M045 · Barriers and critical slowing

Double-Well, Barrier & Separatrix Lab

How does a barrier between two wells change the way a particle moves? It depends on whether the energy clears the barrier. Below the top, the particle rocks back and forth in one well; above it, the particle runs between both wells. Close to the dividing energy it crawls past the top of the barrier, so the period becomes very long. Damping can drain just enough energy to trap a particle that used to cross.

Set the barrier, the well spacing, the tilt and the damping. Compare a particle trapped in one well with one that crosses between wells, approach the separatrix from below and from above, and let damping capture a fast particle on one side. Measure crossings and periods, and check the periods against an independent calculation from the energy integral.

Acquired time $0\,s$

Position $0\,m$

Velocity $0\,\mathrm{m\,s^{-1}}$

Conservative force $0\,N$

Potential energy $0\,J$

Kinetic energy $0\,J$

Mechanical energy $0\,J$

Dissipated energy $0\,J$

Current energy balance defect $0\,J$

Full-record maximum balance defect $0\,J$

Half-step displacement difference · first ten seconds $0\,m$

Actual integration step $0\,ms$

Nearest release-side minimum $0\,m$

Local undamped angular frequency $0\,\mathrm{rad\,s^{-1}}$

Allowed regions at release energy $1$

Release-region left boundary $0\,m$

Release-region right boundary $0\,m$

Conservative quadrature period $0\,s$

Last acquired full turn interval $\text{Await turns}$

Period quadrature resolution difference $0\,s$

Measured minus conservative period $0\,s$

Acquisition status $\text{Complete}$

Fine inspection time $0\,\mathrm{s}$

Signed viscous force $0\,\mathrm{N}$

Signed net force along guide $0\,\mathrm{N}$

Acquired numerical reversals $0$

Amber ruler span $0\,\mathrm{m}$

Amber tick interval $0\,\mathrm{m}$

Barrier above nearest release-side minimum $0\,J$

Saddle position $0\,m$

Current energy above saddle $0\,J$

Acquired numerical saddle crossings $0$

First acquired numerical crossing $\text{Not acquired}$

Release energy classification $\text{Above barrier}$

Acquired dissipative enclosure $\text{Unconfirmed}$

Set the release, then record the motion

How close can you approach the barrier before the motion changes its topology?

Particle vectorsBlue is the conservative force, violet the viscous force and teal the velocity. The parallel display lanes line up with the particle position, and the force and speed scales stay fixed during a trial.Conservative force probesSmall blue arrows sample the given force field along the horizontal guide. They are virtual probes, drawn at the same force scale as the arrow on the particle.Energy-accessible intervalsGrey uses the release energy and teal the current numerical energy; amber marks the edges. The bands also cover separate wells that are not connected. They are not physical walls, and they do not promise that the particle will cross over.Metric rulerThe position, the equilibrium markers and the ruler share one fixed length scale. Particle size and hardware thickness are schematic.Particle mass

m

1.00\,kg

Symmetric barrier scale

B

1.00\,J

Symmetric half separation

a

1.00\,m

Tilt relative to well-loss threshold

f/f_{\mathrm{crit}}

0.00\,

Viscous damping

c

0.00\,kg\,s^{-1}

Release position

x_0

-1.00\,m

Launch kinetic energy / barrier scale

K_0/B

1.08\,

Launch direction

\operatorname{sgn}v_0

1.00\,

Observation window

t_{\mathrm{obs}}

40.00\,s

Requested step

h

5.00\,ms

Acquired record

t/t_{\mathrm{obs}}

0.00\,\%

At the critical energy, the undamped motion has no finite period. Close to the saddle, the computed motion is very sensitive to energy error, so check the energy and step-halving errors before treating a crossing as physical. A particle released exactly at the saddle stays at rest unless something nudges it. The period chart always shows undamped motion measured from the nearest minimum on the release side; damping changes the measured period. Capture is reported only once the recorded energy falls below the saddle by more than the numerical error.

The model is an ideal point particle moving along a horizontal guide in a given polynomial potential, with all quantities in SI units. The 3D force probes and the bands that mark where the particle can go are drawn guides; the Lab does not simulate rolling, a physical barrier or any actuator. The vertical curve is a graph of energy, not a real track. Gray contours are drawn at the release energy, and the recorded motion is teal. Playback recomputes the motion from saved checkpoints. The step-halving comparison covers at most the first ten seconds; the energy check covers every integration step.

