Chapman–Enskog theory
Chapman–Enskog theory provides a framework in which equations of hydrodynamics for a gas can be derived from the Boltzmann equation. The technique justifies the otherwise phenomenological constitutive relations appearing in hydrodynamical descriptions such as the Navier–Stokes equations. In doing so, expressions for various transport coefficients such as thermal conductivity and viscosity are obtained in terms of molecular parameters. Thus, Chapman–Enskog theory constitutes an important step in the passage from a microscopic, particle-based description to a continuum hydrodynamical one. The theory is named for Sydney Chapman and David Enskog, who introduced it independently in  1916 and 1917. Description. The starting point of Chapman–Enskog theory is the Boltzmann equation for the 1-particle distribution function formula_1: where formula_3 is a nonlinear integral operator which models the evolution of formula_4 under interparticle collisions. This nonlinearity makes solving the full Boltzmann equation difficult, and motivates the development of approximate techniques such as the one provided by Chapman–Enskog theory. Given this starting point, the various assumptions underlying the Boltzmann equation carry over to Chapman–Enskog theory as well. The most basic of these requires a separation of scale between the collision duration formula_5 and the mean  free time between collisions formula_6: formula_7. This condition ensures that collisions are well-defined events in space and time, and holds if the dimensionless parameter formula_8 is small, where formula_9 is the range of interparticle interactions and formula_10 is the number density. In addition to this assumption, Chapman–Enskog theory also requires that formula_6 is much smaller than any "extrinsic" timescales formula_12. These are the timescales associated with the terms on the left hand side of the Boltzmann equation, which describe variations of the gas state over macroscopic lengths. Typically, their values are determined by initial/boundary conditions and/or external  fields. This separation of scales implies that the collisional term on the right hand side of the Boltzmann equation is much smaller than the streaming terms on the left hand side. Thus, an approximate solution can be found from It can be shown that the solution to this equation is a Gaussian: where formula_15 is the molecule mass and formula_16 is the Boltzmann constant. A gas is said to be in "local equilibrium" if it satisfies this equation. The assumption of local equilibrium leads directly to the Euler equations, which describe fluids without dissipation, i.e. with thermal conductivity and viscosity  equal to formula_17. The primary goal of Chapman–Enskog theory is to systematically obtain generalizations of the Euler equations which incorporate dissipation. This is achieved by expressing deviations from local equilibrium as a perturbative series in Knudsen number formula_18, which is small if formula_19. Conceptually, the resulting hydrodynamic equations describe the dynamical interplay between free streaming and interparticle collisions. The latter tend to drive the gas "towards" local equilibrium, while the former acts across spatial inhomogeneities to drive the gas "away" from local equilibrium. When the Knudsen number is of the order of 1 or greater, the gas in the  system being considered cannot be described as a fluid. To first order in formula_18 one obtains the Navier–Stokes equations. Second and third orders give rise, respectively, to the Burnett equations and super-Burnett equations. Mathematical formulation. Since the Knudsen number does not appear explicitly in the Boltzmann equation, but rather implicitly in terms of the distribution function and boundary conditions, a dummy variable formula_21 is introduced to keep track of the appropriate orders in the Chapman–Enskog expansion: Small formula_21 implies the collisional term formula_24 dominates the streaming term formula_25, which is the same as saying the Knudsen  number is small. Thus, the appropriate form for the Chapman–Enskog expansion is Solutions that can be formally expanded in this way are known as "normal" solutions to the Boltzmann equation. This class of solutions excludes non-perturbative contributions (such as formula_27), which appear in boundary layers or near internal shock layers. Thus, Chapman–Enskog theory is restricted to situations in which such solutions are negligible. Substituting this expansion and equating orders of formula_21 leads to the hierarchy where formula_30 is an integral operator, linear in both its arguments, which satisfies formula_31 and formula_32. The solution to the first equation  is a Gaussian: for some functions formula_34, formula_35, and formula_36. The expression for formula_37 suggests a connection between these functions and the physical hydrodynamic fields defined as moments of formula_1: From a purely mathematical point of view, however, the two sets of functions are not necessarily the same for formula_40 (for formula_41 they are equal by definition). Indeed, proceeding systematically in the hierarchy, one finds that similarly to formula_37, each formula_43 also contains arbitrary functions of formula_44 and formula_45 whose relation to the physical hydrodynamic fields is "a priori" unknown. One of the key simplifying assumptions of Chapman–Enskog theory  is