  Small-scale challenges to $\Lambda$CDM cosmology require a deeper
understanding of dark matter physics.This paper aims to develop maximum entropy
distributions for dark matter particle velocity (denoted by $X$), speed
(denoted by $Z$), and energy (denoted by $E$) that are especially relevant on
small scales where system approaches full virialization. For systems involving
long-range interactions, a spectrum of halos of different sizes is required to
form to maximize system entropy. While velocity in halos can be Gaussian, the
velocity distribution throughout entire system, involving all halos of
different sizes, is non-Gaussian. With the virial theorem for mechanical
equilibrium, we applied maximum entropy principle to the statistical
equilibrium of entire system, such that maximum entropy distribution of
velocity (the $X$ distribution) could be analytically derived. The halo mass
function was not required in this formulation, but it did indeed result from
the maximum entropy. The predicted $X$ distribution involves a shape parameter
$\alpha$ and a velocity scale, $v_0$. The shape parameter $\alpha$ reflects the
nature of force ($\alpha\rightarrow0$ for long-range force or
$\alpha\rightarrow\infty$ for short-range force). Therefore, the distribution
approaches Laplacian with $\alpha\rightarrow0$ and Gaussian with
$\alpha\rightarrow\infty$. For an intermediate value of $\alpha$, the
distribution naturally exhibits a Gaussian core for $v\ll v_0$ and exponential
wings for $v\gg v_0$, as confirmed by N-body simulations. From this
distribution, the mean particle energy of all dark matter particles with a
given speed, $v$, follows a parabolic scaling for low speeds ($\propto v^2$ for
$v\ll v_0$ in halo core region, i.e., "Newtonian") and a linear scaling for
high speeds ($\propto v$ for $v\gg v_0$ in halo outskirt, i.e., exhibiting
"non-Newtonian" behavior in MOND due to long-range gravity).

Cosmology and Nongalactic Astrophysics;Astrophysics of Galaxies;Fluid Dynamics