Field Equation of Correlation Function of Mass Density Fluctuation for
  Self-Gravitating Systems
  We study the mass density distribution of Newtonian self-gravitating systems.
Modeling the system as a fluid in hydrostatical equilibrium, we obtain from
first principle the field equation and its solution of correlation function
$\xi(r)$ of the mass density fluctuation itself. We apply thid to studies of
the large-scale structure of the Universe within a small redshift range.
  The equation tells that $\xi(r)$ depends on the point mass $m$ and the Jeans
wavelength scale $\lambda_{0}$, which are different for galaxies and clusters.
It explains several longstanding, prominent features of the observed clustering
: that the profile of $\xi_{cc}(r)$ of clusters is similar to $\xi_{gg}(r)$ of
galaxies but with a higher amplitude and a longer correlation length, and that
the correlation length increases with the mean separation between clusters as a
universal scaling $r_0\simeq 0.4d$. Our solution $\xi(r)$ also yields the
observed power-law correlation function of galaxies $\xi_{gg}(r)\simeq
(r_0/r)^{1.7}$ valid only in a range $1<r<10 h^{-1}$Mpc. At larger scales the
solution $\xi(r)$ breaks below the power law and goes to zero around $\sim
50h^{-1}$Mpc, just as the observational data have demonstrated. With a set of
fixed model parameters, the solutions $\xi_{gg}(r)$ for galaxies, the
corresponding power spectrum, and $\xi_{cc}(r)$ for clusters, simultaneously,
agree with the observational data from the major surveys of galaxies, and of
clusters.
