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.

Inspected site number $0$

Inspected longitudinal displacement $0$

Inspected longitudinal velocity $0$

Force from left bond $0$

Force from right bond $0$

Net longitudinal force $0$

Inspected longitudinal acceleration $0$

Drawing displacement / equilibrium spacing scale $0$

Acquired normalized time $0$

Total mechanical energy $0$

Kinetic energy $0$

Bond potential energy $0$

Signed energy defect $0$

Chain momentum $0$

Independent net wall impulse $0$

Momentum minus initial and wall impulse $0$

Half-step displacement difference · first ten time units $0$

Actual normalized integration step $0$

Maximum integration energy defect · full record $0$

Record / model domain $0$

Initial-mode share of harmonic energy $0$

Normalized modal entropy $0$

Sum of harmonic mode energies $0$

Signed nonlinear bond-energy contribution $0$

Lowest acquired initial-mode share $0$

First sampled return · fundamental periods $0$

Departure and return status $0$

Linear fundamental period $0$

Release the chain and record its motion

In a chain of masses and springs, can energy spread among the vibration modes and later return?

Inspected bond forcesTeal shows the left bond's force, gold the right bond's and violet their sum. All three share one force scale at each instant.Inspected velocityBlue arrow with its own scale, separate from force. Rightward is positive.Prepared release outlinesWire outlines mark the starting positions, which stay fixed.Inspect site along the chain

s/L

50.00\,\%

Moving sites

N

16.00\,

Quadratic force coefficient

\alpha

0.25\,

Cubic force coefficient

\beta

0.00\,

Initial sine-mode displacement

A

4.00\,

Initially excited mode

j_0

1.00\,

Observed fundamental periods

N_T

100.00\,

Maximum normalized step

h_{max}

0.04\,

Inspect acquired record

t/t_{end}

0.00\,\%

All quantities are normalized. You can work the rail and the displacement profile directly, or set the sliders from the keyboard. The record stops when the motion leaves the model's valid range or the step limit is reached; the motion itself is never clipped.

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.

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 misconceptionA linear mode returning to the same displacement is the same as nonlinear recurrence.

Better mental modelIn 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 misconceptionThe sum of projected mode energies must be exactly constant.

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

Entropy does not prove thermalization

Typical misconceptionOnce energy spreads over many modes, it stays equally shared forever.

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

Run the experiment

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.
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.
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.
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.
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.
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.

Core mathematical model

Polynomial bond potential

Fixed-end chain dynamics

Linear harmonic projection

Full energy and the nonlinear remainder

Normalized harmonic modal entropy

Common difficulties

Staying put is not coming back

Harmonic energy is not the Hamiltonian

Entropy does not prove thermalization

Run the experiment

See the forces on one bead in 3D

Check the energy and the wall impulse

Run the linear chain first

Record a nonlinear return

Change the mixing strength

Check the numerics