---
title: "FPUT Recurrence Laboratory Interactive Lab | Sandbox Physics"
description: "In a chain of masses and springs, can energy spread among the vibration modes and later return? Yes, in the FPUT chain, a row of masses joined by nonlinear springs. Energy put into one mode spreads to others and then mostly comes back; in a linear chain it would never leave the first mode. Seeing the energy return within a finite run, or seeing it spread widely across modes, does not prove that it will eventually be shared out evenly."
canonical_url: "https://sandboxphysics.com/labs/fput-recurrence/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/fput-recurrence/"
last_modified: "2026-10-22"
---

M089 · Discrete mechanics

# FPUT Recurrence Laboratory

**In a chain of masses and springs, can energy spread among the vibration modes and later return?** Yes, in the FPUT chain, a row of masses joined by nonlinear springs. Energy put into one mode spreads to others and then mostly comes back; in a linear chain it would never leave the first mode. Seeing the energy return within a finite run, or seeing it spread widely across modes, does not prove that it will eventually be shared out evenly.

Start a nonlinear spring chain, fixed at both ends, in a single sine mode. Compare linear, quadratic-force and cubic-force chains and follow the energy in every harmonic mode, the nonlinear part of the energy, the modal entropy and the first sampled return after energy leaves the starting mode. The 3D chain moves lengthwise, and you can check the bond forces on each bead; the full mechanical energy and the wall impulse are shown apart from the modal shares. Drag the release profile in the data row to change its amplitude.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

The 3D apparatus is schematic. Lengthwise displacement is drawn on a scale fixed for each record, separate from the equilibrium spacing, and the bead sizes and spring coils do not model collisions. Colour shows the current local energy relative to the current peak. The model is a normalized, isolated chain of ideal point masses with fixed ends and polynomial springs between nearest neighbours; there are no collisions and no damping. The time unit takes the reference mass and stiffness as one. The harmonic mode energies leave out the nonlinear bond energy and are normalized by their own sum, so that sum is not the full conserved energy. A return means the sampled share of the starting mode first drops below four fifths and later reaches at least nine tenths; a linear mode that never leaves does not count as recurrence. The first linear fundamental period sets the time scale for every starting mode. The entropy describes a finite harmonic distribution and does not prove thermalization. Sampling can shift a threshold return or miss it. A bond with only the quadratic force term is a local polynomial model, so recording stops before any bond reaches the region where the potential no longer holds it. The step size is capped by a conservative bound on the fastest vibration, and a record may take at most 500000 steps; either limit can shorten an extreme record. The half-step comparison covers only the first ten time units.

Physics tutorial

## Energy leaves, spreads and returns

**Background** A nonlinear chain was expected to share its energy quickly among all its modes. The Fermi–Pasta–Ulam–Tsingou experiment found instead that much of the harmonic energy can come back to the mode it started in.

**Why it matters** The 1955 Los Alamos report studied isolated chains whose bond forces are polynomials. This Lab runs a finite test with fixed rules and also tracks the nonlinear part of the energy.

Start with the essentials

- **Focus question**: Is the return still there once you require energy to leave first, compare with a linear chain and use a smaller integration step?
- **One-sentence intuition**: Track the full Hamiltonian separately from the energy in the harmonic modes. Count a later threshold crossing as a return only after the energy has first left the starting mode.

### Core mathematical model

#### Polynomial bond potential

$$
V(r)=\tfrac12r^2+\tfrac\alpha3r^3+\tfrac\beta4r^4,\qquad r_i=u_i-u_{i-1}
$$

The quadratic and cubic force terms come from the same Hamiltonian.

#### Fixed-end chain dynamics

$$
\ddot u_i=V^{\prime}(u_{i+1}-u_i)-V^{\prime}(u_i-u_{i-1}),\qquad u_0=u_{N+1}=0
$$

Nothing drives or damps the chain, and the beads do not collide.

