Very long-term relaxation of harmonic 1D self-gravitating systems

One-dimensional self-gravitating systems admit genuine thermodynamical equilibria. For systems with a strictly monotonic orbital frequency profile, the Landau and Balescu–Lenard theories predict a relaxation time that scales linearly with the number of particles, N , in agreement with simulations. These theories become ill-posed for degenerate frequency profiles, as is the case in the harmonic potential, where all particles have the exact same mean orbital frequency. Using an exact collision-driven 1D integrator, we numerically investigated the self-consistent relaxation of 1D harmonic self-gravitating systems. We show that harmonic systems relax on a timescale that grows quadratically with N . We show that systems that are only partially degenerate display the same quadratic scaling for high N, but transition to the linear non-degenerate behaviour for higher N . The larger the fraction of degenerate orbits, the higher the value of N at which this transition of dynamical regime occurs. Finally, we explored the dynamics of fully non-degenerate systems, but with finite radial support: we confirm that their relaxation time scales linearly with N , but with a substantially larger prefactor than in non-compact systems. Astrophysically, this investigation offers some new clues on the dynamics of density cores, as in the centre of dwarf galaxies.