A new method of solving the system of kinetic equations for gas mixtures (Q2276837)

The solution of the kinetic Boltzmann equation for a monatomic gas is replete with major difficulties which have generated much confusion; only recently has a certain degree of understanding been achieved, and it has become clear, in particular, that: at least two processes - rapid (kinetic) and slow (hydrodynamic) processes - are described by the Boltzmann equation; in its expansion in a series in a small parameter the distribution function must have so-called normal structure for which the first moments of the distribution function are determined only by its zeroth approximation. The natural condition is caused by assumptions of the kinetic theory regarding the conservation of invariants of binary collisions. Many attempts have been made to avoid these fundamental propositions of the theory, and all of them have led to ambiguities. Hilbert's work,which led to erroneous conclusions, is the clearest example. Having this in mind, we should proceed with special care to methods of solution of the still more complicated system of kinetic Boltzmann equations which can be written in dimensionless variables in the form \[ \frac{\partial f_ s}{\partial t}+V^ s\frac{\partial f_ s}{\partial r}=\frac{1}{\epsilon}\sum^{M}_{\tau =1}\iint (f'_ Sf'_ S-f_ Sf_{\tau})g_{s\tau}b dbd\epsilon dV_{\tau} \] or \[ \frac{df_ S}{dt}=\frac{1}{\epsilon}\sum^{M}_{\tau =1}I(f_ Sf_{\tau}). \] We shall first show and then prove that there exist only two essentially different methods of solving this system of kinetic equations which have a completely different physical meaning.