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In this paper, we investigate the existence of higher order dynamics in lattice-based models. We have identified two conditions that determine whether a model would allow some Burnett-like equations when the Chapman-Enskog expansion is used. These two conditions are the member of the conserved quantities as well as the space and time discretization. We shall demonstrate these conditions by discussing (1) pure diffusion equation, and (2) hydrodynamic equations. While the fact that diffusion equation allows the higher order dynamics can be shown easily, we will illustrate that care must be taken when deriving Burnett-like equations for lattice-based hydrodynamics models using the Chapman-Enskog method.