Chapman–Enskog expansion for multi-particle collision models

Particle-based simulation methods for fluid flow follow a discrete time dynamics of subsequent streaming and collision events. The algorithm considered here, called stochastic rotation dynamics, involves collisions between an arbitrary number of partners; all particles that happen to be in the same cell of a randomly positioned grid interact at once by prescribed rules. I show, in two dimensions, how a multi-particle generalization of the Enskog equation can be derived from the Liouville equation and how the hydrodynamic equations can be obtained by a Chapman-Enskog expansion. The resulting macroscopic equations contain a collisional contribution to the transport coefficients, absent in earlier Chapman-Enskog approaches, which agrees exactly with previously known results from other kinetic theories. This approach opens up a powerful, systematic route to deriving hydrodynamic equations for particle-based models, which is generalizable to models with active particles.