FPU Chain Research Articles

Feedback resonance in Fermi–Pasta–Ulam chain

A controlled version of the celebrated Fermi–Pasta–Ulam problem is introduced. The feedback control algorithm based on Speed-gradient approach is proposed and analyzed by extensive computer simulation. It is demonstrated that the feedback controlled system tends to approximate equipartition state much faster than it happens in the open loop (classical) system. Apparently a substantial change in system behavior is achieved with a small intensity control: less than 0.5% of the total system energy. Such a behavior has been observed and analyzed previously for oscillators with one degree-of-freedom (DOF) under the name of feedback resonance (Fradkov A.L., Physica D, 128(1999), 159-168. The results of this paper suggest that feedback resonance phenomenon may be observed in multi-DOF oscillator chains, particularly in FPU chain.

Elucidating the FPU-paradox based on the dynamics of Kuznetzov–Ma breathers

Unlike Zabusky and Kruskal who exploited the Korteweg–de Vries traveling solitons to explain the FPU-recurrence phenomenon, we consider a more robust time periodic Kuznetzov–Ma breather to resolve the paradox. The nonlinear Schrödinger equation is derived from the equation of motion of [Formula: see text]-FPU chain, by using the method of multiple scales combined with a quasi-discreteness approximation. Modulational instability leads to the generation of a nonlinear wave of finite background, known as the Kuznetzov–Ma breather. The spatial localization and time periodic profile of the Kuznetzov–Ma breathers make it ideal in mimicking the FPU-recurrence phenomenon, as underscored by results of numerical simulations.

Traveling wave solutions for the FPU chain: a constructive approach

Traveling waves for the FPU chain are constructed by solving the associated equation for the spatial profile u of the wave. We consider solutions whose derivatives need not be small, may change sign several times, but decrease at least exponentially. This includes multi-bump solutions. Our method of proof is computer-assisted. Unlike other methods, it does not require that the FPU potential has an attractive (positive) quadratic term. But we currently need to restrict the size of that term. In particular, our solutions in the attractive case are all supersonic.

Freezing of the Optical-Branch Energy in a Diatomic FPU Chain

We study the exchange of energy between the modes of the optical branch and those of the acoustic one in a diatomic chain of particles, with masses $m_1$ and $m_2$. We prove that, at small temperature and provided $m_1\gg m_2$, for the majority of the initial data the energy of each branch is approximately constant for times of order $\beta^{S/2}$, where $S=\lfloor \sqrt{m_1/m_2}/2\rfloor$ and $\beta$ is the inverse temperature. The result is uniform in the thermodynamic limit.

Near-Integrability and Recurrence in FPU Chains with Alternating Masses

An important aspect of understanding FPU chains is the existence of invariant manifolds (called “bushes”) in FPU chains. We will focus on the classical periodic FPU chain and on the FPU chain with alternating masses where we show that also in the alternating case nested manifolds (related to bushes) exist. The use of symmetries leads to the emergence of systems of n particles as invariant manifolds of systems with a multiple of n particles. This analysis is followed by examples of existence and stability of special invariant manifolds and phase-space dynamics in the case of 4 and 8 particles. These examples are typical for periodic FPU chains with 4n or 8n particles. It turns out that in the alternating case the dynamics is strongly affected by the choice of the alternating mass m. Normal form calculations help to identify quasi-trapping regions leading to delay of recurrence. The results suggest that equipartition of energy near stable equilibrium is improbable.

Solitary waves in a bistable lattice

The design of architectured materials with bistable building blocks holds exciting possibilities. This new class of metamaterials exploits micro-level structural instabilities to obtain extraordinary physical and mechanical properties. Still, the dynamic behavior of these lattice structures is largely unexplored. Here, we study the dynamic response of a 1-D bistable lattice, i.e. a FPU chain with springs having a non-convex double-well energy potential. In addition to metamaterials, this model-problem is prototypical to a large number of systems, such as unfolding/refolding of proteins, crack propagation, plasticity, and mechanisms underlying martensitic phase transformations. We show that, depending only on the stiffness-ratio of the two energy wells of the bistable springs, the system exhibits two fundamentally different responses to impact; either the impact energy is (almost entirely) trapped in the form of large undulations of the first few springs, or the energy of the impact is (almost completely) transmitted along the chain in the form of a solitary wave that involves transition to the secondary energy-well and back. Focusing our attention to the latter, we reveal, based on analytical treatment and extensive numerical simulations, a universal feature of the solitary wave. Namely, the height of the solitary wave is indifferent to the energy barrier separating the two equilibria of the double well potential. This remarkable feature indicates that the spinodal region of the double-well potential affects the behavior only through its breadth.

