Secular dynamics of long-range interacting particles on a sphere in the axisymmetric limit

We investigate the secular dynamics of long-range interacting particles moving on a sphere, in the limit of an axisymmetric mean-field potential. We show that this system can be described by the general kinetic equation, the inhomogeneous Balescu-Lenard equation. We use this approach to compute long-term diffusion coefficients, that are compared with direct simulations. Finally, we show how the scaling of the system's relaxation rate with the number of particles fundamentally depends on the underlying frequency profile. This clarifies why systems with a monotonic profile undergo a kinetic blocking and cannot relax as a whole under $1/N$ resonant effects. Because of its general form, this framework can describe the dynamics of globally coupled classical Heisenberg spins, long-range couplings in liquid crystals, or the orbital inclination evolution of stars in nearly Keplerian systems.

Figure 1

Illustration of the second-order diffusion coefficient, $N{D}_2\left(u\right)$, for the waterbag DF from Eq. (16), as predicted by Eq. (14), in the absence (Landau) or presence (BL) of collective effects, and compared with $N$-body simulations (that naturally include collective effects). As a result of the presence of neutral modes [see Eq. (B6)], the BL diffusion coefficients locally diverge, as indicated by the vertical dashed lines.

Figure 2

Same as in Fig. 1, but for the DF from Eq. (17).

Figure 3

Dependence of the relaxation rate with the number of particles for the waterbag system from Eq. (16) that has a monotonic frequency profile yielding a kinetic blocking. Top panel: Time dependence of $m_4(N,t)$ for simulations with $N\in \{6,7,8,10,12,14,16,20,24,28,32\}\times {10}^2$ (from light to dark colors) averaged over 100 realizations (dots), and the associated fits (curves). The horizontal lines represent the threshold values ${\overline{m}}_4$ used to measure the crossing times. Bottom panel: Dependence of the crossing time ${\overline{t}}_N$ with the number of particles for different ${\overline{m}}_4$ (light to dark colors). Error bars for the crossing times were estimated by performing 200 bootstrap resamplings over the realizations available: colored dots represent the median value, and error bars the $10\%$ and $90\%$ confidence levels. Errors on the power-law fits were estimated by fitting each bootstrap resampling with a power law, while the plotted fits are the best fit for the median values.

Figure 4

Illustration of the DF from Eq. (19) and the associated nonmonotonic frequency profile $u\mapsto \mathrm{\Omega }\left(u\right)$. In the region of the DF's maximum, the resonance condition $\mathrm{\Omega }\left(u^{\prime }\right)=\mathrm{\Omega }\left(u\right)$ has two solutions, allowing for nonlocal resonant couplings.

Figure 5

Same as in Fig. 3, but for the DF from Eq. (19) that has a nonmonotonic frequency profile, preventing any kinetic blocking.

Figure 6

Illustration of the Nyquist contours $\omega \mapsto det[\mathbf{I}-\widehat{\mathbf{M}}(\omega ,\eta ={10}^{-8})]$ for the DF from Eq. (17) with different dispersions $\sigma$. None of these contours enclose the origin, indicating that these systems are linearly stable. The smaller the dispersion, the closer is the system to being unstable.