The Fermi-Pasta-Ulam problem: 50 years of progress

Abstract

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 considered.

48 pages, no figures, corrected and accepted for publication

References in corpus (4)

• Dynamics of collapsing and exploding Bose-Einstein condensates
• Irregular Dynamics in a One-Dimensional Bose System
• Direction reversing travelling waves in the Fermi-Pasta-Ulam chain
• Long time Evolution of Quantum Averages Near Stationary Points

Cited by in corpus (149)

• From Quantum Chaos and Eigenstate Thermalization to Statistical Mechanics and Thermodynamics
• Relaxation in a Completely Integrable Many-Body Quantum System: An Ab Initio Study of the Dynamics of the Highly Excited States of Lattice Hard-Core Bosons
• Phononics: Manipulating heat flow with electronic analogs and beyond
• Heat Transport in low-dimensional systems
• Prethermalization and universal dynamics in near-integrable quantum systems
• Dynamical obstruction to localization in a disordered spin chain
• Geometrical properties of local dynamics in Hamiltonian systems: the Generalized Alignment Index (GALI) method
• Simulating rare events in dynamical processes
• A route to thermalization in the $\alpha$-Fermi-Pasta-Ulam system
• Holographic Thermalization, stability of AdS, and the Fermi-Pasta-Ulam-Tsingou paradox
• Anomalous Heat Diffusion
• Anomalous Heat Diffusion
• Local Temperature and Universal Heat Conduction in FPU chains
• Cosmic axion thermalization
• q-Breathers and the Fermi-Pasta-Ulam Problem
• Onset of chaos and relaxation in isolated systems of interacting spins-1/2: energy shell approach
• Numerical test of hydrodynamic fluctuation theory in the Fermi-Pasta-Ulam chain
• The Nonlinear Schroedinger Equation with a random potential: Results and Puzzles
• Numerical integration of variational equations
• q-Breathers in Fermi-Pasta-Ulam chains: existence, localization and stability
• Limiting phase trajectories and the origin of energy localization in nonlinear oscillatory chains
• Dynamics of 2D Material Membranes
• Fermi-Pasta-Ulam model with long-range interactions: Dynamics and thermostatistics
• Detecting chaos, determining the dimensions of tori and predicting slow diffusion in Fermi--Pasta--Ulam lattices by the Generalized Alignment Index method
• Conserved quantities and dual turbulent cascades in Anti-de Sitter spacetime
• Thermal transport in the Fermi-Pasta-Ulam model with long-range interactions
• Chaotic Dynamics of N-degree of Freedom Hamiltonian Systems
• FPU physics with nanomechanical graphene resonators: intrinsic relaxation and thermalization from flexural mode coupling
• Holographic Relaxation of Finite Size Isolated Quantum Systems
• Equilibration of Quasi-Integrable Systems
• Stability of Simple Periodic Orbits and Chaos in a Fermi -- Pasta -- Ulam Lattice
• Dynamical freezing of relaxation to equilibrium
• Exact solitary wave solutions for a discrete $\lambda {\varphi }^4$ field theory in 1+1 dimensions
• Dynamics and Statistics of the Fermi--Pasta--Ulam $\beta$--model with different ranges of particle interactions
• Arbitrarily Weak Nonlinearity Can Destroy the Anderson Localization
• Ballistic resonance and thermalization in Fermi-Pasta-Ulam-Tsingou chain at finite temperature
• Nonlinear phononics using atomically thin membranes
• Stability and pre-thermalization in chains of classical kicked rotors
• Triggering waves in nonlinear lattices: Quest for anharmonic phonons and corresponding mean free paths
• Crossover behavior of the thermal conductance and Kramers' transition rate theory
• Wave Turbulence and thermalization in one-dimensional chains
• FPU phenomenon for generic initial data
• Role of initial conditions in $1D$ diffusive systems: compressibility, hyperuniformity and long-term memory
• An Exactly Solvable Model for the Integrability-Chaos Transition in Rough Quantum Billiards
• Application of the GALI Method to Localization Dynamics in Nonlinear Systems
• All-phononic Digital Transistor on the Basis of Gap-Soliton Dynamics in Anharmonic Oscillator Ladder
• Rectification of heat currents across nonlinear quantum chains: A versatile approach beyond weak thermal contact
