A subgrid probability density function , also known as filtered density function (FDF), is the distribution function of scalar at subgrid scales.
The probability of observing values between within the filter volume is
where is the Dirac delta function, is a positive defined filter function with filter width .
The joint subgrid PDF of scalars is
where is the phase space for the scalar variables .
A density weighted FDF, , can be obtained as
and in the same manner for the joint FDF