{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/two-thirds-law-for-pairwise-velocity-and","title":"On the statistical theory of self-gravitating collisionless dark matter flow: Scale and redshift variation of velocity and density distributions","arxiv_id":"2202.06515","date":"2022-02-14","proceeding":null,"authors":["Zhijie Xu"],"abstract":"This paper studies the scale and redshift variation of density and velocity distributions in self-gravitating collisionless dark matter flow by a halo-based non-projection approach. All particles are divided into halo and out-of-halo particles for redshift variation of distributions. 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) 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 at a constant rate $\\varepsilon_u$; iii) On small scales, the even order moments of pairwise velocity $\\Delta u_L$ follow a two-thirds law $\\propto{(-\\varepsilon_ur)}^{2/3}$, while the odd order moments follow a linear scaling $\\langle(\\Delta u_L)^{2n+1}\\rangle=(2n+1)\\langle(\\Delta u_L)^{2n}\\rangle\\langle\\Delta u_L\\rangle\\propto{r}$; 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 $\\Sigma u_L$=$u^{'}_L$+$u_L$. Fully developed velocity fields are never Gaussian on any scale, despite that they can initially be Gaussian; v) On small scales, $u_L$ and $\\Sigma u_L$ can be modeled by a $X$ distribution to maximize the system entropy; vi) On large scales, $\\Delta u_L$ and $\\Sigma u_L$ can be modeled by a logistic or a $X$ distribution; vii) the redshift variation of the velocity distributions follows the evolution of the $X$ distribution involving a shape parameter $\\alpha(z)$ decreasing with time.","url_abs":"https://arxiv.org/abs/2202.06515v3","url_pdf":"https://arxiv.org/pdf/2202.06515v3.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"two-thirds-law-for-pairwise-velocity-and","repo_url":"https://github.com/ZhijieXu2022/dark_matter_flow_dataset","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}