Generalized diffusion equation

Modern analyses of diffusion processes have proposed nonlinear versions of the Fokker–Planck equation to account for nonclassical diffusion. These nonlinear equations are usually constructed on a phenomenological basis. Here, we introduce a nonlinear transformation by defining the q-generating function which, when applied to the intermediate scattering function of classical statistical mechanics, yields, in a mathematically systematic derivation, a generalized form of the advection–diffusion equation in Fourier space. Its solutions are discussed and suggest that the q-generating function approach should be a useful method to generalize classical diffusive transport formulations.