Chapman-Enskog method

A method for finding a solution to the kinetic Boltzmann equation for a single-particle distribution function $ f ( t , r , v ) $, which is the original method of successive approximations, in which the local Maxwell distribution $ f _ { \mathop{\rm loc} } ( r , v \mid n , u , \theta ) $, defined by the standard formula but with local values of the density of the number of particles $ n ( t , r ) $, the hydrodynamic velocity $ u ( t , r ) $ and the temperature $ \theta ( t , r ) $, is used as the zero-th approximation, and where the condition for the existence of a solution for each successive approximation is that the hydrodynamic equation for $ n , u , \theta $ holds in the previous approximation. Since the convolutions with respect to the velocity $ v $ of the Boltzmann collision integral with the quantities $ 1 , v $ and $ v ^ {2} $ vanish, the collision integral does not enter explicitly into the equations of motion for $ n , u $ and $ \theta $. The solution of the Boltzmann equation can be obtained in the form

$$ f ( t , r , v ) = f _ { \mathop{\rm loc} } ( 1 + \phi ( t , r , v ) ) ,\ \ \phi ( t , r , v ) \ll 1 , $$

which reduces to an inhomogeneous integral equation with a linearized collision integral relative to $ \phi $. The inhomogeneous part of the equation contains the quantities $ n ( t , r ) $, $ u ( t , r ) $, $ \theta ( t , r ) $ which are subject to the equation mentioned above. Thus, one is considering six equations at the same time. A solution to the equation for $ \phi $ is sought in the form of an expansion by Sonin polynomials (related to the Laguerre polynomials with half-integer indices) in the velocity space (cf. Laguerre polynomials). It is a characteristic of the entire method that the dependence of the distribution function $ f ( t , r , v ) $ on time enters only by way of its dependence on the local quantities $ n ( t , r ) $, $ u ( t , r ) $ and $ \theta ( t , r ) $. The zero-th approximation $ f = f _ { \mathop{\rm loc} } $ defines the hydrodynamic equations of an ideal fluid (the Euler equations). These are the existence conditions for a first approximation to $ f $, to which correspond the Navier–Stokes equations with explicit expressions for the diffusion coefficients, for the heat conduction and for the two viscosity coefficients. The next stage is the Barnett equation, etc. The parameter of the expansion is, essentially, the relative change in $ n ( t , r ) $, $ u ( t , r ) $ and $ \theta ( t , r ) $ on an interval equal to the average length of free flow (the parameter of inhomogeneity), and so for problems involving a jump in these quantities (shock waves, etc.) the method cannot be applied.

This method of solution, based on a method of solving integral equations due to D. Hilbert (1912), was worked out by D. Enskog (1917) and independently by S. Chapman (1916).

The same solution can also be obtained by Grad's method, which is not as unwieldy as the Chapman–Enskog method. In this case the function $ f $ is represented by a series in the derivatives of $ f _ { \mathop{\rm loc} } ( r , v \mid n , u , \theta ) $ with respect to the velocity component $ v $( which in fact is equivalent to the expansion of the function by Hermite polynomials in the three-dimensional velocity space), and where the coefficients, which depend on $ t $ and $ r $, are the moments of the unknown distribution function, which are also determined by means of the Boltzmann equation. The first approximation to $ f $( which reduces to the Navier–Stokes equation) involves only the second derivative of $ f _ { \mathop{\rm loc} } $.

The above methods of solution for a single-particle distribution function may be obtained immediately from the Bogolyubov chain of equations with a suitably small value for the parameter of inhomogeneity for the hydrodynamic approximation (that is, omitting the kinetic equation). However, since the successive approximations derived in the chain include calculations of higher-order correlations, the results will agree only up to the first order inclusive, since the next approximation for $ f $ already takes account of thirdfold particle correlations, which are not contained in the Boltzmann equation, where the contributions from these terms compete with the terms corresponding to the second approximation for the function $ f $ satisfying the standard Boltzmann equation.

References

[1]	T.D. Cowling, "The mathematical theory of non-uniform gases" , Cambridge Univ. Press (1952)
[2]	G.E. Uhlenbeck, G.V. Ford, "Lectures in statistical mechanics" , Amer. Math. Soc. (1963)
[3]	J.O. Hirschfelder, C.F. Curtiss, R.B. Bird, "The molecular theory of gases and liquids" , Wiley (1954)
[4a]	H. Grad, "Asymptotic theory of the Boltzmann equation I" Physics of Fluids , 6 (1963) pp. 147
[4b]	H. Grad, "Asymptotic theory of the Boltzmann equation II" , Rarified gas dynamics , Acad. Press (1963)
[5]	H. Grad, "Principles of the kinetic theory of gases" S. Flügge (ed.) , Handbuch der Physik , 12 , Springer (1958) pp. 205–294

Comments

For the notions of collision operator (or collision integral) and linearized collision operator (linearized collision integral) cf. Boltzmann equation and Boltzmann equation, linearized.