Turbulence in cascading: Origin of the variance and skewness of density function

Turbulent systems typically exhibit log-normal volume density probability density functions (PDFs) on the $s = \ln\rho$ scale, with their variance ($\sigma^2$) and skewness ($\tau$) empirically regulated by the Mach number ($M$). In this work, we explain both the $\sigma^2$--$M$ and $\tau$--$M$ relationships from a thermodynamic and cascade perspective. Entropy conservation within the compressive modes yields the fiducial relation $\sigma^2 = \ln\left(1 + M^2\right)$, while deviations from a monotonic energy cascade modify this law into a dilogarithm function. Although skewness exerts a negligible impact on the global $\sigma^2$--$M$ scaling, the characteristic skewness parameter $\tau$ obeys a similar scaling law, $\tau \propto \ln\left(1 + M^2\right)$. We demonstrate that the asymmetric wings of the global density PDF originate from underlying low-$s$ and high-$s$ skewed kernels. These kernels are governed by two distinct structural redistribution mechanisms -- the mass-fraction and volume-fraction approaches -- which exhibit a profound duality symmetry. Crucially, the high-$s$ skewed kernels are physically realizable and closely mirror the gravity-driven power-law tails of column density PDFs observed in molecular clouds.