The statistical theory of dark matter flow and high order kinematic and dynamic relations for velocity correlation functions

The statistical theory of dark matter flow and high order kinematic and dynamic relations for velocity correlation functions Statistical theory for self-gravitating collisionless dark matter flow is not fully developed because of 1) intrinsic complexity involving constant divergence flow on small scale and irrotational flow on large scale; 2) lack of self-closed description for peculiar velocity; and 3) mathematically challenging. To better understand dark matter flow, kinematic and dynamic relations among different statistical measures of velocity must be developed for different types of flow. In this paper, a compact derivation is presented to formulate general kinematic relations of any order for incompressible, constant divergence, and irrotational flow. Results are validated by N-body simulation. Dynamic relations can only be determined from self-closed description of velocity evolution. On large scale, we found i) third order velocity correlation can be related to density correlation or pairwise velocity; ii) effective viscosity in adhesion model originates from velocity fluctuations; iii) negative viscosity is due to inverse energy cascade; iv) \(q\)th order velocity correlations follow \(\propto a^{(q+2)/2}\) for odd q and \(\propto a^{q/2}\) for even q; v) overdensity is proportional to density correlation on the same scale, \(\langle\delta\rangle\propto\langle\delta\delta'\rangle\); vi) (reduced) velocity dispersion is proportional to density correlation on the same scale. On small scale, self-closed description for velocity evolution is developed by decomposing velocity into motion in halo and motion of halos. Vorticity, enstrophy, and energy evolution can all be derived from self-closed equation for velocity. Dynamic relation is derived to relate second and third order correlations. Third moment of pairwise velocity is determined by energy cascade rate \(\epsilon_u\) or \(\langle(\Delta u_L)^3\rangle\propto\epsilon_uar\). Finally, combined kinematic and dynamic relations determines the exponential and one-fourth power law velocity correlations on large and small scales, respectively. Condensed slides for all applications "Cascade Theory for Turbulence, Dark Matter, and Supermassive Black Holes" Applications of cascade and statistical theory for dark matter and SMBH evolution: Dark matter particle mass ,size, and properties from energy cascade in dark matter flow: 1) arxiv 2) zenodo slides Origin of MOND acceleration & deep-MOND from acceleration fluctuation & energy cascade: 1) arxiv 2) zenodo slides The baryonic-to-halo mass relation from mass and energy cascade in dark matter flow: 1) arxiv 2) zenodo slides Universal scaling laws and density slope for dark matter halos from rotation curves: 1) arxiv 2) zenodo slides A unified theory for dark matter halo mass function and density profile: 1) arxiv 2) zenodo slides Energy cascade for distribution and evolution of supermassive black holes (SMBHs): 2) zenodo slides The two relevant datasets and accompanying presentation can be found at: Dark matter flow dataset Part I: Halo-based statistics from cosmological N-body simulation Dark matter flow dataset Part II: Correlation-based statistics from cosmological N-body simulation. A comparative study of Dark matter flow & hydrodynamic turbulence and its applications The same dataset also available on Github at: Github: dark_matter_flow_dataset and zenodo at: Dark matter flow dataset from cosmological N-body simulation. Cascade and statistical theory developed by these datasets: Inverse mass cascade in dark matter flow and effects on halo mass functions: 1) arxiv 2) zenodo slides Inverse mass cascade and effects on halo deformation, energy, size, and density profiles: 1) arxiv 2) zenodo slides Inverse energy cascade in dark matter flow and effects of halo shape: 1) arxiv 2) zenodo slides The mean flow, velocity dispersion, energy transfer and evolution of dark matter halos: 1) arxiv 2) zenodo slides Two-body collapse model and generalized stable clustering hypothesis for pairwise velocity 1) arxiv 2) zenodo slides Energy, momentum, spin parameter in dark matter flow and integral constants of motion: 1) arxiv 2) zenodo slides Maximum entropy distributions of velocity, speed, and energy in dark matter flow: 1) arxiv 2) zenodo slides Halo mass functions from maximum entropy distributions in dark matter flow: 1) arxiv 2) zenodo slides The statistical theory of dark matter flow for velocity, density, and potential fields: 1) arxiv 2) zenodo slides High order kinematic and dynamic relations for velocity correlations in dark matter flow: 1) arxiv 2) zenodo slides Evolution of density and velocity distributions and two-thirds law for pairwise velocity: 1) arxiv 2) zenodo slidesPaper and presentation slide