  In the previous paper [Nenashev et al., arXiv:0912.3161] an analytical theory
confirmed by numerical simulations has been developed for the field-dependent
hopping diffusion coefficient D(F) in one-dimensional systems with Gaussian
disorder. The main result of that paper is the linear, non-analytic field
dependence of the diffusion coefficient at low electric fields. In the current
paper, an analytical theory is developed for the field-dependent diffusion
coefficient in three- and two-dimensional Gaussian disordered systems in the
hopping transport regime. The theory predicts a smooth parabolic field
dependence for the diffusion coefficient at low fields. The result is supported
by Monte Carlo computer simulations. In spite of the smooth field dependences
for the mobility and for the longitudinal diffusivity, the traditional Einstein
form of the relation between these transport coefficients is shown to be
violated even at very low electric fields.
