  The self-similar asymptotics for solutions to the drift-diffusion equation
with fractional dissipation, coupled to the Poisson equation, is analyzed in
the whole space. It is shown that in the subcritical and supercritical cases,
the solutions converge to the fractional heat kernel with algebraic rate. The
proof is based on the entropy method and leads to a decay rate in the
$L^1(\mathbb{R}^d)$ norm. The technique is applied to other semilinear
equations with fractional dissipation.
