Transport processes in mixtures of nonpolar gases

In molecular terms, the local values of the diffusion velocity V,(r, t) of the jth component and the local value of the heat flux vector q (r, t) in a volume element dr in a nonuniform N-component mixture of polyatomic gases are given by 

Here n,(r, t) is the local number density of species i, the index / denotes the /th quantum state, wj is the molecular mass, and /j/(Vj, , r, t) is the local value of the distribution function for the molecular velocity v,(r, f) and internal energy Eii, normalized so that 

In order to integrate Eqs. (3.1) and (3.2), it is necessary to know the distribution function fu. In the absence of external forces, the rate of decay of a nonequilibrium velocity distribution to an equilibrium one is given by the generalized Boltzmann equation 

The temperature T, defined by Eqs. (3.6) and (3.7c), implies that a single temperature is sufficient to describe the distribution of energy among the translational and internal degrees of freedom. While this is true in the equilibrium case, it is only strictly true in nonequilibrium cases (with which we are concerned) when the exchange between internal and translational energy is rapid. 

If the nonequilibrium distribution function fu is expressed as a perturbation expansion 

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