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We employ systematic coarse graining techniques to derive hydrodynamic equations from Grad's ten-moment equations. The coarse graining procedure is designed such that it manifestly preserves the thermodynamic structure of the equations. The relevant thermodynamic structure and the coarse graining recipes suggested by statistical mechanics are described in detail and are illustrated by the example of hydrodynamics. A number of mathematical challenges associated with structure-preserving coarse graining of evolution equations for thermodynamic systems as a generalization of Hamiltonian dynamic systems are presented. Coarse graining is a key step that should always be considered before attempting to solve an equation. 1. Introduction. Physicists and engineers want to solve problems, and they hence try to solve equations. The purpose of this paper is to emphasize and to illus- trate that an essential first step to solve a problem may be to coarse grain a given set of equations before studying the solutions. We here consider problems that can be investigated on different levels of description associated with different length and time scales, and we specify a set of rules for systematic coarse graining. From a math- ematical perspective, it is important to specify the structure which should be left invariant in the coarse graining procedure. The relevant structure, which is known as GENERIC ("general equation for the nonequilibrium reversible-irreversible cou- pling") or "metriplectic," is a generalization of Hamiltonian structures to dissipative systems and is of thermodynamic origin. The design of structure-preserving coarse graining procedures and approximation schemes is presented as a general mathemat- ical challenge. It is important to note that coarse graining is fundamentally differ- ent from reduction based on the idea of invariant manifolds because the transition from reversible to irreversible equations, or even increasing degrees of irreversibility, is allowed. The fundamental distinction between reduction (or solving) and coarse graining has been elaborated in detail in (27). We first describe the GENERIC structure. The coarse graining recipes of statis- tical mechanics are then presented and explained in the context of rarefied gas flow. In particular, we shall illustrate the statistical mechanical coarse graining procedure by reducing Grad's ten-moment equations to hydrodynamics. It is well known that this can be done by various methods (16). Our emphasis therefore lies on the method

The Mathematical Procedure of Coarse Graining: From Grad’s Ten‐Moment Equations to Hydrodynamics