A note on the lattice momentum balance in the lattice Boltzmann interaction-framework

In this note, we show how the exploitation of the lattice momentum balance condition allows to envisage an analytical procedure to define the lattice pressure tensor (LPT) for the multi-phase Shan-Chen (SC) lattice Boltzmann method (LBM) with single-range potential. This construction ensures that the LPT normal component to a flat interface is constant to machine precision on each lattice node, i.e., it exactly implements the mechanical equilibrium condition on the lattice. We demonstrate the robustness of the approach by providing analytical expressions for the coexistence curves for different choices of the pseudo-potential and forcing schemes in the SC-LBM. This paper offers a novel, rigorous perspective for controlling the LPT in the SC-LBM, paving the way for its application in more general settings.