Summational invariants

For all microscopic processes (collisions, radiative transitions, and so forth, in which v may change but the macroscopic x and t are fixed for the molecules experiencing the process), certain quantities may be conserved. If ij/. is such a quantity, then it is called a summational invariant because 

---

It may be seen from Eq. (1-51) that if Qx is one of the quantities conserved in a collision (the summational invariants ), its change due to a collision is zero, as expected, by virtue of the appropriate conservation law. 

Summational invariants, 20 Sums of independent random variables, 155... 

Of all the macroscopic quantities in our model, the hydrodynamic density p, flow velocity vector u = (ua), and thermodynamic energy E, have the unique property of being produced by additive invariants of the microscopic motion. The latter, also called sum functions4 and summation invariants,5 occur at an early stage in most treatments. The precise formulation follows. 

The equations for conservation of mass, momentum, and energy for a one-component continuum are well known and are derived in standard treatises on fluid mechanics [l]-[3]. On the other hand, the conservation equations for reacting, multicomponent gas mixtures are generally obtained as the equations of change for the summational invariants arising in the solution of the Boltzmann equation (see Appendix D and [4] and [5]), One of several exceptions to the last statement is the analysis of von Karman [6], whose results are quoted in [7] and are extended in a more recent publication [8] to a point where the equivalence of the continuum-theory and kinetic-theory results becomes apparent [9]. This appendix is based on material in [8]. 

In deriving the species conservation equations, will be set equal to unity, which is not a summational invariant, since the number of molecules need not be conserved in chemical reactions. With ij/- = 1, equation (31) reduces to... 

The conservation equations (2.202), (2.207) and (2.213) are rigorous (i.e., for mono-atomic gases) consequences of the Boltzmann equation (2.185). It is important to note that we have derived the governing conservation equations without knowing the exact form of the collision term, the only requirement is that we are considering summation invariant properties of mono-atomic gases. That is, we are considering properties that are conserved in molecular collisions. 

Considering that (1 + In/) represents a summation invariant property, (2.219) can then be expressed as ... 

It is convenient to introduce the notion of collisional invariants (or summational invariants) (e.g., [55], p. 460 [90], p. 150). The validity of (2.211) is commonly justified by the following arguments Due to the symmetry properties of the collision term expression, interchanging variables gives the following equalities ... 

Analogous to the equation of change of mean molecular properties for dilute gas that was examined in Sect. 2.6, similar macroscopic conservation equations may be derived for dense gas from the Enskog equation. Multiplying the Enskog s equation (2.663) with the summation invariant property, //, and integrating over c, the result... 

If we choose as the summational invariants m, mC and mC, we obtain the Cauchy set of conservation equations named the continuity (2.217), the equation of motion (2.223), and the equation of energy (2.230). The only difference in the final result is that the pressure tensor, p, and the heat flux vector, q, are made up of two parts (i.e., a kinetic and a collisional contribution) ... 

The second last term on the RHS denotes kinetic pressure and the last term the convective rnomenmm flux. From (4.28) it is immediately concluded that 2 (me) = 0 because the me is a summational invariant (i.e., in a particle collision the momentum is conserved). 

---