About the Fermi–Pasta–Ulam–Tsingou Chain
This simulator models a chain of N point masses connected by identical nonlinear springs, with both ends clamped to fixed walls. Each mass i obeys mẍⁱ = V′(xᵢ₊₁−xᵢ) − V′(xᵢ−xᵢ₋₁), integrated with the symplectic velocity Verlet scheme so total energy stays well conserved over long runs. Three spring potentials are selectable: α-FPUT (V = ½x² + ⅓αx³, a small cubic correction to Hooke's law), β-FPUT (V = ½x² + ¼βx⁴, a quartic correction), and the Toda lattice (V = (eᵃˣ − αx − 1)/α², an exactly integrable exponential spring). Setting the nonlinearity slider to 0 recovers the purely harmonic chain, where energy injected into one normal mode stays there forever — normal modes of a linear chain never exchange energy.
At start-up (and on Reset) the chain is excited in a single normal mode k, xᵢ(0) = A·sin(kπi/(N+1)), using the sliders for excitation mode, initial amplitude, and chain length N. The top canvas draws the instantaneous displacement of every mass as a spring-and-bead chain, with bead colour encoding local kinetic energy (blue = cold, red = hot). The bottom canvas is the mode-energy spectrum: at every frame the current state is projected onto the N discrete sine normal modes of the linear chain (a discrete sine transform) and each mode's harmonic energy Eₖ = ½pₖ² + ½ωₖ²qₖ² is accumulated over time, showing how — and whether — energy leaks out of the initially excited mode k into its neighbours. The Stats panel tracks elapsed simulation time, total (conserved) energy, the energy remaining in mode k, and the largest displacement in the chain.
Frequently Asked Questions
What is the FPUT problem?
In 1953 Fermi, Pasta, Ulam, and (uncredited until much later) Tsingou ran one of the first computer experiments, simulating a chain of 64 masses with a small cubic (α) nonlinearity added to otherwise linear springs. They expected the initial single-mode excitation to quickly spread its energy evenly across all normal modes (thermalization/equipartition). Instead they observed near-recurrence: energy shared briefly with a few neighbouring modes, then returned almost entirely to the original mode, again and again, with no sign of thermalizing. This simulator reproduces exactly that setup — excite one mode, watch the mode-energy spectrum, and see whether the energy comes back.
What is the difference between the α-FPUT, β-FPUT, and Toda models?
All three replace the ideal linear spring V=½x² with a slightly nonlinear one. α-FPUT adds a cubic term (asymmetric, softer in one direction and stiffer in the other); β-FPUT adds a quartic term (symmetric, stiffening in both directions); the Toda lattice uses an exponential potential that is exactly solvable (integrable) via inverse scattering and supports exact soliton solutions. Increasing the nonlinearity slider (α or β) strengthens the coupling between normal modes; the Toda model recurs almost perfectly at any amplitude, while α- and β-FPUT eventually break down into genuine chaos and thermalization once the nonlinearity is strong enough.
Why does the energy return to the original mode instead of spreading out?
The resolution, found by Zabusky and Kruskal in 1965, is that the FPUT chain in the continuum limit behaves like the Korteweg–de Vries (KdV) equation, whose solutions are dominated by solitons — solitary waves that pass through each other without losing their identity. The near-recurrence is a discrete-chain echo of soliton dynamics: the system is close enough to an integrable one (Toda, KdV) that the KAM theorem's obstruction to thermalization still largely applies. Only at strong nonlinearity, where enough neighbouring modes overlap and resonate, does the chain lose this near-integrable structure and genuinely thermalize into chaotic, equipartitioned motion.
What does the mode-energy spectrum panel actually plot?
At each animation frame the simulator projects the current positions and velocities onto the N normal modes of the linear chain using a discrete sine transform (qₖ = Σᵢ xᵢ·sin(kπi/(N+1)), and similarly for momenta), then computes each mode's harmonic energy Eₖ = ½pₖ² + ½ωₖ²qₖ² with dispersion relation ωₖ = 2sin(kπ/2(N+1)). These per-mode energies are accumulated into a running total for each of the first 24 modes and shown as bars, with the initially excited mode k highlighted — this is exactly the diagnostic Fermi, Pasta, Ulam, and Tsingou used in their original 1955 report.
Why use velocity Verlet integration instead of a simpler method?
Velocity Verlet is a symplectic integrator: it approximately preserves the Hamiltonian structure of the equations of motion, so total energy oscillates around a constant value instead of drifting away over long simulation times. For a chaotic or near-integrable system like the FPUT chain, a non-symplectic method (e.g. plain Euler) would leak or gain energy artificially and could manufacture false thermalization or false recurrence. The Total Energy stat in this simulator should stay essentially flat over time — that is a direct check of the integrator's quality.
What happens if I turn the nonlinearity slider to zero?
At α=β=0 every model reduces to the purely harmonic chain, whose exact solution is a sum of independent normal modes that never exchange energy. Exciting mode k leaves the mode-energy spectrum showing all the energy permanently parked in bar k, with every other bar at zero — there is no beating, no recurrence, and no chaos, because linear normal modes simply do not interact. This is the baseline the FPUT experiment was designed to test against.
What is the KAM theorem's role here?
The Kolmogorov–Arnold–Moser (KAM) theorem describes how, when a small nonlinear perturbation is added to an integrable Hamiltonian system (like the linear chain or the integrable Toda lattice), most of the quasi-periodic invariant tori survive rather than immediately dissolving into chaos. This is why the α- and β-FPUT chains show long-lived near-recurrence at weak-to-moderate nonlinearity instead of instantly thermalizing: the system stays close to an integrable one. Only once the nonlinearity slider is pushed high enough do enough resonances overlap to destroy the surviving tori and produce genuine chaotic energy sharing across modes.
How does chain length N and excitation mode k affect the dynamics?
Larger N gives finer spatial and spectral resolution and lets higher mode numbers exist, but also increases the number of nearly-resonant mode triplets/quartets that nonlinear terms can couple, generally making thermalization easier at a given nonlinearity. Exciting a low mode (small k, long wavelength) tends to show the cleanest FPUT recurrence, since its nearest neighbours in the mode spectrum are well separated in frequency; exciting a high mode (k close to N) puts the system closer to the Brillouin-zone edge where the dispersion relation flattens and neighbouring modes are more easily driven into resonance.
Why do the beads in the top canvas change colour?
Each bead's colour encodes its instantaneous kinetic energy, ½mvᵢ², scaled relative to the hottest bead in the chain at that moment (blue → low kinetic energy, red → high kinetic energy). Watching the colour pattern is a quick visual proxy for the same energy-sharing question the mode spectrum answers numerically: in the near-recurrent regime the "hot spot" sloshes back and forth in a fairly organized pattern tied to the excited mode's wavelength, while in the chaotic/thermalized regime the hot and cold beads become spatially disordered and roughly uniform in time-average.
Is this the same system Fermi, Pasta, Ulam, and Tsingou originally simulated?
Yes, structurally: a 1-D chain of masses on identical nonlinear springs with fixed (clamped) ends, started from a single sinusoidal normal-mode excitation and evolved forward in time on a computer — precisely the 1955 Los Alamos report's setup (originally with N=32 or 64 masses and a small α or β term). This simulator lets you additionally swap in the exactly integrable Toda potential, which was introduced later (1967) specifically to explain, via soliton theory, why the original FPUT chain refused to thermalize.