Power-law index ﹩\alpha ={\rm{\Delta }}(\mathrm{log}t)/{\rm{\Delta }}(\mathrm{log}{E}_{p})﹩ of the acceleration time as a function of the particle kinetic energy normalized by the proton rest mass energy. The minimum in the curves, α ∼ 0.1, indicates the nearly exponential regime of particle acceleration. Depicted are the models with a steady-state turbulent background of Figure 5, namely the σ ∼ 1 jet at t = 50 L/c run by MGK+21 (black line) and the σ ∼ 1 jet at t = 45 L/c run in this work (blue line). Also shown is α for the nearly exponential regime (between 30L/c < t < 50L/c) of the σ ∼ 1 MHD-PIC model of the top of Figure 6, where particles evolved with the background plasma (red curve).