Critical scaling and supercritical coarsening in Active Model B+

We study critical dynamics and phase-ordering kinetics in Active Model B (AMB) and its minimal extension, Active Model B$+$ (AMB$+$), using deterministic simulations in two dimensions. At criticality $r_c=0$, both models display identical mean-field scaling despite nonequilibrium currents, with order-parameter decay with time as $m(t)\sim t^{-\alpha }$, with $\alpha =\frac{1}{4}$, and dynamical exponent being $z=4$. A generalized equal-area construction yields the binodal densities and phase diagram of AMB$+$. For supercritical quenches, domain size grows as $L(t)\sim t^{1\mathrm{/}3}(1+c\mathrm{/}\ln t)$, revealing logarithmic corrections to the classic $t^{1\mathrm{/}3}$ growth-law; moreover it is consistent with the functional renormalization group predictions for marginal activity in $d=2$. While the logarithmic corrections are quite prominent in AMB, in AMB$+$ they are suppressed as the active current acts against the formation of macro-clusters; the growth is eventually arrested when a long-lived microphase-separated state appears.