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

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We investigate the application of fractional calculus to model stellar dynamics, focusing on resonant relaxation (RR) near a supermassive black hole. 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 CLT and constitutes a superdiffusive Lévy flight. We apply the space-fractional FP equation (space-FFPE), utilizing nonlocal operators, to explore resolutions to observational discrepancies. In transient regimes, the FFPE predicts immediate linear flux (Γ( t ) ∝ t ), consistent with high tidal disruption event (TDE) rates in poststarburst galaxies, whereas local FP models predict significant exponential delay. Furthermore, we demonstrate analytically that nonlocal integral operators permit “barrier jumping,” bypassing bottlenecks like the Schwarzschild Barrier, which local models interpret as severely suppressing extreme-mass-ratio-inspiral (EMRI) rates. We present proof-of-concept N -body simulations that confirm nonlocal 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 nonlocal transport, potentially resolving TDE and EMRI rate questions.