When \(b_0 = 0\) (isotropic Maxwellian), substituting \(Q(f,f) = 0\) into the Vlasov equation and combining with Gauss’s law yields: \(T_\infty \Delta (\log \rho ) = \rho - \rho _{\mathrm{ion}}\).

By the maximum principle on the compact torus, the density \(\rho \) is constant and \(E(x) = 0\) for all \(x\).