How do I determine the equilibrium distribution function to the Korteweg tensor?

Hi, Community
I need to work with a Local Equilibrium Distribution Function in terms of the Korteweg tensor. But I know how to create this distribution function for specific equation differential.
This tensor is,

pi = (B * 1/2 * (nabbla rho)^2 + A * rho * Delta rho)I + C * (nabbla rho) x (nabbla rho)

where B, A, C are constants. This tensor is different from NS-Tensor, for this reason, it is necessary to create a new Local Equilibrium Distribution Function.

Can anyone tell me what bibliography I can use to know how to create this distribution function?

This modification can be implemented at


Thanks for any help