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A fluctuation–relaxation relation for homogeneous, isotropic turbulence

A Liouville-type equation is solved perturbatively for the exact pdf of fluctuating velocities by expansion about an invariant Gaussian measure, which is an approximation to the symmetric part of the pdf and is constrained to give the correct covariance. The antisymmetric part of the pdf is calculated to the lowest nontrivial order and leads to closed equations for the covariance in terms of a response function which arises from the Liouville operator averaged against the ground-state distribution. As the latter is a stationary Gaussian pdf, we obtain an expression for the system response which is linear in the covariance.