Supratransmission in β-Fermi–Pasta–Ulam chains with different ranges of interactions

We study the phenomenon of supratransmission (that is, the sudden transmission of energy above a sharp threshold amplitude of periodic boundary excitations) on an N-particle one-dimensional β-Fermi–Pasta–Ulam (FPU) lattice, under different ranges of interactions. When the system is at rest, we set the first particle in sinusoidal motion at a frequency that lies in the forbidden band-gap of the linear nearest-neighbor FPU chain, and use reliable computational techniques to extend earlier results of energy supratransmission to FPU chains with varying ranges of particle interactions than simply nearest neighbors. We thus discover that supratranmission arises at similar amplitude thresholds as the nearest neighbor case, for a much wider class of short-range interaction systems. However, as the interaction range becomes longer and longer, the threshold amplitude for supratransmission recedes to higher and higher values. It would be highly desirable to verify these results analytically as well as experimentally in superconducting Josephson systems, or arrays of coupled optical waveguides.

Weakly nonlinear localization for a 1-D FPU chain with clustering zones

We study weakly nonlinear spatially localized solutions of a Fermi-Pasta-Ulam model describing a unidimensional chain of particles interacting with a number of neighbors that can vary from site to site. The interaction potential contains quadratic and quartic terms, and is derived from a nonlinear elastic network model proposed by Juanico et al. [1]. The FPU model can be also derived for arbitrary dimensions, under a small angular displacement assumption. The variable interaction range is a consequence of the spatial inhomogeneity in the equilibrium particle distribution. We here study some simple one-dimensional examples with only a few, well defined agglomeration regions. These agglomerations are seen to lead to spatially localized linear modes and gaps in the linear spectrum, which in turn imply a normal form that has spatially localized periodic orbits.

Renormalized dispersion relations of β-Fermi-Pasta-Ulam chains in equilibrium and nonequilibrium states.

We study the nonlinear dispersive characteristics in β-Fermi-Pasta-Ulam (FPU) chains in both thermal equilibrium and nonequilibrium steady state. By applying a multiple scale analysis to the FPU chain, we analyze the contribution of the trivial and nontrivial resonance to the renormalization of the dispersion relation. Our results show that the contribution of the nontrivial resonance remains significant to the renormalization, in particular, in strongly nonlinear regimes. We contrast our results with the dispersion relations obtained from the Zwanzig-Mori formalism and random phase approximation to further illustrate the role of resonances. Surprisingly, these theoretical dispersion relations can be generalized to describe dispersive characteristics well at the nonequilibrium steady state of the FPU chain with driving-damping in real space. Through numerical simulation, we confirm that the theoretical renormalized dispersion relations are valid for a wide range of nonlinearities in thermal equilibrium as well as in nonequilibrium steady state. We further show that the dispersive characteristics persist in nonequilibrium steady state driven-damped in Fourier space.

An Averaging Theorem for FPU in the Thermodynamic Limit

Consider an FPU chain composed of $N\gg 1$ particles, and endow the phase space with the Gibbs measure corresponding to a small temperature $\beta^{-1}$. Given a fixed $K<N$, we construct $K$ packets of normal modes whose energies are adiabatic invariants (i.e., are approximately constant for times of order $\beta^{1-a}$, $a>0$) for initial data in a set of large measure. Furthermore, the time autocorrelation function of the energy of each packet does not decay significantly for times of order $\beta$. The restrictions on the shape of the packets are very mild. All estimates are uniform in the number $N$ of particles and thus hold in the thermodynamic limit $N\to\infty$, $\beta>0$.

Energy exchange and transition to localization in the asymmetric Fermi-Pasta-Ulam oscillatory chain

A finite (periodic) FPU chain is chosen as a convenient point for investigating the energy exchange phenomenon in nonlinear oscillatory systems. As we have recently shown, this phenomenon may occur as a consequence of the resonant interaction between high-frequency nonlinear normal modes. This interaction determines both the complete energy exchange between different parts of the chain and the transition to energy localization in an excited group of particles. In the paper, we demonstrate that this mechanism can exist in realistic (asymmetric) models of atomic or molecular oscillatory chains. Also, we study the resonant interaction of conjugated nonlinear normal modes and prove a possibility of linearization of the equations of motion. The theoretical constructions developed in this paper are based on the concepts of "effective particles" and Limiting Phase Trajectories. In particular, an analytical description of energy exchange between the "effective particles" in the terms of non-smooth functions is presented. The analytical results are confirmed with numerical simulations.

