Ballistic transport and resonance phenomena are elucidated in the one-dimensional $α$-Fermi-Pasta-Ulam-Tsingou (FPUT) model using an approach of computing thermal response functions. The existence of periodic oscillations in spatially sinusoidal temperature profiles seen in previous studies is confirmed. However, the results obtained using response functions enable a more complete understanding. In particular, it is shown that resonance involves beats between normal modes which tend to reinforce in a one-dimensional chain. Anharmonic scattering acts to destroy phase coherence across the statistical ensemble, and with increasing anharmonicity, transport is driven towards the diffusive regime. These results provide additional insight into anomalous heat transport in low-dimensional systems. Normal-mode scattering is also explored using time correlation functions. Interestingly, these calculations, in addition to demonstrating loss of phase coherence across an ensemble of simulations, appear to show evidence of so-called q-breathers in conditions of strong anharmonicity. Finally, we describe how the approach outlined here could be developed to include quantum statistics and also also first-principles estimates of phonon scattering rates to elucidate second sound and ballistic transport in realistic materials at low temperatures.
Understanding the interplay between different wave excitations, such as phonons and localized solitons, is crucial for developing coarse-grained descriptions of many-body, near-integrable systems. We treat the Fermi-Pasta-Ulam-Tsingou (FPUT) non-linear chain and show numerically that at short timescales, relevant to the largest Lyapunov exponent, it can be modeled as a random perturbation of its integrable approximation -- the Toda chain. At low energies, the separation between two trajectories that start at close proximity is dictated by the interaction between few soliton modes and an intrin…
One-dimensional particle chains are fundamental models to explain anomalous thermal conduction in low-dimensional solids like nanotubes and nanowires. In these systems the thermal energy is carried by phonons, i.e. propagating lattice oscillations that interact via nonlinear resonance. The average energy transfer between the phonons is described by the wave kinetic equation (WKE), derived directly from the microscopic dynamics. Here, we use the spatially nonhomogeneous WKE of the prototypical $β-$ Fermi-Pasta-Ulam-Tsingou (FPUT) model, equipped with thermostats able to set different temperatur…
We study conversion of thermal energy to mechanical energy and vice versa in $α$-Fermi-Pasta-Ulam-Tsingou~(FPUT) chain with spatially sinusoidal profile of initial temperature. We show analytically that coupling between macroscopic dynamics and quasiballistic heat transport gives rise to mechanical vibrations with growing amplitude. This new phenomenon is referred to as "ballistic resonance". At large times, these mechanical vibrations decay monotonically, and therefore the well-known FPUT recurrence paradox occurring at zero temperature is eliminated at finite temperatures.
In 1955, Fermi, Pasta, Ulam, and Tsingou reported recurrence over time of energy between modes in a one-dimensional array of nonlinear oscillators. Subsequently, there have been myriad numerical experiments using homogenous FPUT arrays, which consist of chains of ideal, nonlinearly-coupled oscillators. However, inherent variations --- e.g., due to manufacturing tolerance --- introduce heterogeneity into the parameters of any physical system. We demonstrate that such tolerances degrade the observance of recurrence, often leading to complete loss in moderately sized arrays. We numerically simula…
We study the thermalization dynamics of one-dimensional diatomic lattices (which represents the simplest system possessing multi-branch phonons), exemplified by the famous Fermi-Pasta-Ulam-Tsingou (FPUT)-$β$ and the Toda models. Here we focus on how the system relaxes to the equilibrium state when part of highest-frequency optical modes are initially excited, which is called the anti-FPUT problem comparing with the original FPUT problem (low frequency excitations of the monatomic lattice). It is shown numerically that the final thermalization time $T_{\rm eq}$ of the diatomic FPUT-$β$ chain de…
It is reported that the numerical simulations of the Fermi-Pasta-Ulam problem were performed by a young lady, Mary Tsingou. After 50 years of omission, it is time for a proper recognition of her decisive contribution to the first ever numerical experiment, central in the solitons and chaos theories, but also one of the very first out-of-equilibrium statistical mechanics study. Let us quote from now on the Fermi-Pasta-Ulam-Tsingou problem.
