hermite polynomials, gauss quadratures, lattice dimensions

Dear all,
I’ve a doubt concerning the link between the gauss quadrature nodes and the lattice space discretization.
According to the derivation in the PhD thesis of Dr. Malaspinas and the work of Chen (2006) the lattice boltzmann equation can be derived by projecting the bolzmann equation on Hermite bases. Everything is fine up to the moment when the original equation
\frac{\partial f}{\partial t}+\xi\times\grad f =-\omega(f-f^{eq})

is replaced by it’s discrete version

\frac{\partial f_a}{\partial t}+\xi_a\times\grad f_a =-\omega(f_a-f^{eq}_a) being a=1,…,9 in the D2Q9

Now my point is the following. The quantity \xi_a is in fact your quadrature node which is not unity (e.g. in the D2Q9 this it has magnite sqrt(3)).
How do I move from this formula to the lattice units? I can understand that f^{eq}_a can be rewritten by a proper rescaling, however I still have the quantity on the left hand side
\xi_a\times\grad f_a
that I don’t see how can this be rescaled… (in particular \xi_a is not a unit velocity)

and this last point is quite critical since then we will have

where \xi_a is not the unit lenght.

In other words, I’m missing part of the step to go from the quadrature nodes to the lattice nodes, that is
f(x+c_a,t+1)-f(x,t)=-\omega(f_a-f^{eq}_a) (being c_a the unit vector)

I hope I’ve been clear.
Thanks in advance to everyone!