A.N. Gorban, I.V. Karlin
Quasi-Equilibrium Closure Hierarchies for the Boltzmann Equation
(576K, pdf)
ABSTRACT. In this paper, explicit method of constructing approximations
the Triangle Entropy Method is developed for nonequilibrium
problems. This method enables one to treat any complicated
nonlinear functionals that fit best the physics of a problem
(such as, for example, rates of processes) as new independent
variables.
The work of the method is demonstrated on the Boltzmann's-type
kinetics. New macroscopic variables are introduced (moments of
the Boltzmann collision integral, or scattering rates). They are
treated as independent variables rather than as infinite moment
series. This approach gives the complete account of rates of
scattering processes. Transport equations for scattering rates
are obtained (the second hydrodynamic chain), similar to the
usual moment chain (the first hydrodynamic chain). Various
examples of the closure of the first, of the second, and of the
mixed hydrodynamic chains are considered for the hard sphere
model. It is shown, in particular, that the complete account of
scattering processes leads to a renormalization of transport
coefficients.
The method gives the explicit solution for the closure problem,
provides thermodynamic properties of reduced models, and can be
applied to any kinetic equation with a thermodynamic Lyapunov
function.