Collisionless dark-matter Vlasov equation

For comoving position $\mathbf{x}$, canonical momentum $\mathbf{p}=am{\mathbf{x}}^{\prime }$, and Newtonian potential $\varphi$, the dark-matter phase-space density obeys

The velocity-dispersion tensor is the second central moment of a phase-space distribution,It contributes the stress-divergence term ${\rho }^{-1}{\partial }_j(\rho {\sigma }^{ij})$ to the Euler equation and vanishes in the ideal single-stream limit.