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Mesoscopic formulas of linear and angular momentum fluxes

Many approaches of coarse-graining have been developed under the names of Cosserat theory or polar-fluid theory, for those materials in which some component elements undergo non-affine deformations, such as elastic materials with inclusions or granular matters. For the complex elements such as living cells, however, the microscopic variables and their dynamics are often unknown, and there have been no systematic theory of coarse-graining from the microscales, nor the formulas like Irving-Kirkwood formula that constitutes the macroscopic stress or couple-stress in terms of some microscale quantities. We show that, for the quasi-steady states, the coarse-graining procedure must generally provides with the Cosserat-type balance equations as long as the procedure keeps track of the conservation of linear and angular momenta, and that the fluxes of these conserved quantities should generally be expressed in the Irving-Kirkwood-type formulas, where the inter- particle distance or forces/torques should be replaced by those associated to the pair of neighboring coarse-graining volumes. This framework, which refers to no particular micro-variables or dynamics, is valid for active complex matters out of equilibrium and with any multi-body interactions.