# Subgrid PDF

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A subgrid probability density function $\bar{P}(\eta)$ , also known as filtered density function (FDF), is the distribution function of scalar $Z$ at subgrid scales.

The probability of observing values between $\eta < Z < \eta + d\eta$ within the filter volume is $\bar{P}(\eta) d\eta$

$\bar{P}(\eta) \equiv \int_V \delta \left( Z(\mathbf{x'},t) - \eta \right) G \left( \mathbf{x} -\mathbf{x'}, \Delta \right) dV'$

where $\delta$ is the Dirac delta function, $G$ is a positive defined filter function with filter width $\Delta$.

The joint subgrid PDF of $N$ scalars is

$\bar{P}(\underline{\psi}) \equiv \int_V \prod_i^N \delta \left( \Phi_j(\mathbf{x'},t) - \psi_j \right) G \left( \mathbf{x} -\mathbf{x'}, \Delta \right) dV'$

where $\underline{\psi} = ( \psi_1,\psi_2,.. \psi_N)$ is the phase space for the scalar variables $\underline{\phi} = ( \phi_1,\phi_2,.. \phi_N)$.

A density weighted FDF, $\tilde{P}(\eta)$, can be obtained as

$\bar{\rho} \tilde{P}(\eta) \equiv \int_V \rho \delta \left( Z(\mathbf{x'},t) - \eta \right) G \left( \mathbf{x} -\mathbf{x'}, \Delta \right) dV'$

and in the same manner for the joint FDF

$\bar{\rho} \tilde{P}(\underline{\psi}) \equiv \int_V \rho \prod_i^N \delta \left( \Phi_j(\mathbf{x'},t) - \psi_j \right) G \left( \mathbf{x} -\mathbf{x'}, \Delta \right) dV'$