Pasta Ulam Chain Research Articles

Variation of the Specific Heat in the Fermi–Pasta–Ulam Chain due to Energy Localization

Variation of the Specific Heat in the Fermi–Pasta–Ulam Chain due to Energy Localization

The [formula omitted] resonance in a particle chain

We consider four masses in a circular configuration with nearest-neighbour interaction, generalising the spatially periodic Fermi–Pasta–Ulam-chain where all masses are equal. We identify the mass ratios that produce the 1:2:4 resonance — the normal form in general is non-integrable already at cubic order. Taking two of the four masses equal allows to retain a discrete symmetry of the fully symmetric Fermi–Pasta–Ulam-chain and yields an integrable normal form approximation. The latter is also true if the cubic terms of the potential vanish. We put these cases in context and analyse the resulting dynamics, including a detuning of the 1:2:4 resonance within the particle chain.

Structural localization in the classical and quantum Fermi-Pasta-Ulam model.

We study the statistics and short-time dynamics of the classical and the quantum Fermi-Pasta-Ulam chain in the thermal equilibrium. We analyze the distributions of single-particle configurations by integrating out the rest of the system. At low temperatures, we observe a systematic increase in the mobility of the chain when transitioning from classical to quantum mechanics due to zero-point energy effects. We analyze the consequences of quantum dispersion on the dynamics at short times of configurational correlation functions.

Experimental demonstration of excitation and propagation of intrinsic localized modes in a mass–spring chain

Excitation and propagation of nonlinear localized oscillations are demonstrated experimentally by using a mass–spring chain to emulate those in the celebrated Fermi–Pasta–Ulam chain of β type. Two types of experiments are performed, one corresponding to a boundary-value problem and the other to an initial-value one. In the former experiment, one end of the chain is driven sinusoidally in the direction of the chain above a cut-off frequency. In the latter experiment, only a number of weights centered around the weight in the middle of the chain are held displaced initially and released suddenly. The experimental results show quantitatively good agreements with the numerical ones. Two videos linked also show visually mobile and stationary localized oscillations.

Numerical simulation of the nonlinear dynamics of harmonically driven Riesz-fractional extensions of the Fermi–Pasta–Ulam chains

In this work, we introduce a spatially discrete model that is a modification of the well-known α-Fermi–Pasta–Ulam chain with damping. The system is perturbed at one end by a harmonic disturbance irradiating at a frequency in the forbidden band-gap of the classical regime, and a nonlocal coupling between the oscillators is considered using discrete Riesz fractional derivatives. We propose fully discrete expressions to approximate an energy functional of the system, and we use them to calculate the total energy of fractional chains over a relatively long period of time [Fract. Diff. Appl. 4 (2004) 153–162]. The approach is thoroughly tested in the case of local couplings against known qualitative results, including simulations of the process of nonlinear recurrence in the traditional chains of anharmonic oscillators. As an application, we provide evidence that the process of supratransmission is present in spatially discrete Fermi–Pasta–Ulam lattices with Riesz fractional derivatives in space. Moreover, we perform numerical experiments for small and large amplitudes of the harmonic disturbance. In either case, we establish the dependency of the critical amplitude at which supratransmission begins as a function of the driving frequency. Our results are in good agreement with the analytic predictions for the classical Fermi–Pasta–Ulam chain.

Heat Conduction in the α−β Fermi–Pasta–Ulam Chain

Recent simulation results on heat conduction in a one-dimensional chain with an asymmetric inter-particle interaction potential and no onsite potential found non-anomalous heat transport in accordance to Fourier’s law. This is a surprising result since it was long believed that heat conduction in one-dimensional systems is in general anomalous in the sense that the thermal conductivity diverges as the system size goes to infinity. In this paper we report on detailed numerical simulations of this problem to investigate the possibility of a finite temperature phase transition in this system. Our results indicate that the unexpected results for asymmetric potentials is a result of insufficient chain length, and does not represent the asymptotic behavior.