• Signatures of Chaos in Time Series Generated by Many-Spin Systems at High Temperatures
• Resonant Hamiltonian systems and weakly nonlinear dynamics in AdS spacetimes
• Equilibration and Thermalization of Classical Systems
• Spatially quasi-periodic water waves of infinite depth
• Universal law of thermalization for one-dimensional perturbed Toda lattices
• Interaction-disorder competition in a spin system evaluated through the Loschmidt Echo
• Renormalized phonons in nonlinear lattices: A variational approach
• Discrete Symmetry and Stability in Hamiltonian Dynamics
• Duration of classicality in highly degenerate interacting Bosonic systems
• Behavior and Breakdown of Higher-Order Fermi-Pasta-Ulam-Tsingou Recurrences
• Coupled Oscillator Model for Nonlinear Gravitational Perturbations
• Linear superposition in nonlinear wave dynamics
• Thermal conduction and interface effects in nanoscale Fermi-Pasta-Ulam conductors
• Understanding Divergent Thermal Conductivity in Single Polythiophene Chains using Modal Analysis and Sonification
• Geometry of quantum observables and thermodynamics of small systems
• Dynamical Transition for a class of integrable models coupled to a bath
• Dynamics of nonlinear excitations of helically confined charges
• Anharmonicity-induced excited-state quantum phase transition in the symmetric phase of the two-dimensional limit of the vibron model
• Dynamics of Time-Modulated, Nonlinear Phononic Lattices
• Fermi-Pasta-Ulam phenomena and persistent breathers in the harmonic trap
• Quasi-periodic traveling gravity-capillary waves
• Internal dissipation of a polymer
• Stability of Nonlinear Normal Modes in the FPU-$\beta$ Chain in the Thermodynamic Limit
• Loschmidt echo in many-spin systems: a quest for intrinsic decoherence and emergent irreversibility
• Energy returns in global AdS_4
• Long-lived Solitons and Their Signatures in the Classical Heisenberg Chain
• Thermalization of local observables in the $\alpha$-FPUT chain
• Unusual ergodic and chaotic properties of trapped hard rods
• Chaotic dynamics in a quantum Fermi-Pasta-Ulam problem
• Simply improved averaging for coupled oscillators and weakly nonlinear waves
• The $\beta$ Fermi-Pasta-Ulam-Tsingou Recurrence Problem
• Solitons in one-dimensional mechanical linkage
• The closeness of localised structures between the Ablowitz-Ladik lattice and Discrete Nonlinear Schrödinger equations II: Generalised AL and DNLS systems
• Energy localization and excess fluctuations from long-range interactions in equilibrium molecular dynamics
• Nekhoroshev theorem for the periodic Toda lattice
• Transition fronts and their universality classes
• An extension of Bogoliubov theory for a many-body system with a time scale hierarchy: the quantum mechanics of second Josephson oscillations
• Variability in Fermi--Pasta--Ulam--Tsingou Arrays Prevents Recurrences
• Novel effective ergodicity breaking phase transition in a driven-dissipative system
• Divergence of the Floquet-Magnus expansion in a periodically driven one-body system with energy localization
• Classical route to ergodicity and scarring in collective quantum systems
• On the foundations of statistical mechanics
• Nonlinear Topological Edge States: from Dynamic Delocalization to Thermalization
• Existence of exponentially spatially localised breather solutions for lattices of nonlinearly coupled particles: Schauder's fixed point theorem approach
• Time-periodicities in holographic CFTs
• Localization in the Discrete Non-Linear Schrödinger Equation and geometric properties of the microcanonical surface
• The Fermi-Pasta-Ulam recurrence and related phenomena for 1D shallow-water waves in a finite basin
• Quantum damping of Fermi-Pasta-Ulam revivals in ultracold Bose gases
• Localized Thermal States
• Some breathers and multi-breathers for FPU-type chains
• Three- and four-wave resonances in the nonlinear quadratic Kelvin lattice
• Spontaneously formed phonon frequency combs in van der Waals solid CrXTe${}_3$ (X=Ge,Si)
• The anti-Fermi-Pasta-Ulam-Tsingou problem in one-dimensional diatomic lattices
