Transport by normal diffusion can be decomposed into hydrodynamic modes whi
ch relax exponentially toward the equilibrium state. In chaotic systems wit
h 2 degrees of freedom, the fine scale structures of these modes are singul
ar and fractal, characterized by a Hausdorff dimension given in tens of Rue
lle's topological pressure. For long-wavelength modes, we relate the Hausdo
rff dimension to the diffusion coefficient and the Lyapunov exponent. This
relationship is tested numerically on two Lorentz gases, one with hard repu
lsive forces, the other with attractive, Yukawa forces. The agreement with
theory is excellent.