Covariant Lyapunov vectors

Recent years have witnessed a growing interest in covariant Lyapunov vectors (CLVs) which span local intrinsic directions in the phase space of chaotic systems. Here, we review the basic results of ergodic theory, with a specific reference to the implications of Oseledets’ theorem for the properties of the CLVs. We then present a detailed description of a ‘dynamical’ algorithm to compute the CLVs and show that it generically converges exponentially in time. We also discuss its numerical performance and compare it with other algorithms presented in the literature. We finally illustrate how CLVs can be used to quantify deviations from hyperbolicity with reference to a dissipative system (a chain of Hénon maps) and a Hamiltonian model (a Fermi–Pasta–Ulam chain).This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Lyapunov analysis: from dynamical systems theory to applications’.

Kinetic Relations for a Lattice Model of Phase Transitions

The aim of this article is to analyse travelling waves for a lattice model of phase transitions, specifically the Fermi–Pasta–Ulam chain with piecewise quadratic interaction potential. First, for fixed, sufficiently large subsonic wave speeds, we rigorously prove the existence of a family of travelling wave solutions. Second, it is shown that this family of solutions gives rise to a kinetic relation which depends on the jump in the oscillatory energy in the solution tails. Third, our constructive approach provides a very good approximate travelling wave solution.

DISCRETE SYMMETRY AND STABILITY IN HAMILTONIAN DYNAMICS

In the present tutorial we address a problem with a long history, which remains of great interest to date due to its many important applications: It concerns the existence and stability of periodic and quasiperiodic orbits in N-degree of freedom Hamiltonian systems and their connection with discrete symmetries. Of primary importance in our study is what we call nonlinear normal modes (NNMs), i.e. periodic solutions which represent continuations of the system's linear normal modes in the nonlinear regime. We examine questions concerning the existence of such solutions and discuss different methods for constructing them and studying their stability under fixed and periodic boundary conditions. In the periodic case, we find it particularly useful to approach the problem through the discrete symmetries of many models, employing group theoretical concepts to identify a special type of NNMs which we call one-dimensional "bushes". We then describe how to use linear combinations of s ≥ 2 such NNMs to construct s-dimensional bushes of quasiperiodic orbits, for a wide variety of Hamiltonian systems including particle chains, a square molecule and octahedral crystals in 1, 2 and 3 dimensions. Next, we exploit the symmetries of the linearized equations of motion about these bushes to demonstrate how they may be simplified to study the destabilization of these orbits, as a result of their interaction with NNMs not belonging to the same bush. Applying this theory to the famous Fermi Pasta Ulam (FPU) chain, we review a number of interesting results concerning the stability of NNMs and higher-dimensional bushes, which have appeared in the recent literature. We then turn to a newly developed approach to the analytical and numerical construction of quasiperiodic orbits, which does not depend on the symmetries or boundary conditions of our system. Using this approach, we demonstrate that the well-known "paradox" of FPU recurrences may in fact be explained in terms of the exponential localization of the energies Eq of NNM's being excited at the low part of the frequency spectrum, i.e. q = 1, 2, 3, …. These results indicate that it is the stability of these low-dimensional compact manifolds called q-tori, that is related to the persistence or FPU recurrences at low energies. Finally, we discuss a novel approach to the stability of orbits of conservative systems, expressed by a spectrum of indices called GALI k, k = 2, …, 2N, by means of which one can determine accurately and efficiently the destabilization of q-tori, leading, after very long times, to the breakdown of recurrences and, ultimately, to the equipartition of energy, at high enough values of the total energy E.

The effect of cubic potentials on discrete breathers in a mixed Klein–Gordon/Fermi–Pasta–Ulam chain

Nonlinearity has a crucial impact on the symmetry properties of dynamical systems. This paper studies a one-dimensional mixed Klein–Gordon/Fermi–Pasta–Ulam diatomic chain using the expanded rotating plane-wave approximation and numerical calculations to determine the effect of cubic potentials on the symmetry properties of discrete breathers in this system. The results will be very useful to researchers in the field of numerical calculations on discrete breathers.

COMPUTATIONAL STUDY OF THE TRANSMISSION OF ENERGY IN A TWO-DIMENSIONAL LATTICE WITH NEAREST-NEIGHBOR INTERACTIONS

In this work, we establish computationally the existence of the process of nonlinear supratransmission in the simplest generalization of the classical β-Fermi–Pasta–Ulam chain to two-space dimensions, subject to harmonic boundary data in one end of the Dirichlet form. Our simulations employ a finite-difference scheme with multiple properties of consistency in the domains of the solutions, the local energy density and the total energy of the system. Moreover, our results establish the presence of the phenomenon of nonlinear infratransmission or lower-transmission, thus proving the existence of a bistable region where a conducting and an insulating regimes coexist.

