanisms are subsequently combined, yielding vibrational selftrapping. The experiments unambiguously prove that both the N-H and the C=O band of crystalline acetanilide (ACN), a model system for proteins, show vibrational self-trapping. The C=O band is self-trapped only at low enough temperature, while thermally induced disorder destroys the mechanism at room temperature. The binding energy of the N-H band, on the other hand, is considerably larger and self-trapping survives thermal fluctuations even at room temperature.

This report, initially prepared to introduce a discussion on “breathers in biomolecules”, discusses some experimental results and their connection with theory. Classical vibrations are considered first and quantum effects are discussed in a second part. The paper does not intend to be a complete review of nonlinear localisation in molecules, but instead it introduces and discusses a few examples with the aim of stimulating thoughts and hopefully further experimental studies or theoretical investigations paying particular attention to the experimental aspects.

The paper discusses the role of nonlinear excitations at finite temperatures. The motivating principle is the description of one-dimensional thermodynamic instabilities (e.g. the Peyrard-Bishop (PB) model of DNA denaturation) as driven by the thermal statistical properties of domain-walls of macroscopic energy. The possible relevance of independent local modes to the finite-temperature spectra of the PB model is discussed. Finally, a minimal model of the helix-coil transition in polypeptides is presented; the model exhibits cooperativity comparable to that observed experimentally in polyalanine and, as such, it offers an interesting perspective for identifying finite-energy nonlinear structures as “agents” of the helix-coil transition.

We review nonlinearity-induced localization in discrete systems with emphasis on theory and numerical calculations on models used in materials physics to describe interatomic interactions. We discuss how the concept of discrete breather or intrinsic localized mode could become an important tool for understanding nanoscale phenomena in molecules and solids. A particularly attractive field of application for nonlinear localized excitations is the controlled and directed transport of energy. We discuss the recently proposed targeted transfer of localized excitations based on the concept of nonlinear resonance and its potential applications. As an area for such applications, we present directional ultrafast electron transfer at low temperatures using this selective transfer mechanism and we give examples from biological electron transfer. We finally discuss possible applications in nanotechnology and biomolecules.

The following sections are included:

• INDEX