A probability density or distribution function of turbulence has been thought to be symmetric due to the symmetry of the partial differential equations from the first principles. However, the experimental data have been shown otherwise by a so-called Taylor correlation function, and this is an unresolved issue. A recent study shows that this probability density function can be asymmetric analytically by introducing negative eigenvalues. Here, in this work, we show the mathematical basis for this asymmetry, although the partial differential equations follow the symmetry of the Lie groups. We also demonstrate a complete solution of partial differential equations, including the exponential terms and negative eigenvalues, which plays a vital part in transient phenomena. Our analysis shows that the asymmetry is from the partition of velocities of the same or opposing direction, not from the negative eigenvalues. Fundamentally, the loss of rotational symmetry is caused by the exponential terms for the transient solution, which we demonstrate by the derivation of the complete solution. The new correlation produces excellent agreement with the experimental data. The universality and limitations of the correlation function are discussed, and through the parameter study, the variations and statistical nature of the probability function are clarified. The asymmetry probability function should have wider applicability than the symmetric Gaussian distribution, which is the special case of the asymmetry probability function.
