Fractality of the hydrodynamic modes of diffusion

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.