When all particles orbit at the same frequency, relaxation doesn't speed up. It slows down dramatically.
One-dimensional self-gravitating systems are among the few long-range interacting systems that admit genuine thermodynamic equilibrium. For generic potentials, where each orbit has a distinct frequency, the standard kinetic theories (Landau, Balescu-Lenard) predict a relaxation timescale proportional to N, the number of particles. More particles means more interactions but each interaction is weaker, and the two effects produce a clean linear scaling. Simulations confirm this.
But Watanabe and Tashiro (arXiv:2603.11238) show that harmonic potentials — where every particle orbits at exactly the same frequency — break this picture completely. The standard theories become ill-posed at frequency degeneracy. The resonance denominators that drive Balescu-Lenard theory diverge. The formalism doesn't just give the wrong answer; it gives no answer.
The numerical answer, found with an exact collision-driven integrator: relaxation scales as N². Not N. Squaring the timescale. A system of 10,000 particles that would relax in 10,000 dynamical times under a generic potential takes 100 million dynamical times in a harmonic one.
The mechanism: in non-degenerate systems, resonances between orbits of different frequencies drive energy exchange. Particles with slightly different frequencies gradually phase-mix, and the slight frequency mismatch is the lever through which the system relaxes. In a harmonic potential, that lever vanishes. All particles oscillate at the same frequency, so they never go out of phase. The mechanism that drives relaxation has no foothold.
Partial degeneracy interpolates: at low N, the system behaves quadratically (the degenerate orbits dominate), then transitions to linear at large N (when the non-degenerate orbits take over). The crossover N depends on the fraction of degenerate orbits.
The astrophysical implication: dark matter density cores — the flat-bottomed potentials observed in dwarf galaxies — have approximately harmonic central regions. Core relaxation may be vastly slower than models predict.
Watanabe and Tashiro, "Very long-term relaxation of harmonic 1D self-gravitating systems," arXiv:2603.11238 (2026).