I'm following the book Lattice Gas Cellular Automata and Lattice Boltzmann models which refers to this paper to explain how to discretize the Boltzmann equation (BE) into the Lattice Boltzmann equation (LBE), but in this paper they ommit the term for forces acting on the particles. In all the documents I'm seeing this term is neglected, but I'd like to see how is discretized as well. In the paper, they claim to discretize the speeds in i speeds, so there will be $f_i(x,t)$ distributions associated each to each $v_i$. But, how do you apply this to the term of the forces?
$$ F\partial_pf(x,p,t) $$$$ \frac{\partial f(r,p,t)}{\partial t} + \frac{p}{m}\nabla_r f(r,p,t) + F\nabla_p f(r,p,t)) = \frac{f^0-f}{\tau} $$
is this the dimensionles:
$$ F'Fr\partial_{v'_i}f^*_i(x,t) $$$$ \begin{equation} \frac{\partial f^*_i(r,t)}{\partial t} + c_i\nabla_r f_i^*(r,t) + \frac{F_0L}{mU^2}F^*\nabla_{c_i} f_i^*(r,t)) = \frac{f_i^{*0}-f_i^*}{\epsilon\tau^*} \end{equation} $$
? and how do you discretize then the force term?
where F' is a$^*$ are normalized forceunits with a characteristic force/time/speed/lenght/time... in the system, Fr is the Froude number, and $f_i^*$ is the rescaled $f_i$ and $v_i'$ is the rescaled velocity $i$. Maybe it is a stupid question, but I just wanted to be sure this has any mathematical sense, mostly because the derivative with respect the speed in the force term.
Also, I'm curious about one fact. The BE is presented for a diluted gas, but LBE is used for liquids as well. Is all the difference hidden in the collision term, which at the end doesn't mind because you use the BGK approximation?