A computational technique with multiple properties of consistency in the study of modified [formula omitted]-Fermi–Pasta–Ulam chains

In this work, we present a numerical method to approximate solutions of a continuous β -Fermi–Pasta–Ulam-like medium which, amongst other features, includes the presence of damping and a fourth-order partial derivative, obtained when approximating the continuous limit of the classical β -Fermi–Pasta–Ulam chain. The method includes a discrete energy scheme which consistently approximates the continuous energy formula; moreover, it is shown that the discrete rate of change of energy consistently approximates its continuous counterpart. Computational experiments are conducted in order to show that the method preserves energy in the case of conservative systems, and it loses energy in the case of dissipative media. An application is conducted in order to approximate the occurrence of the process of nonlinear supratransmission in the classical β -Fermi–Pasta–Ulam chain and in discrete sine-Gordon systems, obtaining results which are in good agreement with the theory available in the literature.

Global dynamical properties of the Fermi–Pasta–Ulam system

In this paper we give a statistical description of the phase space of a Fermi–Pasta–Ulam chain using the Fast Lyapunov Indicator, looking for properties valid in the thermodynamic limit.

The Fermi–Pasta–Ulam problem: Periodic orbits, normal forms and resonance overlap criteria

Fermi, Pasta and Ulam observed that the excitation of a low frequency normal mode in a nonlinear acoustic chain leads to localization in normal mode space on large time scales. Fast equipartition (and thus complete delocalization) in the Fermi–Pasta–Ulam chain is restored if relevant intensive control parameters exceed certain threshold values. We compare recent results on periodic orbits (in the localization regime) and resonant normal forms (in a weak delocalization regime), and relate them to various resonance overlap criteria. We show that the approaches quantitatively agree in their estimate of the localization–delocalization threshold. A key ingredient for this transition are resonances of overtones.

Asymptotic analysis of combined breather–kink modes in a Fermi–Pasta–Ulam chain

We find approximations to travelling breather solutions of the one-dimensional Fermi–Pasta–Ulam (FPU) lattice. Both bright breather and dark breather solutions are found. We find that the existence of localised (bright) solutions depends upon the coefficients of cubic and quartic terms of the potential energy, generalising an earlier inequality derived by James [G. James, Existence of breathers on FPU lattices, C. R. Acad. Sci. Paris 332 (2001) 581–586]. We use the method of multiple scales to reduce the equations of motion for the lattice to a nonlinear Schrödinger equation at leading order and hence construct an asymptotic form for the breather. We show that in the absence of a cubic potential energy term, the lattice supports combined breathing-kink waveforms. The amplitude of breathing-kinks can be arbitrarily small, as opposed to the case for traditional monotone kinks, which have a nonzero minimum amplitude in such systems. We also present numerical simulations of the lattice, verifying the shape and velocity of the travelling waveforms, and confirming the long-lived nature of all such modes.

Bifurcations of discrete breathers in a diatomic Fermi–Pasta–Ulam chain

Discrete breathers are time-periodic, spatially localized solutions of the equations of motion for a system of classical degrees of freedom interacting on a lattice. Such solutions are investigated for a diatomic Fermi–Pasta–Ulam chain (FPU) i.e. a chain of alternate heavy and light masses coupled by anharmonic forces. For hard interaction potentials, discrete breathers in this model are known to exist either as ‘optic breathers’ with frequencies above the optic band or as ‘acoustic breathers’ with frequencies in the gap between the acoustic and the optic band. In this paper, bifurcations between different types of discrete breathers are found numerically, with the mass ratio m and the breather frequency ω as bifurcation parameters. We identify a period tripling bifurcation around optic breathers, which leads to new breather solutions with frequencies in the gap, and a second local bifurcation around acoustic breathers. These results provide new breather solutions of the FPU system which interpolate between the classical acoustic and optic modes. The two bifurcation lines originate from a particular ‘corner’ in parameter space (ω, m). As parameters lie near this corner, we prove by means of a centre manifold reduction that small amplitude solutions can be described by a four-dimensional reversible map. This allows us to derive formally a continuum limit differential equation which characterizes at leading order the numerically observed bifurcations.

