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

Turbulence is a complex phenomenon that plays a critical role in the interstellar medium (ISM). Previous simulations and observations show that the probability density function (PDF) of gas density in isothermal and compressible systems under turbulence exhibits a near lognormal shape, with a strong empirical relation between the variance ($\sigma^2$) and Mach number ($M$). In this work, we aim to explain the $\sigma^2$-$M$ relation and the deviation from the lognormal shape from a thermodynamic and cascading perspective. By introducing a virtual dissipation process, during which turbulent entropy and structural dissipation are assumed to be coupled, we derive the empirical relation $\sigma^2 = \ln(1 + M)^2$. Additionally, by introducing a delay parameter $q$ for the local gas temperature, we derive the deviation from the empirical relation at high $M$. We further argue that the exponential tails of PDFs (on the $s = \ln(\rho)$ scale) arise from the convolution of PDF kernels, which can be skewed at both the low-$s$ and high-$s$ ends. Skewness has limited influence on the $\sigma^2$-$M$ relation. Two density-fraction strategies--the mass-fraction and volume-fraction approaches--are introduced to explain the physical origins of the low-$s$ and high-$s$ skewed PDF kernels. These two types of PDF kernels are dual to each other and exhibit highly symmetric mathematical structures. We speculate that the high-$s$ skewed PDF kernels are physical and may be analogous to the high-density tails of column-density PDFs in molecular clouds, which are influenced by gravity. Inspired by this, we propose a form of an "isothermal" turbulent system that likely favors the volume-fraction strategy.