  In this work, a third-order Chapman-Enskog analysis of the
multiple-relaxation-time (MRT) pseudopotential lattice Boltzmann (LB) model for
multiphase flow is performed for the first time. The leading terms on the
interaction force, consisting of an anisotropic and an isotropic term, are
successfully identified in the third-order macroscopic equation recovered by
the lattice Boltzmann equation (LBE), and then new mathematical insights into
the pseudopotential LB model are provided. For the third-order anisotropic
term, numerical tests show that it can cause the stationary droplet to become
out-of-round, which suggests the isotropic property of the LBE needs to be
seriously considered in the pseudopotential LB model. By adopting the classical
equilibrium moment or setting the so-called "magic" parameter to 1/12, the
anisotropic term can be eliminated, which is found from the present third-order
analysis and also validated numerically. As for the third-order isotropic term,
when and only when it is considered, accurate continuum form pressure tensor
can be definitely obtained, by which the predicted coexistence densities always
agree well with the numerical results. Compared with this continuum form
pressure tensor, the classical discrete form pressure tensor is accurate only
when the isotropic term is a specific one. At last, in the framework of the
present third-order analysis, a consistent scheme for third-order additional
term is proposed, which can be used to independently adjust the coexistence
densities and surface tension. Numerical tests are subsequently carried out to
validate the present scheme.
