We discuss the long time behaviour of a finite energy wave packet in nonlinear Hamiltonians on infinite lattices at arbitrary dimension, exhibiting linear Anderson localization. Strong arguments both mathematical and numerical, suggest for infinite models that small amplitude wave packets may generate stationary quasiperiodic solutions (KAM tori) almost undistinguishable from linear wave packets. The probability of this event is non vanishing at small enough amplitude and goes to unity at amplitude zero. Most other wave packets (non KAM tori) are chaotic. We discuss the Arnold diffusion conjecture (recently partially proven) and propose a modified Boltzmann statistics for wave packets valid in generic models. The consequence is that the probability that a chaotic wave packet spreads to zero amplitude is zero. It must always remain focused around one or few chaotic spots which moves randomly over the whole system and generates subdiffusion.We study a class of Ding Dong models also generating subdiffusion where the nonlinearities are replaced by hard core potentials. Then we prove rigorously that spreading is impossible for any initial wave packet.

Electron Transfer (ET) reactions are modeled by the dynamics of a quantum two-level system (representing the electronic state) coupled to a thermalized bath of classical harmonic oscillators (representing the nuclei degrees of freedom). Unlike for the standard Marcus theory, the complex amplitudes of the electronic state are chosen as reaction coordinates. Then, the dynamical equations at non vanishing temperature become those of an effective Hamiltonian submitted to damping terms and their associated Langevin random forces. The advantage of this new formalism is to extend the original theory by taking into account both ionic and covalent interactions. The standard theory is recovered only when covalent interactions are neglected. Increasing these covalent interactions from zero, the energy barrier predicted by the standard theory first depresses, next vanish (or almost vanish) and for stronger covalent interactions, covalent bond formation takes place of ET. In biochemistry, the standard Marcus theory often fails to explain the enzymatic reactions especially those with non Arrhenius behavior which are barrierless and also dissipate little heat. We claim that this improved theory should yield an interesting tool for understanding them.

We propose a new and general formalism for elementary chemical reactions where quantum electronic variables are used as reaction coordinates. This formalism is in principle applicable to all kinds of chemical reactions ionic or covalent. Our theory reveals the existence of an intermediate situation between ionic and covalent which may be almost barrierless and isoenegetic and which should be of high interest for understanding biochemistry.

When nonlinearity is added to an infinite system with purely discrete linear spectrum, Anderson modes become coupled with one another by terms of higher order than linear, allowing energy exchange between them. It is generally believed, on the basis of numerical simulations in such systems, that any initial wave-packet with finite energy spreads down chaotically to zero amplitude with second moment diverging as a power law of time, slower than standard diffusion (subdiffusion). We present results which suggest that the interpretation of spreading cannot be described as initially believed and that new questions arise and still remain opened. We show that an initially localized wave-packet with finite norm may generate two kinds of trajectories both obtained with nonvanishing probability. The first kind consists of KAM trajectories which are recurrent and do not spread. Empirical investigations suggest that KAM theory may still hold in infinite systems under two conditions: (1) the linearized spectrum is purely discrete, (2) the considered solutions are square summable and not too large in amplitude. We check numerically that in appropriate regions of the parameter space, indeed many initial conditions can be found with finite probability that generate (nonspreading) infinite dimension tori (almost periodic solutions) in a fat Cantor set in (projected) phase space. The second kind consists of trajectories which look initially chaotic and often spread over long times. We first rigorously prove that initial chaos does not necessarily imply complete spreading e.g. for large norm initial wave-packet. Otherwise, in some modified models, no spreading at all is proven to be possible, despite the presence of initial chaos in contradiction with early beliefs. The nature of the limit state is still unknown. However, we attempt to present empirical arguments suggesting that if a trajectory starts chaotically spreading, there will necessarily exist (generally large) critical spreading distances that depend on the disorder realization where the trajectory will be sticking to a dense set of KAM tori. This effect should induce drastic slowing down of the spreading which could be viewed as "inverse Arnold diffusion" since the trajectory approaches KAM tori regions instead of leaving them. We suggest that this effect should self-organize the chaotic behavior and that at long time, the wave-packet might not be spread down to zero, but could have a limit profile with marginal chaos (with singular continuous spectrum), despite a long spatial tail. Further analytical and numerical investigations are required.

