We consider diffusion processes with a spatially varying diffusivity giving
rise to anomalous diffusion. Such heterogeneous diffusion processes are
analysed for the cases of exponential, power-law, and logarithmic dependencies
of the diffusion coefficient on the particle position. Combining analytical
approaches with stochastic simulations, we show that the functional form of the
space-dependent diffusion coefficient and the initial conditions of the
diffusing particles are vital for their statistical and ergodic properties. In
all three cases a weak ergodicity breaking between the time and ensemble
averaged mean squared displacements is observed. We also demonstrate a
population splitting of the time averaged traces into fast and slow diffusers
for the case of exponential variation of the diffusivity as well as a particle
trapping in the case of the logarithmic diffusivity. Our analysis is
complemented by the quantitative study of the space coverage, the diffusive
spreading of the probability density, as well as the survival probability.Comment: 16 pages, 20 figures, RevTeX