#### Linear harmonic projection

$$
Q_j=\sqrt{\frac2{N+1}}\sum_{i=1}^N u_i\sin\frac{ij\pi}{N+1},\qquad E_j=\tfrac12(\dot Q_j^2+\omega_j^2Q_j^2)
$$

Even when the bonds are nonlinear, the harmonic modes are still a useful way to see where the energy is.

#### Full energy and the nonlinear remainder

$$
H=\sum_jE_j+\sum_i\left(\tfrac\alpha3r_i^3+\tfrac\beta4r_i^4\right)
$$

Only the full Hamiltonian, remainder included, is the conserved mechanical energy.

#### Normalized harmonic modal entropy

$$
p_j=E_j/\sum_kE_k,\qquad S=-\frac{\sum_jp_j\log p_j}{\log N}
$$

Energy concentrated in a few harmonic modes gives low entropy. The nonlinear remainder is left out of this measure.

### Common difficulties

#### Staying put is not coming back

**Typical misconception** A linear mode returning to the same displacement is the same as nonlinear recurrence.

**Better mental model** In the linear chain the mode never loses its energy. Here a return counts only after the starting mode’s share has first dropped below four fifths.

#### Harmonic energy is not the Hamiltonian

**Typical misconception** The sum of projected mode energies must be exactly constant.

**Better mental model** The nonlinear bond energy can be positive or negative, and it trades energy with the harmonic part.

#### Entropy does not prove thermalization

**Typical misconception** Once energy spreads over many modes, it stays equally shared forever.

**Better mental model** Energy can still come back within the finite record. Entropy alone cannot show that the chain reaches equilibrium in the long run.

### Run the experiment

1. 01

   #### See the forces on one bead in 3D

   Pause, choose an inspected site and use the fine motion step. Compare the teal and gold bond forces with the violet sum and the acceleration readout.

   **What to observe:** The beads move only along the chain. The displacement profile below is a graph; bead size and drawing magnification are schematic.

2. 02

   #### Check the energy and the wall impulse

   Open Energy & wall impulse and scrub the acquired record, then return to Motion & modes.

   **What to observe:** The full energy terms and the signed numerical energy error follow the same cursor. The fixed walls exchange momentum with the chain but do no work.

3. 03

   #### Run the linear chain first

   Acquire the linear preset and inspect the modal history.

   **What to observe:** One mode keeps its energy, and the energy never leaves it.

4. 04

   #### Record a nonlinear return

   Use the default quadratic-force preset and acquire the full record.

   **What to observe:** Energy spreads to other modes and the starting mode’s share falls. Later, the sampled share climbs back past the return threshold.

5. 05

   #### Change the mixing strength

   Compare cubic-force and stronger releases, then inspect the entropy.

   **What to observe:** Different runs need not return within the same observation window.

6. 06

   #### Check the numerics

   Reduce the step and compare the full energy defect and early refinement readout.

   **What to observe:** Convergence over the early part of the run and the energy error over the whole record are two different kinds of evidence.

Physical conclusion

A return within a finite run and a high modal entropy do not show that the energy ends up equally shared in the long run.

Motion & dynamics · Starter path

## From a falling ball to an orbit

Start by measuring gravity and splitting motion into components. Then add contact forces and collisions, and follow the same laws all the way to an orbit.

[Start here → **Free Fall & Measuring Gravity** Capture a few drops and fit the camera positions to estimate gravitational acceleration. Then add air damping and watch the fit drift from the truth. Observe · 14 min](https://sandboxphysics.com/labs/free-fall/)

1. [1 Free Fall & Measuring Gravity](https://sandboxphysics.com/labs/free-fall/)
2. [2 Projectile Flight Designer](https://sandboxphysics.com/labs/projectile-flight/)
3. [3 Inclined Plane & Friction](https://sandboxphysics.com/labs/inclined-plane/)
4. [4 One-Dimensional Collision Bench](https://sandboxphysics.com/labs/collision-bench/)
5. [5 Mass–Spring Oscillator](https://sandboxphysics.com/labs/mass-spring/)
6. [6 How Orbits Take Shape](https://sandboxphysics.com/labs/orbit/)

[View the full path →](https://sandboxphysics.com/topics/mechanics/#starter-path)