We suggest that KAM theory could be extended for certain infinite-dimensional systems with purely discrete linear spectrum. We provide empirical arguments for the existence of square summable infinite-dimensional invariant tori in the random discrete nonlinear Schrödinger equation, appearing with a finite probability for a given initial condition with sufficiently small norm. Numerical support for the existence of a fat Cantor set of initial conditions generating almost periodic oscillations is obtained by analyzing i) sets of recurrent trajectories over successively larger time scales, and ii) finite-time Lyapunov exponents. The norm region where such KAM-like tori may exist shrinks to zero when the disorder strength goes to zero and the localization length

We investigate the long-time behavior of a wave packet initially localized at a single site n₀ in translationally invariant harmonic and anharmonic chains with random interactions. In the harmonic case, the energy profile averaged on time and disorder decays for large |n-n₀| as a power law ≈ C|n-n₀|(⁻η), where η=⁵/₂ and ³/₂ for initial displacement and momentum excitations, respectively. The prefactor C depends on the probability distribution of the harmonic coupling constants and diverges in the limit of weak disorder. As a consequence, the moments of the energy distribution averaged with respect to disorder diverge in time as t(β(ν)) for ν ≥ 2, where β=ν+1-η for ν>η-1 . Molecular-dynamics simulations yield good agreement with these theoretical predictions. Therefore, in this system, the second moment of the wave packet diverges as a function of time despite the wave packet is not spreading. Thus, this only criterion, often considered earlier as proving the spreading of a wave packet, cannot be considered as sufficient in any model. The anharmonic case is investigated numerically. It is found for intermediate disorder that the tail of the energy profile becomes very close to those of the harmonic case. For weak and strong disorders, our results suggest that the crossover to the harmonic behavior occurs at much larger |n-n₀| and larger time.

We suggest that KAM theory could be extended for certain infinite-dimensional systems with purely discrete linear spectrum. We provide empirical arguments for the existence of square summable infinite-dimensional invariant tori in the random discrete nonlinear Schrödinger equation, appearing with a finite probability for a given initial condition with sufficiently small norm. Numerical support for the existence of a fat Cantor set of initial conditions generating almost-periodic oscillations is obtained by analyzing (i) sets of recurrent trajectories over successively larger time scales, and (ii) finite-time Lyapunov exponents. The norm region where such KAM-like tori may exist shrinks to zero when the disorder strength goes to zero and the localization length diverges.

The propagation of pressure fronts (impact solutions) in 1D chains of atoms coupled by anharmonic potentials between nearest neighbor and submitted to damping forces preserving uniform motion, is investigated. Travelling fronts between two regions at different uniform pressures are found numerically and well approximate analytically. It is proven that there are three analytical relations between the impact velocity, the compression, the front velocity and the energy dissipation which only depend on the coupling potential and are \textit{independent} of the damping. Such travelling front solutions cannot exist without damping.

