A Sling for a Particle: How Delay Helps It Escape the Pit ✛

Original: "Time-delayed feedback turns Arrhenius escape logarithmic"
arXiv:2608.30624v1 · 2026-08-31 · CC BY 4.0 · 1 min
A delayed force swings a particle and flings it out of the pit faster than ordinary thermal escape.

A pea sits in a hollow. To roll out, it needs a strong kick, and such kicks are rare, especially if the hollow is deep. But if you shake the hollow in a special way, with a delay, so that the force responds to the pea's position not immediately but a moment later, the pea starts to swing more and more strongly and eventually flies out. This is like a sling: the hand whirls the stone, building up energy, and then lets go.

A delay means that the restoring force is computed from the particle's position not now, but a fraction of a second earlier. This happens in optical traps where a computer controls the laser.

At first, the particle sits at the bottom of the pit, and thermal noise nudges it slightly. If the delay is small, the pit dampens these nudges, and the particle stays put. But when the delay exceeds a threshold, the pit becomes unstable. Now even a tiny random fluctuation grows on its own, like a snowball rolling downhill. The particle swings more and more, and after some time it is thrown over the barrier.

The key finding is that the escape time now depends on the barrier height not exponentially but logarithmically, meaning it grows very slowly. You can tune the delay so that the particle escapes faster than if there were no pit at all. This could help speed up chemical reactions or measure trap properties.

The optimal delay allows the particle to leave the well faster than if the well were absent altogether: free diffusion to the boundary takes longer. This resembles a "slingshot" effect: the delayed force rocks the particle and flings it out, using the energy of the trap itself.

\frac{\langle T \rangle}{\tau_{\text{relax}}} = A(s) \ln\left(\frac{\Delta U}{D}\right) + B(s)
The escape time depends logarithmically on the ratio of barrier to noise; the slope A(s) is determined only by the delay, while the constant B(s) reflects the potential shape.
A(s) = \frac{1}{2 \operatorname{Re} z}
The slope is inversely proportional to twice the real part of the leading root of the characteristic equation, which sets the exponential growth rate of the instability.
\mu T = \ln\left(\frac{L}{R}\right) - \ln\left|\cos(\omega T - \phi)\right|
The escape time is the sum of the envelope crossing time (logarithm of the ratio of boundary to seed amplitude) and a correction for oscillations.
Concepts
lead time delayed oscillator theory second law of thermodynamics arrow of time temperature difference Bernoulli's principle carnot limit efficiency Bjerknes feedback positive feedback loop limit control theory law of conservation of energy signal delay feedback loop relaxation relaxation time