Breather stability and longevity in thermally relaxing nonlinear arrays depend sensitively on their interactions with other excitations. We review the relaxation of breathers in Fermi-Pasta-Ulam arrays, with a specific focus on the different relaxation channels and their dependence on the interparticle interactions, dimensionality, initial condition, and system parameters.
Shortcuts to adiabaticity (STAs) have been used to make rapid changes to a system while eliminating or minimizing excitations in the system's state. In quantum systems, these shortcuts allow us to minimize inefficiencies and heating in experiments and quantum computing protocols, but the theory of STAs can also be generalized to classical systems. We focus on one such STA, approximate counter-diabatic (ACD) driving, and numerically compare its performance in two classical systems: a quartic anharmonic oscillator and the $β$ Fermi-Pasta-Ulam-Tsingou (FPUT) lattice. In particular, we modify an e…
In the framework of the Fermi-Pasta-Ulam (FPU) model, we show a simple method to give an accurate analytical estimation of the maximal Lyapunov exponent at high energy density. The method is based on the computation of the mean value of the modulational instability growth rates associated to unstable modes. Moreover, we show that the strong stochasticity threshold found in the $β$-FPU system is closely related to a transition in tangent space: the Lyapunov eigenvector being more localized in space at high energy.
We study the asymptotic dynamics of breathers in finite Fermi-Pasta-Ulam chains at zero and non-zero temperatures. While such breathers are essentially stationary and very long-lived at zero temperature, thermal fluctuations tend to lead to breather motion and more rapid decay.
The problem of detecting specific features of microscopic dynamics in the macroscopic behavior of a many-degrees-of-freedom system is investigated by analyzing the position and momentum time series of a heavy impurity embedded in a chain of nearest-neighbor anharmonic Fermi-Pasta-Ulam oscillators. Results obtained in a previous work [M. Romero-Bastida, Phys. Rev. E {\bf69}, 056204 (2004)] suggest that the impurity does not contribute significantly to the dynamics of the chain and can be considered as a probe for the dynamics of the system to which the impurity is coupled. The ($r,τ$) entropy, …
Studies of thermal transport in long-range (LR)interacting systems are currently particularly challenging. The main difficulties lie in the choice of boundary conditions and the definition of heat current when driving systems in an out-of-equilibrium state by the usual thermal reservoirs. Here, by employing a reverse type of thermal baths that can overcome such difficulties, we reveal the intrinsic features of thermal transport underlying a LR interacting Fermi-Pasta-Ulam chain. We find that under an appropriate range value of LR exponent $σ=2$, while a \emph{nonballistic} power-law length ($L…
Recent progress on thermal transport in one-dimensional long-range interacting Fermi-Pasta-Ulam-Tsingou lattice systems
Minjiang Collaborative Center for Theoretical Physics, College of Physics and Electronic Information Engineering, Minjiang University（闽江大学物理与电子信息工程学院理论物理协同中心）；Institute of Systems Science, Department of Physics, College of Information Science and Engineering, Huaqiao University（华侨大学信息科学与工程学院物理系系统科学研究所）；MOE Key Laboratory of Advanced Micro-Structured Materials, China-EU Joint Laboratory for Nanophononics, Center for Phononics and Thermal Energy Science, School of Physics Science and Engineering, Tongji University（同济大学物理科学与技术学院声子学与热能科学中心中纳音子学中欧联合实验室教育部先进微结构材料重点实验室）
We study the dynamics of Fermi-Pasta-Ulam chains with both harmonic and anharmonic power-law long-range interactions. We show that the dynamics is described in the continuum limit by a generalized fractional Boussinesq differential equation, whose derivation is performed in full detail. We also discuss a version of the model where couplings are alternating in sign.
The dynamics of energy relaxation in thermalized one- and two-dimensional arrays with nonlinear interactions depend in detail on the interactions and, in some cases, on dimensionality. We describe and explain these differences for arrays of the Fermi-Pasta-Ulam type. In particular, we focus on the roles of harmonic contributions to the interactions and of breathers in the relaxation process.