We investigate the dynamics of a macroscopic system which consists of an anharmonic subsystem embedded in an arbitrary harmonic lattice, including quenched disorder. The coupling between both parts is bilinear. Elimination of the harmonic degrees of freedom leads to a nonlinear Langevin equation with memory kernels Gamma(t) and noise term zeta(t) for the anharmonic coordinates q(t) = (q(alpha) (t)). For zero temperature, i.e. for zeta(t) equivalent to 0, we prove that the support of the Fourier transform of Gamma(t) and of the time averaged velocity-velocity correlation functions K(t) of the anharmonic system cannot overlap. As a consequence, the asymptotic solutions can be constant, periodic, quasiperiodic or almost periodic, and possibly weakly chaotic. For a sinusoidal trajectory q(t) with frequency Omega we find that the energy E(T) transferred to the harmonic system up to time T is proportional to T(alpha). If Omega equals one of the phonon frequencies omega(nu), it is alpha = 2. We prove that there is a zero measure set L such that for Omega in its full measure complement R \ L, it is alpha = 0, i.e. there is no energy dissipation. Under certain conditions L contains a subset L' such that for Omega is an element of L' the dissipation rate is nonzero and may be subdissipative (0 <= alpha < 1) or superdissipative (1 < alpha <= 2), compared to ordinary dissipation (alpha = 1). Consequently, the harmonic bath does act as an anomalous thermostat, in variance with the common belief that elimination of a macroscopically large number of degrees of freedom always generates dissipation, forcing convergence to equilibrium. Intraband discrete breathers are such solutions which do not relax. We prove for arbitrary anharmonicity and small but finite coupling that intraband discrete breathers with frequency Omega exist for all Omega in a Cantor set C(k) of finite Lebesgue measure. This is achieved by estimating the contribution of small denominators appearing for G(t; Omega), related to Gamma(t). For Omega is an element of C(k) the small denominators do not lead to divergencies such that G(t; Omega) is a smooth and bounded function in t. (C) 2009 Elsevier B.V. All rights reserved.

We study energy propagation in locally time-periodically driven disordered nonlinear chains. For frequencies inside the band of linear Anderson modes, three different regimes are observed with increasing driver amplitude: 1) Below threshold, localized quasiperiodic oscillations and no spreading; 2) Three different regimes in time close to threshold, with almost regular oscillations initially, weak chaos and slow spreading for intermediate times, and finally strong diffusion; 3) Immediate spreading for strong driving. The thresholds are due to simple bifurcations, obtained analytically for a single oscillator, and numerically as turning-points of the nonlinear response manifold for a full chain. Generically, the threshold is nonzero also for infinite chains.

We study the spreading of an initially localized wave packet in two nonlinear chains (discrete nonlinear Schrödinger and quartic Klein-Gordon) with disorder. Previous studies suggest that there are many initial conditions such that the second moment of the norm and energy density distributions diverges with time. We find that the participation number of a wave packet does not diverge simultaneously. We prove this result analytically for norm-conserving models and strong enough nonlinearity. After long times we find a distribution of nondecaying yet interacting normal modes. The Fourier spectrum shows quasiperiodic dynamics. Assuming this result holds for any initially localized wave packet, we rule out the possibility of slow energy diffusion. The dynamical state could approach a quasiperiodic solution (Kolmogorov-Arnold-Moser torus) in the long time limit.

for an Invited Paper for the MAR08 Meeting of The American Physical Society A nonadiabatic and nonlinear theory for electron transfer SERGE AUBRY, Laboratoire Léon Brillouin, CEA Saclay We propose a general theory both non adiabatic and nonlinear which extends those used for the standard theory of electron transfer (ET) in chemistry but also becomes equivalent to it far from the inversion point. In the vicinity of the inversion point, the model parameters may be finely tuned such that large amplitude electronic oscillations between the donor and an extrasite, associated with large amplitude and collective phonon oscillations at the same frequency, are spontaneously generated (coherent electronphonon oscillator or CEPO). This extrasite is not a true acceptor but could play the role of a catalyst because by the CEPO it may trigger irreversible and ultrafast ET at low temperature toward a third site which is a real acceptor (while in the absence of catalyst, ET cannot occur). Such a trimer system may be regulated by small perturbations and behaves as a molecular transistor. We illustrate this idea by explicit numerical simulations on trimer models of the type donor-catalyst-acceptor. We discuss the relevance of our approach for understanding the ultrafast electron transfer experimentally observed in biosystems such as the photosynthetic reaction center.