Dispersive stability of infinite-dimensional Hamiltonian systems on lattices

We derive dispersive stability results for oscillator chains like the FPU chain or the discrete Klein–Gordon chain. If the nonlinearity is of degree higher than four, then small localized initial data decay exactly as in the linear case. For this, we provide sharp decay estimates for the linearized problem using oscillatory integrals and avoiding the nonoptimal interpolation between different ℓ p spaces.

Anomalous Heat Conduction in One-Dimensional Quantum Fermi–Pasta–Ulam Lattice: Semiquantal Approach

We investigate the mechanism of heat conduction in one-dimensional (1D) quantum FPU chain with quantum fluctuation in the framework of semiquantal molecular dynamics. In the semiquantal many-body simulations, a Hartree-type many-body wave function is adopted for a whole chain and a single-particle state of a particle is represented by a trial wave function with Jackiw–Kerman (JK) form. With the help of the Dirac's time-dependent variational principle (TDVP), a set of equations of the variational parameters contained in the JK wave packet is obtained, and it describes the quantum dynamics of the nonlinear lattices approximately. These equations not only prove highly efficient in recovering dynamics of classical heat conduction, but also allow exploring the case with quantum mechanical characteristics. As a consequence, we find the thermal conductivity diverges with system size as well as in the classical version when considering quantum fluctuation. Furthermore, in three different temperature regions it is...

Resonant normal form for even periodic FPU chains

We investigate periodic FPU chains with an even number of particles. We show that near the equilibrium point, any such chain admits a resonant Birkhoff normal form of order four which is completely integrable —an important fact which helps explain the numerical experiments of Fermi, Pasta, and Ulam. We analyze the moment map of the integrable approximation of an even FPU chain. Unlike the case of odd FPU chains these integrable systems (generically) exhibit hyperbolic dynamics. As an application we prove that any FPU chain with Dirichlet boundary conditions admits a Birkhoff normal form up to order four and show that a KAM theorem applies.

Results on Normal Forms for FPU Chains

In this paper we prove, among other results, that near the equilibirum position, any periodic FPU chain with an odd number N of particles admits a Birkhoff normal form up to order 4, whereas any periodic FPU chain with N even admits a resonant normal form up to order 4. This resonant normal form of order 4 turns out to be completely integrable. Further, for N odd, we obtain an explicit formula of the Hessian of its Hamiltonian at the fixed point.

Temporal asymmetry of fluctuations in the nonequilibrium FPU model

The large deviation theory recently developed by Bertini, De Sole, Gabrielli, Jona-Lasinio and Landim is meant to extend the Onsager–Machlup theory to nonequilibrium phenomena, and predicts that the fluctuations of densities and currents in certain stochastic processes are not symmetric with respect to the time reversal operation. In this paper, several notions of fluctuations are introduced for deterministic systems, and it is observed that temporally asymmetric (not necessarily large) fluctuations are ubiquitous, even in the context of time reversal invariant dynamics of particle systems, in nonequilibrium states. To illustrate these ideas, the nonequilibrium FPU chain devised by Lepri, Livi and Politi is studied in detail.

Symmetry and Resonance in Periodic FPU Chains

The symmetry and resonance properties of the Fermi Pasta Ulam chain with periodic boundary conditions are exploited to construct a near-identity transformation bringing this Hamiltonian system into a particularly simple form. This `Birkhoff-Gustavson normal form' retains the symmetries of the original system and we show that in most cases this allows us to view the periodic FPU Hamiltonian as a perturbation of a nondegenerate Liouville integrable Hamiltonian. According to the KAM theorem this proves the existence of many invariant tori on which motion is quasiperiodic. Experiments confirm this qualitative behaviour. We note that one can not expect it in lower-order resonant Hamiltonian systems. So the FPU chain is an exception and its special features are caused by a combination of special resonances and symmetries.

Heat conduction in the diatomic Toda lattice revisited

The problem of the diverging thermal conductivity in one-dimensional (1D) lattices is considered. By numerical simulations, it is confirmed that the thermal conductivity of the diatomic Toda lattice diverges, which is the opposite of the current popular belief. Also, the diverging exponent is found to be almost the same as the FPU chain. It is reconfirmed that the diverging thermal conductivity is universal in 1D systems, where the total momentum preserves.