Discrete nonlinear Schrödinger approximation of a mixed Klein–Gordon/Fermi–Pasta–Ulam chain: Modulational instability and a statistical condition for creation of thermodynamic breathers

We analyze certain aspects of the classical dynamics of a one-dimensional discrete nonlinear Schrödinger model with inter-site as well as on-site nonlinearities. The equation is derived from a mixed Klein–Gordon/Fermi–Pasta–Ulam chain of anharmonic oscillators coupled with anharmonic inter-site potentials, and approximates the slow dynamics of the fundamental harmonic of discrete small-amplitude modulational waves. We give explicit analytical conditions for modulational instability of travelling plane waves, and find in particular that sufficiently strong inter-site nonlinearities may change the nature of the instabilities from long-wavelength to short-wavelength perturbations. Further, we describe thermodynamic properties of the model using the grand-canonical ensemble to account for two conserved quantities: norm and Hamiltonian. The available phase space is divided into two separated parts with qualitatively different properties in thermal equilibrium: one part corresponding to a normal thermalized state with exponentially small probabilities for large-amplitude excitations, and another part typically associated with the formation of high-amplitude localized excitations, interacting with an infinite-temperature phonon bath. A modulationally unstable travelling wave may exhibit a transition from one region to the other as its amplitude is varied, and thus modulational instability is not a sufficient criterion for the creation of persistent localized modes in thermal equilibrium. For pure on-site nonlinearities the created localized excitations are typically pinned to particular lattice sites, while for significant inter-site nonlinearities they become mobile, in agreement with well-known properties of localized excitations in Fermi–Pasta–Ulam-type chains.

Stability of an oscillator chain from high-frequency initial conditions

We study the stability of the Fermi–Pasta–Ulam chain of coupled oscillators with a quartic coupling nonlinearity (FPU- β system), for initial conditions that give rise primarily to high-frequency oscillations. We consider initial conditions that are either close to a high-frequency mode of the linearized system or to the nonlinear equilibria that can be constructed from an envelope approximation for the high-frequency oscillations. The theory indicates that broad mode-like initial conditions are unstable and the instability structure can often be approximately calculated from the stability analysis of the π -mode equilibrium in which there is exact alternation of the phases of the individual oscillators. In contrast, examination of the stability of breather-like structures indicates that they are probably stable to the fast parametric instability. These theoretical observations are generally confirmed by numerical integration of the exact equations of motion. The study clarifies previous observations of the formation of nonlinear structures, and the relative stability of those structures on time scales that are long compared to the growth times of parametric instabilities calculated within the envelope approximation.

Tsallis thermostatistics for finite systems: a Hamiltonian approach

The derivation of the Tsallis generalized canonical distribution from the traditional approach of the Gibbs microcanonical ensemble is revisited (Phys. Lett. A 193 (1994) 140). We show that finite systems whose Hamiltonians obey a generalized homogeneity relation rigorously follow the nonextensive thermostatistics of Tsallis. In the thermodynamical limit, however, our results indicate that the Boltzmann–Gibbs statistics is always recovered, regardless of the type of potential among interacting particles. This approach provides, moreover, a one-to-one correspondence between the generalized entropy and the Hamiltonian structure of a wide class of systems, revealing a possible origin for the intrinsic nonlinear features present in the Tsallis formalism that lead naturally to power-law behavior. Finally, we confirm these exact results through extensive numerical simulations of the Fermi–Pasta–Ulam chain of anharmonic oscillators.

Effective Hamiltonian for travelling discrete breathers

Hamiltonian chains of oscillators in general probably do not sustain exact travelling discrete breathers. However solutions which look like moving discrete breathers for some time are not difficult to observe in numerics. In this paper we propose an abstract framework for the description of approximate travelling discrete breathers in Hamiltonian chains of oscillators. The method is based on the construction of an effective Hamiltonian enabling one to describe the dynamics of the translation degree of freedom of moving breathers. Error estimate on the approximate dynamics is also studied. The concept of the Peierls–Nabarro barrier can be made clear in this framework. We illustrate the method with two simple examples, namely the Salerno model which interpolates between the Ablowitz–Ladik lattice and the discrete nonlinear Schrodinger system, and the Fermi–Pasta–Ulam chain.