Time-structure invariance criteria for closure approximations

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Abstract
In many areas of physics, time evolution equations for moments of distributions are expressed in terms of higher-order moments. Closure approximations are then introduced in an ad hoc fashion to reduce the higher-order moments to functions of the lower-order ones. Herein, the time-structure invariance of the Poisson bracket as manifested through the Jacobi identity is used to derive constraint relationships on these approximations. These constraints severely limit the allowable functionality of general closures and help to define the boundaries within which future investigations should concentrate.