Dynamical instability and statistical behaviour of N-body systems

In this paper, we argue about a synthetic characterization of the qualitative properties of generic many-degrees-of-freedom (mdf) dynamical systems (DS's) by means of a geometric description of the dynamics [Geometro-Dynamical Approach (GDA)]. We exhaustively describe the mathematical framework needed to link geometry and dynamical (in)stability, discussing in particular which geometrical quantity is actually related to instability and why some others cannot give, in general, any indication of the occurrence of chaos. The relevance of the Schur theorem to select such Geometrodynamic Indicators (GDI) of instability is then emphasized, as its implications seem to have been underestimated in some of the previous works. We then compare the analytical and numerical results obtained by us and by Pettini and coworkers concerning the FPU chain, verifying a complete agreement between the outcomes of averaging the relevant GDI's over phase space (Casetti and Pettini, 1995) and our findings (Cipriani, 1993), obtained in a more conservative way, time-averaging along geodesics. Along with the check of the ergodic properties of GDI's, these results confirm that the mechanism responsible for chaos in realistic DS's largely depends on the fluctuations of curvatures rather than on their negative values, whose occurrence is very unlikely. On these grounds we emphasize the importance of the virialization process, which separates two different regimes of instability. This evolutionary path, predicted on the basis of analytical estimates, receives clear support from numerical simulations, which, at the same time, confirm also the features of the evolution of the GDI's along with their dependence on the number of degrees of freedom, N , and on the other relevant parameters of the system, pointing out the scarce relevance of negative curvature (for N ⪢ 1) as a source of instability. The general arguments outlined above, are then concretely applied to two specific N-body problems, obtaining some new insights into known outcomes and also some new results The comparative analysis of the FPU chain and the gravitational N-body system allows us to suggest a new definition of strong stochasticity, for any DS. The generalization of the concept of dynamical time-scale, t D, is at the basis of this new criterion. We derive for both the mdf systems considered the ( N , ε)-dependence of t D (ε being the specific energy) of the system. In light of this, the results obtained (Cerruti-Sola and Pettini, 1995), indeed turn out to be reliable, the perplexity there raised originating from the neglected N -dependence of t D, and not to an excessive degree of approximation in the averaged equations used. This points out also the peculiarities of gravitationally bound systems, which are always in a regime of strong instability; the dimensionless quantity L 1 = γ 1 · t D [γ 1 is the maximal Lyapunov Characteristic Number (LCN)] being always positive and independent of ε, as it happens for the FPU chain only above the strong stochasticity threshold (SST). The numerical checks on the analytical estimates about the ( N , ε)-dependence of GDI's, allow us to single out their scaling laws, which support our claim that, for N ⪢ 1, the probability of finding a negative value of Ricci curvature is practically negligible, always for the FPU chain, whereas in the case of the Gravitational N-body system, this is certainly true when the virial equilibrium has been attained. The strong stochasticity of the latter DS is clearly due to the large amplitude of curvature fluctuations. To prove the positivity of Ricci curvature, we need to discuss the pathologies of mathematical Newtonian interaction, which have some implications also on the ergodicity of the GDI's for this DS. We discuss the Statistical Mechanical properties of gravity, arguing how they are related to its long range nature rather than to its short scale divergencies. The N -scaling behaviour of the single terms entering the Ricci curvature show that the dominant contribution comes from the Laplacian of the potential energy, whose singularity is reflected on the issue of equality between time and static averages. However, we find that the physical N-body system is actually ergodic where the GDI's are concerned, and that the Ricci curvature associated is indeed almost everywhere (and then almost always) positive, as long as N ⪢ 1 and the system is gravitationally bound and virialized. On these grounds the equality among the above mentioned averages is restored, and the GDA to instability of gravitating systems gives fully reliable and understandable results. Finally, as a by-product of the numerical simulations performed, for both the DS's considered, it emerges that the time averages of GDI's quickly approach the corresponding canonical ones, even in the quasi-integrable limit, whereas, as expected, their fluctuations relax on much longer timescales, in particular below the SST.

Localization and equipartition of energy in the β-FPU chain: Chaotic breathers

The evolution towards equipartition in the β-FPU chain is studied considering as initial condition the highest frequency mode. Above an analytically derived energy threshold, this zone-boundary mode is shown to be modulationally unstable and to give rise to a striking localization process. The spontaneously created excitations have strong similarity with moving exact breathers solutions. But they have a finite lifetime and their dynamics is chaotic. These chaotic breathers are able to collect very efficiently the energy in the chain. Therefore their size grows in time and they can transport a very large quantity of energy. These features can be explained analyzing the dynamics of perturbed exact breathers of the FPU chain. In particular, a close connection between the Lyapunov spectrum of the chaotic breathers and the Floquet spectrum of the exact ones has been found. The emergence of chaotic breathers is convincingly explained by the absorption of high frequency phonons whereas a breather's metastability is for the first time identified. The lifetime of the chaotic breather is related to the time necessary for the system to reach equipartition. The equipartition time turns out to be dependent on the system energy density ε only. Moreover, such time diverges as ε −2 in the limit ε → 0 and vanishes as ε −1 4 for ε → ∞.