Dark matter, if it exists, accounts for five times as much as ordinary baryonic matter. To better understand the self-gravitating collisionless dark matter flow on different scales, a statistical theory involving kinematic and dynamic relations must be developed for different types of flow, e.g., incompressible, constant divergence, and irrotational flow. This is mathematically challenging because of the intrinsic complexity of dark matter flow and the lack of a self-closed description of flow velocity. This paper extends our previous work on second-order statistics Xu [Phys. Fluids 35, 077105 (2023)] to kinematic relations of any order for any type of flow. Dynamic relations were also developed to relate statistical measures of different orders. The results were validated by N-body simulations. On large scales, we found that (i) third-order velocity correlations can be related to density correlation or pairwise velocity; (ii) the pth-order velocity correlations follow proportional to a((p+2)/2) for odd p and proportional to a(p/2 )for even p, where a is the scale factor; (iii) the overdensity delta is proportional to density correlation on the same scale, <delta > proportional to <delta delta '>; (iv) velocity dispersion on a given scale r is proportional to the overdensity on the same scale. On small scales, (i) a self-closed velocity evolution is developed by decomposing the velocity into motion in haloes and motion of haloes; (ii) the evolution of vorticity and enstrophy are derived from the evolution of velocity; (iii) dynamic relations are derived to relate second- and third-order correlations; (iv) while the first moment of pairwise velocity follows <Delta u(L)> = -Har (H is the Hubble parameter), the third moment follows <(Delta u(L))(3)> proportional to epsilon(u)ar that can be directly compared with simulations and observations, where epsilon(u) approximate to 10(-7) m(2)/s(3) is the constant rate for energy cascade; (v) the pth order velocity correlations follow proportional to a((3p-5)/4 )for odd p and proportional to a(3p/4 )for even p. Finally, the combined kinematic and dynamic relations lead to exponential and one-fourth power-law velocity correlations on large and small scales, respectively. (c) 2024 Author(s).
