Time-structure invariance criteria for closure approximations

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.