  In this paper, we focus on the velocity and acceleration fluctuations and
distributions of cold dark matter particles that involve long-range
gravitational interactions. For comparison, in the kinetic theory of gases,
molecules undergo random elastic collisions involving a short-range
interaction, where only velocity fluctuations are relevant that follows a
Maxwellian-Boltzmann distribution. Long-range gravity requires a broad size of
haloes to be formed. Hierarchical structure formation proceeds through the
merging of smaller haloes to form larger haloes, which facilitates a continuous
energy cascade from small- to large-scale haloes at a constant rate of
$\varepsilon_u\approx -10^{-7}$m$^2$/s$^3$. The velocity fluctuation involves a
critical velocity $u_c$, and the acceleration fluctuation involves a critical
acceleration $a_c$. The two critical quantities are related by the rate of the
energy cascade as $\varepsilon_{u}\approx -{a_c u_c/[2(3\pi)^2]}$, where the
factor $3\pi$ is from the angle of incidence during the merging. With a typical
velocity dispersion of $u_c$ on the order of 500km/s at $z=0$, the critical
acceleration is found to be $a_{c0}\equiv a_c(z=0) \approx 10^{-10}$m/s$^2$ at
$z=0$ with a redshift dependence $a_{c} \propto (1+z)^{3/4}$ that is greater at
higher redshift. This suggests that the critical acceleration $a_c$ might
explain the universal acceleration $a_0 \approx 10^{-10}$m/s$^2$ in the
empirical Tully-Fisher relation or modified Newtonian dynamics (MOND). The
redshift evolution of $a_0 \propto (1+z)^{3/4}$ is also proposed and validated
by this work, Magneticum, and EAGLE simulations. Future work on the
high-redshift Tully-Fisher relation will provide more insight. A notable
coincidence of dark energy density $\rho_{\Lambda} \propto {a_{c0}^{2}/G}$
might suggest an entropic origin of dark energy from the fluctuations of dark
matter accelerations.
