Mean convected derivative second form

The expected value of the convected derivative of Q can also be written in a second independent form starting with15 

Note that A, and , will, in general, depend on multi-point information from the random fields U and 0. For example, they will depend on the velocity/scalar gradients and the velocity/scalar Laplacians. Since these quantities are not contained in the one-point formulation for U(x, t) and 0(x, t), we will lump them all into an unknown random vector Z(x, f).16 Denoting the one-point joint PDF of U, 0, and Z by /u, / ,z(V, ip, z x, t), we can express it in terms of an unknown conditional joint PDF and the known joint velocity, composition PDF  

15 Note that the extension to other flow quantities is formally trivial application of the chain mle would add the corresponding terms on the right-hand side of the final equality. 

16 Conceptually, since a random field can be represented by a Taylor expansion about the point (x, t), the random vector Z(x, t) could have as its components the infinite set of partial derivatives of U and / of all orders greater than zero with respect to x, X2, X3, and t evaluated at (x, l). 

The expected value of the term involving A, can then be written as 

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