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A new definition is given for the ``ideal random function'' (derivative of the Wiener function), which separates out infinite factors by fullest exploitation of the possibilities of the Dirac delta function. By allowing all integrals to be written formally as sums, this facilitates the definition and manipulation of the Wiener-Hermite functionals, especially for vector random processes of multiple argument. Expansion of a random function in Wiener-Hermite functionals is discussed. An expression is derived for the expectation value of the product of any number of Wiener-Hermite functionals; this is all that is needed in principle to obtain full statistical information from the Wiener-Hermite functional expansion of a random function. The method is illustrated by the calculation of the first correction to the flatness factor (measure of Gaussianity) of a nearly-Gaussian random function.

A Wiener-Hermite functional expansion is used to treat a random initial value process involving the Burgers model equation. The nonlinear model equation has many of the characteristics of the Navier-Stokes equation. It is found that the functional expansion converges better the larger the separation variable in the correlation function (the nearer to joint normal is the distribution). To the present order, the treatment is similar to a quasinormal assumption. The computations show that the correlation function quickly approaches an equilibrium form for quite different initial values. The power spectrum function approaches an equilibrium form also, where it falls off like the inverse second power of the wavenumber.

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The Wiener-Hermite functional expansion, which is the expansion of a random function about a Gaussian function, is here substituted into the Burgers one-dimensional model equation of turbulence. The result is a hierarchy of equations which (along with initial conditions) determine the kernel functions which play the role of expansion coefficients in the series. Initial conditions are postulated, based on physical reasoning, criteria of simplicity, and the assumption that the series is to represent the late decay stage (in which the Gaussian correction is small and also decreasing with time). These are shown to justify an iterative solution to the equations. The first correction to the Gaussian approximation is calculated. This is then tested by evaluating the correction to the flatness factor, which for an exactly Gaussian function has the value 3, but which has been found by experiment (in real three-dimensional fluids, of course) to have a value which deviates from the Gaussian value increasingly rapidly with the order of the derivative. We utilize this effect as a test of the inherent ability of the Wiener-Hermite expansion to bring to realization the physical properties implicit in the Navier-Stokes or Burgers equations. The various contributions to the flatness-factor deviation, when computed, do show a potential capability of providing a theoretical basis for the effect.