We review some results concerning small adiabatic polarons, bipolarons and many polaron-bipolaron structures which were described in a series of works of the present author during recent decades.We first investigate the existence of the single small polaron in models with short range interactions as a function of dimensionality. We show by the variational method that it always exists in one dimension (1D) while in two and more dimensions, it appears discontinuously only beyond a critical coupling as a small polaron. When a magnetic field is added in 2D and 3D, large polarons now exist at small coupling and there is a first order transition versus electron phonon coupling between large and small polarons which disappears beyond a critical magnetic field. We also mention commensurability effects generated when the number of quantum fluxes per plaquette is rational. We next discuss the existence of multipeaked polarons and show that they do not exist as ground-states in a variation of the Holstein model, in contradiction with early claims. We also mention the possible existence, of polaro-breathers where ail electron may become trapped by ail anharmonic and localized lattice vibration.Next we investigate the possible mobility of small polarons and emphasize the role of the Peierls-Nabarro (PN) energy barrier which could be strongly depressed near first order transitions of the polaron. These ideas are applied to the bipolaron in the adiabatic Hostein-Hubbard model. We show that, depending oil the Hubbard repulsion, the bipolaron may be single site or multipeaked. Then, it, consists of two polarons bonded by a magnetic interaction. In 2D, more complex bipolarons which are spin quadrisinglet may become ground-state.Next we describe theorems obtained for the adiabatic Holstein-Hubbard model near the anticontinuous limit where the electronic transfer integral is zero. We prove that the bipolaronic and polaronic structures with many electrons which trivially exist at this limit, persist by continuity as insulating structures when the transfer integral becomes nonzero. We also prove that the system ground-state belongs to this family of solutions. It could be either a Bipolaronic Charge Density Wave when the Hubbard term is not too large or a polaronic charge density wave when it is large enough with a superposed spin ordering.In the 1D adiabatic Holstein models with small transfer integral, the ground-state is ail insulating bipolaronic charge density wave which may be commensurate or incommensurate whether the band filling is rational or irrational. When the transfer integral increases, the incommensurate bipolaronic CDWs undergo a reverse transition by breaking of analyticity where the CDW loses its bipolaronic character and becomes a conducting Peierls-Frohlich CDW.Finally, we briefly discuss the role of quantum fluctuations on the bipolaronic and polaronic structure. We argue that; this role becomes essential when the PN energy barrier becomes small. Then the spatial ordering of the bipolaronic structure may disappear and instead we could expect Bose condensation of the bipolarons which are hard core bosons, that is bipolaronic superconductivity. We also briefly mention the role of adiabatic polarons in chemistry in chemical reactions which consist in electron transfer between molecules. Exceptional phenomena such as ultrafast electron transfer could be described with polaronic models which are nonadiabatic.

We present a partial review of results concerning the theory of Discrete Breathers in nonlinear hamiltonian and discrete systems. These special time-periodic solutions gained much interest during the last decade because they may be involved in numerous and various phenomena in physics and biophysics where they could produce nontrivial effects of energy focusing and transfer. We first review the principles which govern their existence and which are used in the existence proofs available up to now. Next we discuss their linear stability and the interaction of Discrete Breathers with small amplitude waves, showing also how they could grow or decay. We also briefly discuss the existence of intraband DBs in systems with a linear discrete spectrum which are not spatially periodic (random or otherwise) and where linear waves cannot propagate. We also show that nonlinearity may restore the existence of solutions that propagate energy. We review some results concerning energy transportation by DBs and especially the main features associated with their mobility. We briefly discuss new perspectives opened up by the theory for applications that, in particular, look especially interesting for biophysics.