In the present work we study the Fermi--Pasta--Ulam (FPU) $β$--model involving long--range interactions (LRI) in both the quadratic and quartic potentials, by introducing two independent exponents $α_1$ and $α_2$ respectively, which make the {forces decay} with distance $r$. Our results demonstrate that weak chaos, in the sense of decreasing Lyapunov exponents, and $q$--Gaussian probability density functions (pdfs) of sums of the momenta, occurs only when long--range interactions are included in the quartic part. More importantly, for $0\leq α_2<1$, we obtain extrapolated values for $q \equiv …
We investigate anomalous energy transport processes in the Fermi-Pasta-Ulam $β$ lattice. They are determined by the maximum sound velocity of the relevant weakly damped energy carriers. That velocity can be numerically resolved by measuring the propagating fronts of the correlation functions of energy/momentum fluctuations at different times. The numerical results are compared with the predictions for solitons and effective (renormalized) phonons, respectively. Excellent agreement has been found for the prediction of effective long wavelength phonons, giving strong evidence that the energy car…
A brief review of the Fermi-Pasta-Ulam (FPU) paradox is given, together with its suggested resolutions and its relation to other physical problems. We focus on the ideas and concepts that have become the core of modern nonlinear mechanics, in their historical perspective. Starting from the first numerical results of FPU, both theoretical and numerical findings are discussed in close connection with the problems of ergodicity, integrability, chaos and stability of motion. New directions related to the Bose-Einstein condensation and quantum systems of interacting Bose-particles are also consider…
We consider a damped $β$-Fermi-Pasta-Ulam chain, driven at one boundary subjected to stochastic noise. It is shown that, for a fixed driving amplitude and frequency, increasing the noise intensity, the system's energy resonantly responds to the modulating frequency of the forcing signal. Multiple peaks appear in the signal to noise ratio, signalling the phenomenon of stochastic resonance. The presence of multiple peaks is explained by the existence of many stable and metastable states that are found when solving this boundary value problem for a semi-continuum approximation of the model. Stoch…
The largest Lyapunov exponent of a system composed by a heavy impurity embedded in a chain of anharmonic nearest-neighbor Fermi-Pasta-Ulam oscillators is numerically computed for various values of the impurity mass $M$. A crossover between weak and strong chaos is obtained at the same value $ε_{_T}$ of the energy density $ε$ (energy per degree of freedom) for all the considered values of the impurity mass $M$. The threshold $\epsi lon_{_T}$ coincides with the value of the energy density $ε$ at which a change of scaling of the relaxation time of the momentum autocorrelation function of the impu…
The Peierls equation is considered for the Fermi-Pasta-Ulam $β$ lattice. Explicit form of the linearized collision operator is obtained. Using this form the decay rate of the normal mode energy as a function of wave vector $k$ is estimated to be proportional to $k^{5/3}$. This leads to the $t^{-3/5}$ long time behavior of the current correlation function, and, therefore, to the divergent coefficient of heat conductivity. These results are in good agreement with the results of recent computer simulations. Compared to the results obtained though the mode coupling theory our estimations give the …
We present a numerical study of dynamical correlations (structure factors) of the long-range generalization of the Fermi-Pasta-Ulam oscillator chain, where the strength of the interaction between two lattice sites decays as a power $α$ of the inverse of their distance. The structure factors at finite energy density display distinct peaks, corresponding to long-wavelength propagating modes, whose dispersion relation is compatible with the predictions of the linear theory. We demonstrate that dynamical scaling holds, with a dynamical exponent $z$ that depends weakly on $α$ in the range $1<α<3$. …
The linear response to temperature variations is well characterised for equilibrium systems but a similar theory is not available, for example, for inertial heat conducting systems, whose paradigm is the Fermi-Pasta-Ulam (FPU) model driven by two different boundary temperatures. For models of inertial systems out of equilibrium, including relaxing systems, we show that Andersen thermostats are a natural tool for studying the thermal response. We derive a fluctuation-response relation that allows to predict thermal expansion coefficients or the heat capacitance in nonequilibrium regimes. Simula…
