  We systematically analyze the tensorial structure of the lattice pressure
tensors for a class of multi-phase lattice Boltzmann models (LBM) with
multi-range interactions. Due to lattice discrete effects, we show that the
built-in isotropy properties of the lattice interaction forces are not
necessarily mirrored in the corresponding lattice pressure tensor. This finding
opens a different perspective for constructing forcing schemes, achieving the
desired isotropy in the lattice pressure tensors via a suitable choice of
multi-range potentials. As an immediate application, the obtained LBM forcing
schemes are tested via numerical simulations of non-ideal equilibrium
interfaces and are shown to yield weaker and less spatially extended spurious
currents with respect to forcing schemes obtained by forcing isotropy
requirements only. From a general perspective, the proposed analysis yields an
approach for implementing forcing symmetries, never explored so far in the
framework of the Shan-Chen method for LBM. We argue this will be beneficial for
future studies of non-ideal interfaces.
