Fractional diffusion equation on fractals: Self-similar stationary solutions in a force field derived from a logarithmic potential

Abstract

The fractional diffusion equation of Giona and Roman is extended for diffusion processes in force fields. For Euclidean diffusion self-similar stationary concentration profiles may exist provided that the force fields derive from logarithmic potentials. The physically significant potentials obey a non-linear differential equation which can be integrated analytically. By using these potentials a fractional Euclidean diffusion equation is derived. By using the heuristic method of Giona and Roman this equation is extended for diffusion on fractals. The self-similar stationary concentration profile is computed in the fractal case and a relationship among the different scaling exponents is derived. If the stationary concentration profile corresponds to thermodynamic equilibrium then a generalized Einstein equation may be derived: it establishes a connection between the fractal diffusion coefficient and the generalized mobility which depends on the anomalous diffusion exponent but is independent of the fractal dimension of the system. The fluctuations of the concentration profile for finite systems of non-interacting particles are analysed by using the theory of random point processes. A statistical correlation of the positions of different particles exists which is due to the finite number of particles; it vanishes as the total number of particles tends to infinity. In this context a fractal analogue of the thermodynamic limit is introduced.