Gap-controlled thermalization in a chain of alternating nonlinear springs: an SSH model version of the Fermi-Pasta-Ulam-Tsingou problem
Posgrado en Ciencias Físicas, Universidad Nacional Autónoma de México（墨西哥国立自治大学物理科学研究生院）；Depto. de Sistemas Complejos, Instituto de Física, Universidad Nacional Autónoma de México (UNAM)（墨西哥国立自治大学物理研究所复杂系统系）
We introduce and numerically study a long-range-interaction generalization of the one-dimensional Fermi-Pasta-Ulam (FPU) $β-$ model. The standard quartic interaction is generalized through a coupling constant that decays as $1/r^α$ ($α\ge 0$)(with strength characterized by $b>0$). In the $α\to\infty$ limit we recover the original FPU model. Through classical molecular dynamics computations we show that (i) For $α\geq 1$ the maximal Lyapunov exponent remains finite and positive for increasing number of oscillators $N$ (thus yielding ergodicity), whereas, for $0 \le α<1$, it asymptotically decre…
Recent work has developed a nonlinear hydrodynamic fluctuation theory for a chain of coupled anharmonic oscillators governing the conserved fields, namely stretch, momentum, and energy. The linear theory yields two propagating sound modes and one diffusing heat mode. In contrast, the nonlinear theory predicts that, at long times, the sound mode correlations satisfy Kardar-Parisi-Zhang (KPZ) scaling, while the heat mode correlations satisfies Levy-walk scaling. In the present contribution we report on molecular dynamics simulations of Fermi-Pasta-Ulam chains to compute various spatiotemporal co…
We show numerically that including soft-sphere type collisions in the celebrated Fermi-Pasta-Ulam ($FPU$) chain completely alters the thermal transport characteristics. The resulting $FPU^C$ chains, while being momentum preserving, satisfy the Fourier's law and do not show anomalous thermal transport behavior. Collisions play a significant role in reducing the boundary jumps typically observed in $FPU$ chains. The thermal conductivity of the $FPU^C$ chains is significantly smaller than the $FPU$ chains at low temperatures due to the fast redistribution of energy from the lowest mode of vibrati…
Random initial data and average shock time in the Fermi-Pasta-Ulam-Tsingou chain
Scuola Internazionale di Studi Superiori Avanzati（高级研究国际学院）；INFN, Sezione di Firenze（意大利国家核物理研究所佛罗伦萨分部）；Department of Mathematics “T. Levi-Civita”, University of Padova（帕多瓦大学“T. 莱维-奇维塔”数学系）；ISC-CNR（凝聚态物理信息系统中心（CNR））；École Normale Supérieure de Lyon（里昂高等师范学院）
We study heat conduction in one, two and three dimensional anharmonic lattices connected to stochastic Langevin heat baths. The inter-atomic potential of the lattices is double-well type, i.e., $V_{\rm DW}(x)=k_2x^2/2+k_4 x^4/4$ with $k_2<0$ and $k_4>0$. We observe two different temperature regimes of transport: a high-temperature regime where asymptotic length dependence of nonequilibrium steady state heat current is similar to the well-known Fermi-Pasta-Ulam lattices with an inter-atomic potential, $V_{\rm FPU}(x)=k_2x^2/2+k_4 x^4/4$ with $k_2,k_4>0$. A low temperature regime where heat cond…
Currently, effective phonons (renormalized or interacting phonons) rather than solitary waves (for short, solitons) are regarded as the energy carriers in nonlinear lattices. In this work, by using the approximate soliton solutions of the corresponding equations of motion and adopting the Boltzmann distribution for these solitons, the average velocities of solitons are obtained and are compared with the sound velocities of energy transfer. Excellent agreements with the numerical results and the predictions of other existing theories are shown in both the symmetric Fermi-Pasta-Ulam-$β$ lattices…
The pioneering computer simulations of the energy relaxation mechanisms performed by Fermi, Pasta and Ulam can be considered as the first attempt of understanding energy relaxation and thus heat conduction in lattices of nonlinear oscillators. In this paper we describe the most recent achievements about the divergence of heat conductivity with the system size in 1d and 2d FPU-like lattices. The anomalous behavior is particularly evident at low energies, where it is enhanced by the quasi-harmonic character of the lattice dynamics. Remakably, anomalies persist also in the strongly chaotic region…
