Nonthermal electron acceleration and transition probability density in turbulent wakefields driven by an intense...

Nonthermal power-law spectra with index -2 are ubiquitously observed in turbulent wakefield acceleration, yet their statistical origin remains unresolved. Using one-dimensional particle-in-cell simulations with particle tracking, we analyze the stochastic energy evolution of electrons accelerated by turbulent wakefields and construct the transition probability density (TPD) of energy jumps. We show that the particle population separates into three dynamical regimes. Low-energy particles exhibit a superposition of Brownian diffusion and Lévy-type jump process, moderate-energy particles are dominated by Lorentzian-type TPDs indicating a nonlocal Lévy-type jump process in momentum space, and high-energy particles undergo ballistic acceleration. The Lorentzian-type TPD in the nonthermal regime naturally leads to a fractional Fokker-Planck description with fractional index α=1, whose analytical solution yields a robust f(γ)∝γ^{-2} spectrum. Comparison with the κ distribution confirms consistency with strongly nonthermal plasmas while revealing fractional transport as the dynamical origin of the universal index -2 heavy tail.