The regulatory enzyme Aspartate transcarbamylase (ATCase), the first enzyme of the pyrimidine pathway, catalyses the carbamylation of the amino group of aspartate by carbamylphosphate. This reaction is feedback inhibited by the end products of the pathway, CTP and UTP. In contrast, it is stimulated by ATP which binds to a regulatory site distant from the catalytic site by 60 A. The structural features specifically involved in this effect of ATP delineate a path for the transmission of this regulatory signal over that distance. The nature of these structural features suggests the possible implication of polarons in the transmission of a long-distance electromechanical signal. (c) 2006 Elsevier B.V. All rights reserved.

The energy propagation through a chain of coupled anharmonic oscillators under the influence of a harmonic driving force applied at one end is numerically investigated using the nonlinear response manifold method and real time simulations. For driving frequencies in the band gap of the linear spectrum, where energy propagation is impossible in linear systems, the existence of propagation thresholds for the amplitude of the driving force is related to the existence of turning points in the nonlinear response manifold and can be associated with discrete breathers at the driving frequency. A rich dynamical behavior involving several mechanisms of nonlinear transmission is exhibited that depends to a great extent on the local response of the forced nonlinear oscillator at the edge of the system. At low force amplitudes, a localized quasilinear response of this first site blocks energy propagation. Above a certain amplitude threshold, which is determined by the stability of the mode localized on the first site, the chain exhibits dissipative response and weak propagation occurs through linear modes of the system. Increasing further the driving amplitude, this phonon radiation remains the only propagation process, up to a second threshold amplitude at which energy transmission dramatically increases by a few orders of magnitude. This large energy flow is due to large amplitude nonlinear waves which in some cases appear as mobile discrete breathers propagating throughout the chain.

We suggest a new mechanism where sonoluminescence is produced by the tremendously large adiabatic pressure pulse (shock wave) generated by the close to supersonic (or above) impact of the fluid on the hard core bubble. The light flash is mostly emitted by the fluid surrounding the bubble. More generally, the emission spectrum of any material submitted to a large adiabatic compression is (roughly) globally dilated by some Grüneisen coefficient γ̄. Temperature simultaneously increases by the same factor, which increases the power of the emitted radiation by a factor γ̄4. A rigorous lower bound for the sound velocity in the compressed region at impact is obtained with purely kinematic arguments only assuming the existence of a non-negative Van der Waals volume for the fluid. For supersonic impacts, the increase of the sound velocity reaches at least one order of magnitude (and, with reasonable assumptions, much more), which yields an estimation of the Grüneisen coefficient γ̄ and indicates it may become very large. Then, during the pressure pulse, the thermal infrared (IR) radiation of the compressed fluid can be extended up to visible–ultraviolet (UV) simultaneously with an intense brightness. The dynamics of collapsing bubbles have been analyzed taking into account fluid compressibility. Shock waves are generated when the bubble, at a minimum radius, suddenly becomes almost incompressible. For impacts close to supersonic (or above), an intense pressure is briefly generated in a sphere which extends beyond the central bubble and which thus mostly contains surrounding fluid compacted to near its Van der Waals volume. This compacted fluid generates an intense emission of UV–visible light which suddenly disappears when the fluid expands from its Van der Waal volume. This situation occurs when the sphere of compacted fluid reaches a critical size of a few minimum bubble radii. Next, this pressure pulse radially propagates through the fluid, initially at highly supersonic velocities, which decay to the normal sound velocity as it simultaneously spreads out.

Standard Kramers theory of chemical reactions involves a coupling with a Langevin thermal bath which intrinsically forbids the possible existence of Discrete Breathers (DBs) (i.e. local modes). However, it is now known that in complex systems, that energy may focus for long time as Discrete Breathers (local mode). In very special systems, targeted energy transfer may occur subsequently to another selected site and induces an ultraselective chemical reaction operating at low temperature. The dynamics of the reaction is non brownian but highly coherent along a specific path in the phase space where the system is nearly integrable (chemical expressway).A simple toy model illustrating this idea is reduced to a Rotor weakly coupled to a Morse oscillator (supposed to represent two specific local modes in a complex system) which are appropriately tuned for targeted energy transfer. When the Rotor is initially rotating with a frequency resonant with those of the Morse oscillator at rest, the energy of the Rotor is almost completely transferred to the Morse oscillator and induces chemical dissociation. The periodic oscillations of the Rotor and Morse oscillator remain coherent and their frequencies simultaneously vary, but always remain resonant.This process is analytically described within an integrable approximation. Numerical investigations of this model confirm that in the appropriate conditions, the particle in the Morse oscillator is indeed promptly ejected at infinity with a finite velocity (chemical dissociation) despite some chaotic transient manifesting imperfect integrability.

