The statistics of velocity and density fields are crucial for cosmic structure formation and evolution. This paper extends our previous work on the two-point second-order statistics for the velocity field [Xu, Phys. Fluids 35, 077105 (2023)] to one-point probability distributions for both density and velocity fields. The scale and redshift variation of density and velocity distributions are studied by a halo-based non-projection approach. First, all particles are divided into halo and out-of-halo particles so that the redshift variation can be studied via generalized kurtosis of distributions for halo and out-of-halo particles, respectively. Second, without projecting particle fields onto a structured grid, the scale variation is analyzed by identifying all particle pairs on different scales r. We demonstrate that: (i) the Delaunay tessellation can be used to reconstruct the density field. The density correlation, spectrum, and dispersion functions were obtained, modeled, and compared with the N-body simulation; (ii) the velocity distributions are symmetric on both small and large scales and are non-symmetric with a negative skewness on intermediate scales due to the inverse energy cascade on small scales with a constant rate epsilon u; (iii) on small scales, the even-order moments of pairwise velocity Delta u(L) follow a two-thirds law proportional to(-epsilon ur)(2/3), while the odd-order moments follow a linear scaling <(Delta u(L))(2n+1)>=(2n+1)<(Delta u(L))(2n)><Delta u(L)>proportional to r; (iv) the scale variation of the velocity distributions was studied for longitudinal velocities u(L) or u(L)('), pairwise velocity (velocity difference) Delta u(L) = u(L)(') - u(L), and velocity sum Sigma uL = u(L)(') + u(L). Fully developed velocity fields are never Gaussian on any scale, despite that they can initially be Gaussian; (v) on small scales, uL and Sigma u(L) can be modeled by a X distribution to maximize the entropy of the system. The distribution of Delta u(L) can be different; (vi) on large scales, Delta u(L) and Sigma u(L) can be modeled by a logistic or a X distribution, while u(L) has a different distribution; and (vii) the redshift variation of the velocity distributions follows the evolution of the X distribution involving a shape parameter alpha(z) decreasing with time.
