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

TL;DR

This paper applies fractional calculus to model stellar dynamics near supermassive black holes, revealing non-local transport mechanisms that explain observed transient phenomena and challenge traditional local theories.

Contribution

It introduces a fractional Fokker-Planck approach to account for non-Gaussian, superdiffusive stellar relaxation, providing new insights into galactic nucleus phenomena.

Findings

Transient flux predicted as linear in time, matching observed TDE rates

Non-local operators enable barrier jumping, bypassing the Schwarzschild Barrier

Proof-of-concept simulations confirm non-local RR transport mechanisms

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 L\'{e}vy 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 ($\Gamma(t) \propto t$), 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.