In his famous pioneering work published in 1940, Hendrik Kramers understood chemical reactions, as the climbing of an energy barrier between two energy wells in a large configuration space where the first well represents the state of the reactant molecules and the second well those of the product molecules. In standard theories, chemical reactions are still essentially governed by the thermal (Brownian) motion in a potential of a point defined by the reaction coordinates which represents the configuration of the whole system. Many detailed investigations have been performed during the last decades for specific chemical reactions where energy landscape, basin and saddle point in energy (transition states) were accurately calculated. However, these models assume that the diffusion process in the phase space is incoherent and ignore possible coherent phenomena which could bias the reaction process. Actually, numerous studies show that the polyatomic molecules seen as a set of non-linear coupled oscillators is not an ergodic system, but instead, they form a mixed phase space with regular and chaotic regions [1,2]. Particularly, it is now well-known that spontaneous energy localization may occur in complex molecular systems. Transitions from normal modes (extended motions) to local modes were first observed spectroscopically in molecules. These local modes in finite size systems are known to exist as well in infinite discrete and nonlinear lattices [3,4]. They appear as localized oscillations with large amplitudes well above the thermal noise and may persist out of thermal equilibrium over unexpected long life time. Their existence requires both the discreteness and the nonlinearity of the system but does not require its spatial periodicity. Many open problems concern DNA transcription through bubble opening, protein folding and biological machines which involve bond breaking/formation with a high degree of selectivity and specificity in conformational changes. Energy localisation phenomena could play an essential role in these processes. There are several major problems to understand before practical application to those complex systems. How such localized energy packet could be spontaneously created and how it could be transported selectively from one place to another without being spread out thus favouring specific reactions? Assuming that a local mode has been already produced by another mechanism (for example energy has been released at a specific location by ATP), our aim is to show on a simple toy model that then a coherent energy transfer may occur spontaneously at a selected site and induce a chemical reaction. Local modes in complex system may be viewed as (almost) isolated nonlinear oscillators, the frequency of which depends on their amplitude. Their stability require in principle that they are nonresonant with the normal modes of the system but nevertheless coherent energy transport requires (special) resonances. Indeed, when two resonant harmonic oscillators are weakly coupled, it is well-known that any amount of energy injected on the first one is completely transferred to the second one after it is completely transferred (and subsequently oscillates back and forth). In contrast, such resonant energy transfer generally cannot occur for two weakly coupled anharmonic oscillators because the frequencies of these oscillators do not remain equal during the whole transfer. However, there are special situations where the anharmonic oscillators are well tuned one with each other (and are said to be conjugated). Beside the condition of linear resonance at the initial time, a precise condition is required on the nonlinearity [5]. Then, when a selected amount of energy is injected on the first one, this energy is completely transferred to the second one after some time while the frequencies of the two nonlinear oscillators both vary but persistently remain equal. We called this phenomena Targeted Energy Transfer (TET). Applying our theory, we have shown that such a situation is met when a rotor chosen with an appropriate inertia momentum and an appropriate angular momentum is weakly coupled with a Morse oscillator [6]. TET trajectories of this nonintegrable model can be analytically calculated with a very good accuracy. Numerical simulations shown fig1 confirms that when the TET conditions are fulfilled, there is initially almost complete energy transfer according to the theory, but in addition due to the nonintegrability