  we study, in this paper, the non-Gaussian features of the mass density field
of neutral hydrogen fluid and the Ly-alpha transmitted flux of QSO absorption
spectrum from the point-of-view of self-similar log-Poisson hierarchy. It has
been shown recently that, in the scale range from the onset of nonlinear
evolution to dissipation, the velocity and mass density fields of cosmic baryon
fluid are extremely well described by the She-Leveque's scaling formula, which
is due to the log-Poisson hierarchical cascade. Since the mass density ratio
between ionized hydrogen to total hydrogen is not uniform in space, the mass
density field of neutral hydrogen component is not given by a similar mapping
of total baryon fluid. Nevertheless, we show, with hydrodynamic simulation
samples of the concordance $\Lambda$CDM universe, that the mass density field
of neutral hydrogen, is also well described by the log-Poisson hierarchy. We
then investigate the field of Ly$\alpha$ transmitted flux of QSO absorption
spectrum. Due to redshift distortion, Ly$\alpha$ transmitted flux fluctuations
are no longer to show all features of the log-Poisson hierarchy. However, some
non-Gaussian features predicted by the log-Poisson hierarchy are not affected
by the redshift distortion. We test these predictions with the high resolution
and high S/N data of quasars Ly$\alpha$ absorption spectra. All results given
by real data, including $\beta$-hierarchy, high order moments and scale-scale
correlation, are found to be well consistent with the log-Poisson hierarchy. We
compare the log-Poisson hierarchy with the popular log-normal model of the
Ly$\alpha$ transmitted flux. The later is found to yield too strong
non-Gaussianity at high orders, while the log-Poisson hierarchy is in agreement
with observed data.
