Abstract
<title>Abstract</title> <p>The cascade phenomenon is fundamental to turbulence, a typical non-equilibrium system. In turbulence, energy is cascaded from large to small scales, where it is dissipated by fluid viscosity. Using Illustris and Virgo simulations, this paper presents the cascade in a different non-equilibrium system, the self-gravitating collisionless dark matter flow. During gravitational collapse and hierarchical structure formation, the cosmic energy of dark matter decreases continuously over time, as if it were "dissipated" by the expanding background. This is facilitated by the energy cascade from large to small haloes via halo merging and from large to small scales in individual haloes via particle migration in fluctuating non-uniform gravitational potential, such that the cosmic energy is cascaded to and "dissipated" on small scales. We identify an inverse mass cascade across haloes of different sizes, which leads to a random walk of haloes in halo mass space. The halo mass function (double-$\lambda$) is naturally given by the corresponding Fokker-Planck equation for halo random walk. Similarly, the random walk of particles in haloes leads to the distribution of particles. The halo density profile (double-$\gamma$) can be analytically derived from the corresponding Fokker-Planck equation for particle random walk. Universal scaling laws were identified that exhibit small-scale permanence for the halo density $\rho_h\propto r^{-4/3}$. The different inner density slopes of simulated haloes can be explained by the nonzero net mass and energy flux in individual haloes. The mass and energy cascade in dark matter flow establishes a statistically steady state to continuously release the system energy and maximize the system entropy. The key feature of this statistically steady state is scale-independent rates of cascade such that the statistical structures of the haloes are self-similar and scale-free, and there is no net accumulation of mass and energy on any intermediate scales.</p>