The Fermi-Pasta-Ulam (FPU) one-dimensional Hamiltonian includes a quartic term which guarantees ergodicity of the system in the thermodynamic limit. Consistently, the Boltzmann factor $P(ε) \sim e^{-βε}$ describes its equilibrium distribution of one-body energies, and its velocity distribution is Maxwellian, i.e., $P(v) \sim e^{- βv^2/2}$. We consider here a generalized system where the quartic coupling constant between sites decays as $1/d_{ij}^α$ $(α\ge 0; d_{ij} = 1,2,\dots)$. Through {\it first-principle} molecular dynamics we demonstrate that, for large $α$ (above $α\simeq 1$), i.e., shor…
We address the question of the effect of disorder on heat conduction in an anharmonic chain with interactions given by the Fermi-Pasta-Ulam (FPU) potential. In contrast to the conclusions of an earlier paper [Phys. Rev. Lett. 86, 63 (2001)] which found that disorder could induce a finite thermal conductivity at low temperatures, we find no evidence of a finite temperature transition in conducting properties. Instead, we find that at low temperatures, small system size transport properties are dominated by disorder but the asymptotic system size dependence of current is given by the usual FPU r…
The recently discovered phenomenon of nonlinear supratransmission consists in a sudden increase of the amplitude of a transmitted wave triggered by the excitation of nonlinear localized modes of the medium. We examine this process for the Fermi-Pasta-Ulam chain, sinusoidally driven at one edge and damped at the other. The supratransmission regime occurs for driving frequencies above the upper band-edge and originates from direct moving discrete breather creation. We derive approximate analytical estimates of the supratransmission threshold, which are in excellent agreement with numerics. When …
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…
We employ the heat perturbations correlation function to study thermal transport in the one-dimensional (1D) Fermi-Pasta-Ulam-$β$ lattice with both nearest-neighbor and next-nearest-neighbor couplings. We find that such a system bears a peculiar phonon dispersion relation, and thus there exists a competition between phonon dispersion and nonlinearity that can strongly affect the heat correlation function' shape and scaling property. Specifically, for small and large anharmoncity, the scaling laws are ballistic and superdiffusive types, respectively, which are in good agreement with the recent …
We study analytical solutions of the Fractional Boussinesq Equation (FBE), which is an effective model for the Fermi-Pasta-Ulam (FPU) one-dimensional lattice with long-range couplings. The couplings decay as a power-law with exponent s, with 1 < s < 3, so that the energy density is finite, but s is small enough to observe genuine long-range effects. The analytic solutions are obtained by introducing an ansatz for the dependence of the field on space and time. This allows to reduce the FBE to an ordinary differential equation, which can be explicitly solved. The solutions are initially localize…
We demonstrate that in the atomic-scale limit the thermal conductance $\mathcal K$ of the FPU model and its variants strongly deviates from the mesoscopic behavior due to the relevance of contact resistance. As a result, atomic chains follow $\log \mathcal K = ν\log T$, where the power law coefficient $ν$ is exactly two times larger than the mesoscopic value. We smoothly interconnect the atomic and mesoscopic limits, and demonstrate that this turnover behavior takes place in other nonlinear FPU-like models. Our results are significant for nanoscale applications, manifesting an atomic thermal c…
We study the thermal transport properties of the one dimensional Fermi-Pasta-Ulam model ($β$-type) with long-range interactions. The strength of the long-range interaction decreases with the (shortest) distance between the lattice sites as ${distance}^{-δ}$, where $δ\ge 0$.Two Langevin heat baths at unequal temperatures are connected to the ends of the one dimensional lattice via short-range harmonic interactions that drive the system away from thermal equilibrium. In the nonequilibrium steady state the heat current, thermal conductivity and temperature profiles are computed by solving the equ…
We study the equilibrium time correlations for the conserved fields of classical anharmonic chains and argue that their dynamic correlator can be predicted on the basis of nonlinear fluctuating hydrodynamics. In fact our scheme is more general and would cover also other one-dimensional hamiltonian systems, for example classical and quantum fluids. Fluctuating hydrodynamics is a nonlinear system of conservation laws with noise. For a single mode it is equivalent to the noisy Burgers equation, for which explicit solutions are available. Our focus is the case of several modes. No exact solutions …