Physics tutorial

A separatrix separates confined and crossing orbits

BackgroundA quartic double well has two stable minima with a saddle, the top of the barrier, between them. Without damping, a particle with less energy than the saddle stays on one side; with more, it can reach both wells. At exactly the saddle energy, a moving particle creeps ever closer to the saddle and never completes a cycle. That orbit is the separatrix.

Why it mattersThe energy and phase-space picture is standard classical mechanics for motion along one coordinate. The barrier height and the largest allowed tilt are worked out directly from the quartic potential shown. With viscous damping, the model counts every bit of kinetic energy the damping removes.

Start with the essentials

Focus question: Why does crossing a little above the barrier take so much longer?
One-sentence intuition: As the energy approaches the top of the barrier, the particle lingers longer and longer near the saddle. Damping can drain enough energy that an orbit that used to cross gets trapped on one side.

Core mathematical model

Tilted double well

U(x)=B\left[\left(\frac{x}{a}\right)^2-1\right]^2+fx

Without tilt, the barrier scale is the height of the barrier above either minimum. With tilt, the barrier shown for the release side is recomputed from where the minima and the saddle actually are.

When both minima survive

|f|<f_{\mathrm{crit}}=\frac{8B}{3\sqrt3\,a},\qquad U\prime(x_*)=0

The tilt is kept below the saddle-node threshold, so the two minima and the saddle between them always stay distinct.

Energy balance with viscous damping

m\ddot x=-U\prime(x)-c\dot x,\qquad H(t)+Q(t)=H(0),\qquad Q(t)=\int_0^t c\dot x^2\,dt

Each damping substep shrinks the velocity exponentially and adds the exact kinetic energy it removes to the heat. Errors from the conservative part of the step still show up as a mismatch in the balance.

Energy above the barrier and the undamped period

\Delta H=H-U(x_s),\qquad T(H)=\sqrt{2m}\int_{x_-}^{x_+}\frac{dx}{\sqrt{H-U(x)}}

The orbit at exactly the critical energy has no finite period, so the plot leaves a gap between the branches below and above the barrier.

Motion near the saddle

U(x)\simeq U(x_s)-\frac{m\lambda_s^2}{2}(x-x_s)^2,\qquad \lambda_s^2=-\frac{U\prime\prime(x_s)}{m}

The period grows logarithmically near the saddle, which magnifies small energy errors. A particle released exactly at the saddle stays at rest unless something nudges it.

Common difficulties

Crossings at the critical energy come from the numerics

Typical misconceptionEvery crossing recorded at the critical energy really happens.

Better mental modelThe exact separatrix does not cross the saddle in any finite time. Error from a finite time step can tip it into crossing or turning back, so compare the energy error with the half-step difference.

With tilt, the barrier scale is not the real barrier

Typical misconceptionThe symmetric barrier setting always equals the barrier height on the launch side.

Better mental modelTilt shifts the energies of both minima and of the saddle. The readout uses the actual difference between them.

Damping changes the orbit

Typical misconceptionAfter the particle loses energy, the measured period still equals the undamped reference period.

Better mental modelAs energy drains away, the orbit keeps changing. Capture is reported only once the recorded energy sits below the saddle by more than the numerical uncertainty.

Run the experiment

01
Compare trapped and crossing motion
Acquire full records for Below the barrier and Across both wells, then look at the phase portrait and the recorded saddle crossings.
What to observe: Without damping, only motion above the barrier reaches both wells.
02
Approach the separatrix
Compare Approach from below with Approach from above, then select Critical energy.
What to observe: Both periods get longer. At the critical energy the reference shows no finite period, and the result becomes much more sensitive to numerical error.
03
Let damping capture the particle
Acquire a full record with Dissipative capture and look at heat plus mechanical energy.
What to observe: Heat plus mechanical energy stays equal to the starting energy, and the orbit ends up trapped on one side.
04
Tilt without erasing a well
Change the tilt and drag the release point in phase space. Compare the actual saddle energy with the symmetric barrier scale.
What to observe: The saddle shifts and the barrier is higher from one side than from the other, yet both minima remain.