Fractional Dynamics in Galactic Nuclei: Non-Local Transport, Transient Phenomena and the Nullification of the Schwarzschild Barrier

TL;DR

This work argues that Resonant Relaxation near supermassive black holes is a non-local, Lévy-flight process with infinite variance, invalidating standard local Fokker-Planck descriptions. By deriving and applying space-fractional FFPEs from the CTRW framework, it explains immediate transient TDE refilling in post-starburst galaxies and enables non-local barrier jumping across the Schwarzschild Barrier, potentially elevating EMRI rates. Analytic results and proof-of-concept N-body simulations support the non-local transport picture, though rigorous confirmation requires larger-N studies and direct measurements of jump statistics. The fractional approach provides a mathematically consistent, physically motivated framework that may reconcile observed high TDE rates and EMRI production with the true stochastic nature of RR in galactic nuclei, at the cost of adopting non-local numerical methods.

Abstract

We investigate the application of fractional calculus to model stellar dynamics, focusing on Resonant Relaxation (RR) near a supermassive black hole (SMBH). Standard theories use the local Fokker-Planck (FP) equation, restricted to Gaussian processes under the Central Limit Theorem (CLT). We argue this is inadequate for RR. We demonstrate that gravitational interactions inherently produce infinite variance in stochastic torques, violating the CLT. Consequently, RR is governed by the Generalized Central Limit Theorem (GCLT) and constitutes a superdiffusive Lévy flight. We apply the space-fractional Fokker-Planck equation (FFPE), utilizing non-local operators, to explore resolutions to observational discrepancies. In transient regimes, the FFPE predicts immediate, linear flux ($Γ(t) \propto t$), consistent with high Tidal Disruption Event (TDE) rates in post-starburst galaxies, whereas local FP models predict significant exponential delay. Furthermore, we demonstrate analytically that non-local integral operators permit ``barrier jumping,'' bypassing bottlenecks like the Schwarzschild Barrier (SB), which local models interpret as severely suppressing Extreme Mass-Ratio Inspiral (EMRI) rates. We present proof-of-concept $N$-body simulations that confirm non-local RR transport, although the resolution must be improved to rule out enhanced Two-Body Relaxation in the small-N setup. The fractional framework offers a compelling alternative description for non-local transport, potentially resolving TDE and EMRI rate questions.

Figures (4)

• Figure 1: Comparison of the fundamental solutions (propagators) for local diffusion (Gaussian, $q=2$) and superdiffusion (Lévy stable, $q=1.5$), assuming $D=1, t=1$. Left panel (linear scale) shows the overall shape. Right panel (log-log scale) highlights the behavior of the tails. The Gaussian propagator exhibits rapid exponential decay (appearing parabolic in log-log). The Lévy propagator possesses a heavy power-law tail (linear shape in log-log, following the asymptotic behavior $(\Delta J)^{-2.5}$). This heavy tail significantly enhances the probability of non-local jumps (Lévy flights); for instance, at $|\Delta J|=5$ ($\approx 3.5\sigma$), the Lévy probability density is approximately 13 times higher than the Gaussian probability density.
• Figure 2: Evolution of the transient flux into the loss cone following a perturbation, normalized by the diffusion timescale $T_{\rm diff}$. The log-log scale highlights the behavior at early times. The local model (blue line) exhibits exponential suppression ($\Gamma(t) \propto \exp(-C/t)$), leading to a significant delay. It reaches 1% of the steady-state flux only at $t \approx 0.17\, T_{\rm diff}$. The fractional model (red dashed line) exhibits immediate linear growth ($\Gamma(t) \propto t$) due to non-local jumps, reaching the 1% flux level approximately 17 times faster at $t = 0.01\, T_{\rm diff}$.
• Figure 3: Impact of the Schwarzschild Barrier ($J_{\text{SB}}$) on the steady-state angular momentum flux $\mathcal{F}(J)$. The local diffusion coefficient $D(J)$ (gray dashed line) drops by 95% (a factor of 20) for $J < J_{\text{SB}}$ due to Adiabatic Invariance. In the local model (blue solid line), the flux is limited by the integrated resistance ($\int 1/D(J) dJ$), resulting in an 88% reduction of the global flux ($\mathcal{F}_{\rm local} \approx 0.12$). The fractional model (red dash-dotted line) shows that the flux is maintained ($\mathcal{F}_{\rm frac} = 1.0$) by non-local "barrier jumping" from $J > J_{\text{SB}}$. The fractional flux is $\sim 8.3$ times higher than the local flux.
• Figure 4: Time-averaged presence density of stellar-mass black holes in the $a$ vs. $1-e$ phase space, generated from 300 $N$-body simulations of 400 particles each, focusing on the critical milliparsec scales. Left panel: The purely Newtonian case (no PN terms). Particles are driven to high eccentricities (low $1-e$) by relaxation processes, filling the phase space. Right panel: The relativistic (PN) case without spin. The solid blue line marks the theoretical location of the Schwarzschild Barrier ($\tau_{\rm SS} \approx \tau_{\rm coh}$), and the dashed blue line marks the gravitational wave capture limit ($\tau_{\rm GW} < \tau_{\rm rlx}$). The particle density is high on both sides of the SB (solid line), with no visible depletion or barrier to transport in this TBR-dominated regime.