Universal shape of diffusion fronts in inhomogeneous media from linear response requirements

We show that the asymptotic behaviour of the diffusion front in disordered or fractal media is related to the response of the random walk to an applied bias. In general this diffusion front takes the scaled form P(R, t) ∼ 1tvdF ƒ (Rtv), where ν is the diffusion exponent and ƒ(u) behaves as exp (−ub) for large u. On regular lattices the requirement that a velocity appears in presence of a bias yields b = 1/(1 − ν). On a fractal lattice, one has to take into account the tortuous nature of the medium which leads to b = dF/(d̂ − νdF), where d̂ is the spreading dimension of the fractal. This is a new derivation of the result recently obtained in ref. [1], which establishes its deep physical origin.