REPRESENTATION THEORY AND RELATED TOPICS SEMINAR

Alexander Gorban

Abstract: Derivation of the hydrodynamic equations from a microscopic description is the classical problem of physical kinetics. The famous Chapman-Enskog method provides an opportunity to represent a solution of this problem from the Boltzmann kinetic equation as a formal series in powers of the Knudsen number, Kn.

If the Chapman-Enskog expansion is truncated at a certain order, we obtain subsequently: the Euler hydrodynamics, the Navier-Stokes hydrodynamics, the Burnett hydrodynamics, etc. The post-Navier-Stokes terms extend the hydrodynamic description beyond the hydrodynamic limit Kn<<1.

However, even in the simplest regime (one-dimensional linear deviations around the global equilibrium), the Burnett hydrodynamic equations violate the basic physics: sufficiently short acoustic waves are increasing with time instead of decaying. The situation does not improve in the higher order approximations.

Up to now, the problem of the exact relationship between kinetics and hydrodynamics remains unsolved. All the methods used to establish this relationship are not rigorous, and involve approximations. In this work, we have considered situations where hydrodynamics is the exact consequence of kinetics, and in that respect, a new class of exactly solvable models of statistical physics has been established.

We are looking for kinetic systems which allow exact reduction to hydrodynamics.  Grad's systems are considered as the minimal kinetic models where the Chapman-Enskog method can be studied exactly. The Chapman-Enskog method is the Taylor series expansion approach to solving the equation for slow invariant manifold (invariance equation). In a simple case these series could be summarised exactly. The invariance equation for this example can be solved directly, without any expansion. This example serves also as a benchmark for comparison of various reduction methods. Various approximation techniques for the invariance equation, such as the method of partial summation, Pade approximants, and invariance principle are compared both in linear and nonlinear situations.