Finite-Dimensional Characteristic Functions, Pdfs and Cdfs Based on the Dirac Distribution

The standard statistical approach is covered in this chapter, starting with the definition of the coarse-grained C(umulative)d(istribution)f(function) and the derivation of its transport pde using the Heaviside step function and the Dirac pseudo-function. The role of pressure is analyzed on the basis of Poisson’s pde and Green’s functions. Multidimensional transport equations for Cdfs are derived and the individual terms related to the underlying physics using simple flow situations as examples. The popular Pdf equations are established as a derivative of the Cdf equation. Homogeneous turbulence is solved for a simple example.