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

Abstract

One-dimensional self-gravitating systems admit genuine thermodynamical equilibria. For systems with strictly monotonic orbital frequency profile, the Landau and Balescu-Lenard theories predict a relaxation time scaling linearly with the number of particles, $N$, in agreement with simulations. Yet, these theories become ill-posed for degenerate frequency profiles, as is the case in the harmonic potential, where all particles share the exact same mean orbital frequency. Using an exact collision-driven 1D integrator, we investigate numerically 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 low $N$, but transition to the linear, non-degenerate behaviour for larger $N$. The larger the fraction of degenerate orbits, the larger the value of $N$ at which this transition of dynamical regime occurs. Finally, we explore the dynamics of fully non-degenerate systems, albeit with finite radial support: we confirm that their relaxation time scales linearly with $N$, though with a substantially larger prefactor than in non-compact systems. Astrophysically, this investigation should offer some new clues on the dynamics of density cores, as in the center of dwarf galaxies.

Figures (8)

• Figure 1: Time evolution of the KS distance (equation \ref{['eq:def_ks']}) for Plummer, Compact, and Harmonic, with ${ N \!=\! 50 }$ and averaged over ${ N_{\mathrm{r}} \!=\! 2\, 000 }$ realizations. The bottom dashed line is the characteristic noise level, ${ 1 / \sqrt{N_{\mathrm{r}} \!\times\! N} }$. The threshold associated with ${ D_{0} \!=\! 0.015 }$ is given by the dot-dashed line. It is used to estimate the relaxation time, $t_{\mathrm{rel}}$ (equation \ref{['eq:introduction_D0']}). The relaxation times vary greatly between the different potentials, with Harmonic being the slowest of them all.
• Figure 2: Dependence of the relaxation time, $t_{\mathrm{rel}}$, with the number of particles, $N$, for Plummer, Compact and Harmonic (Table \ref{['tab:Potentials']}). Both Plummer and Compact relax on a timescale of order ${ t_{\mathrm{rel}} \!\propto\! N t_{\mathrm{dyn}} }$ (solid lines), for $N$ large enough. On the contrary, Harmonic relaxes on a timescale of order ${ t_{\mathrm{rel}} \!\propto\! N^2 t_{\mathrm{dyn}} }$ (dashed line). For small $N$, measurements are polluted by small-$N$ effects.
• Figure 3: Same as Figure \ref{['fig:scaling_plummer_harmonic']} for Anharmonic and various values of $\epsilon$, the level of dynamical degeneracy. As one increases $\epsilon$, the fraction of non-degenerate orbits, the clusters relax faster. Partially degenerate clusters display two relaxation regimes: a relaxation time consistent with ${ N^2 t_{\mathrm{dyn}} }$ (resp. ${ N t_{\mathrm{dyn}} }$) for small (resp. large) $N$. Increasing $\epsilon$ lowers the value of $N$ for which the transition occurs.
• Figure 4: Relative error in the total energy (in red) and the total momentum (in blue) during the integration of one Harmonic cluster with ${ N \!=\! 50 }$, as considered in Figure \ref{['fig:ks']}. The errors grow linearly in time (as highlighted by the dashed black line). At ${ t \!=\! 2 \!\times\! 10^7 t_{\mathrm{dyn}} }$, the integrator went through ${ \!\sim\! 6.5 \!\times\! 10^{9} }$ collisions for that run.
• Figure 5: Thermalization of the (ensemble-averaged) Plummer (top), Compact (middle) and Harmonic (bottom) clusters used to produce Figure \ref{['fig:ks']}. Each cluster relaxes towards the ${ N \!=\! 50 }$ thermodynamical distribution, (see Appendix \ref{['app:Thermodynamical_distribution']}). The relaxation timescale greatly differs between the considered potentials. For Harmonic, the sharp edges of the initial density profile are quickly smoothed out.