• Black hole spectra in holography: consequences for equilibration of dual gauge theories
• Stabilities of one-dimensional stationary states of Bose-Einstein condensates
• Numerical study of Fermi-Pasta-Ulam recurrence for water waves over finite depth
• Birkhoff normal form for the periodic Toda lattice
• Open-system eigenstate thermalization in a noninteracting integrable model
• Energy transfer mechanisms in a dipole chain: From energy equipartition to the formation of breathers
• The fifth-order partial differential equation for the description of the α+β Fermi-Pasta-Ulam model
• Role of energy uncertainties in ergodicity breaking induced by competing interactions and disorder. A dynamical assessment through the Loschmidt echo
• Structural localization in the Classical and Quantum Fermi-Pasta-Ulam Model
• Energy-recurrence Breakdown and Chaos in Disordered Fermi-Pasta-Ulam-Tsingou Lattices
• Dynamical mechanisms leading to equilibration in two-component gases
• Higher order splitting methods with modified integrators for a class of Hamiltonian systems
• Nonlinear evolution of disturbances in higher time-derivative theories
• Nonlinearity accelerates the thermalization of the quartic FPUT model with stochastic baths
• Analysis of classical phase space and energy transfer for two rotating dipoles in an electric field
• Prethermalization in Fermi-Pasta-Ulam-Tsingou chains
• Dynamics of Nonlinear Waves on Bounded Domains
• Energy cascade and Burgers turbulence in the Fermi-Pasta-Ulam-Tsingou chain
• Efficient detection of chaos through the computation of the Generalized Alignment Index (GALI) by the multi-particle method
• Second Josephson excitations beyond mean field as a toy model for thermal pressure: exact quantum dynamics and the quantum phase model
• Extensive packet excitations in FPU and Toda lattices
• Collision of Polymers in a Vacuum
• Ergodic and non-ergodic many-body dynamics in strongly nonlinear lattices
• q-Breathers and thermalization in acoustic chains with arbitrary nonlinearity index
• Phase transition from localization to chaos in classical many-body system
• Negative Differential Heat Conductivity in a Harmonic Chain Coupled to a Particle Reservoir
• Renormalized waves and thermalization of the Klein-Gordon equation: What sound does a nonlinear string make?
• Compression and information entropy of binary strings from the collision history of three hard balls
• Thermalization is typical in large classical and quantum harmonic systems
• Ideal Linear Chain Polymers with Fixed Angular Momentum
• Phase transitions from linear to nonlinear information processing in neural networks
• Nonlinear dynamics determines the thermodynamic instability of condensed matter in vacuo
• Stability Properties of 1-Dimensional Hamiltonian Lattices with Non-analytic Potentials
• Solitons in lattice field theories via tight-binding supersymmetry
• Symbolic Computations for Nonlinear Resonances
• Stochastic Dissipative Euler's equations for a free body
• Intrinsic Dimensionality of Fermi-Pasta-Ulam-Tsingou High-Dimensional Trajectories Through Manifold Learning: A Linear Approach
• Revisiting a Low-Dimensional Model with Short Range Interactions and Mean Field Critical Behavior
• From Near-Integrable to Far-from-Integrable: A Unified Picture of Thermalization and Heat Transport
• The equilibrium classical scatter spectrum of waves
• Exploring the properties of quantum scars in a toy model
• Some Comments on Unitary Qubit Lattice Algorithms for Classical Problems
• Probability distribution in the Toda system: The singular route to a steady state
• The hidden role of coupled wave network topology on the dynamics of nonlinear lattices
• Nonlinear dynamics and chaos in multidimensional disordered Hamiltonian systems
• A molecular dynamics simulation of thermalization of crystalline lattice with harmonic interaction
• Stages of dynamics in the Fermi-Pasta-Ulam system as probed by the first Toda integral
• Supratransmission in Lattices with Purely Nonlinear Coupling
• Recurrence in a periodically driven and weakly damped Fermi-Pasta-Ulam-Tsingou chain