to assume that these otherwise arbitrary functions can be written in terms of the exact hydrodynamic fields and their spatial gradients. In other words, the space and time dependence of formula_4 enters only implicitly through the hydrodynamic fields. This statement is physically plausible because small Knudsen numbers correspond to the hydrodynamic regime, in which the state of the gas is determined solely by the hydrodynamic fields. In the case of formula_37, the functions formula_34, formula_35, and formula_36 are assumed exactly equal to the physical hydrodynamic fields. While these assumptions are physically plausible, there is the question of whether solutions  which satisfy these properties actually exist. More precisely, one must show that solutions exist satisfying Moreover, even if such solutions exist, there remains the additional question of whether they span the complete set of normal solutions to the Boltzmann equation, i.e. do not represent an artificial restriction of the original expansion in formula_21. One of the key technical achievements of Chapman–Enskog theory is to answer both of these questions in the positive. Thus, at least at the formal level, there is no loss of generality in the Chapman–Enskog approach. With these formal considerations established, one can proceed to  calculate formula_53. The result is where formula_55 is a vector and formula_56 a tensor, each a solution of a linear inhomogeneous integral equation that can be solved explicitly by a polynomial expansion. Here, the colon denotes the double dot product, formula_57 for tensors formula_58, formula_59. Predictions. To first order in the Knudsen number, the heat flux formula_60 is found to obey Fourier's law of heat conduction, and the momentum-flux tensor formula_62 is that of a Newtonian fluid, with formula_64 the identity tensor. Here, formula_65 and formula_66 are the thermal conductivity and viscosity. They can be calculated explicitly in  terms of molecular parameters by solving a linear integral equation; the table below summarizes the results for a few important molecular models (formula_15 is the molecule mass and formula_16 is the Boltzmann constant). With these results, it is straightforward to obtain the Navier–Stokes equations. Taking velocity moments of the Boltzmann equation leads to the "exact" balance equations for the hydrodynamic fields formula_69, formula_70, and formula_71: formula_72 As in the previous section the colon denotes the double dot product, formula_57. Substituting the Chapman–Enskog expressions for formula_74 and formula_75, one arrives at the Navier–Stokes equations. Comparison with experiment. An  important prediction of Chapman–Enskog theory is that viscosity is independent of density (this can be seen for each molecular model in table 1, but is actually model-independent). This counterintuitive result traces back to James Clerk Maxwell, who inferred it in 1860 on the basis of more elementary kinetic arguments. It is well-verified experimentally for gases at ordinary densities. On the other hand, the theory predicts that formula_66 does depend on temperature. For rigid elastic spheres, the predicted scaling is formula_77, while other models typically show greater variation with temperature. For instance, for molecules repelling each other with  force formula_78 the predicted scaling is formula_79, where formula_80. Taking formula_81, corresponding to formula_82, shows reasonable agreement with the experimentally observed scaling for helium. For more complex gases the agreement is not as good, most likely due to the neglect of attractive forces. Indeed, the Lennard-Jones model, which does incorporate attractions, can be brought into closer agreement with experiment (albeit at the cost of a more opaque formula_83 dependence; see the Lennard-Jones entry in table 1). Chapman–Enskog theory also predicts a simple relation between formula_65 and formula_66 in the form formula_86, where formula_87 is the specific heat  at constant volume and formula_4 is a purely numerical factor. For spherically symmetric molecules, its value is predicted to be very close to formula_89 in a slightly model-dependent way. For instance, rigid elastic spheres have formula_90, and molecules with repulsive force formula_91 have formula_92 (the latter deviation is ignored in table 1). The special case of Maxwell molecules (repulsive force formula_93) has formula_94 exactly. Since formula_65, formula_66, and formula_87 can be measured directly in experiments, a simple experimental test of Chapman–Enskog theory is to measure formula_4 for the spherically symmetric noble gases. Table 2 shows that there is  reasonable agreement between theory and experiment. Extensions. The basic principles of Chapman–Enskog theory can be extended to more diverse physical models, including gas mixtures and molecules with internal degrees of freedom. In the high-density regime, the theory can be adapted to account for collisional transport of momentum and energy, i.e. transport over a molecular diameter "during" a collision, rather than over a mean free path ("in between" collisions). Including this mechanism predicts a density dependence of the viscosity at high enough density, which is also observed experimentally. One can also carry out the theory to higher order in  the Knudsen number. In particular, the second-order contribution formula_99 has been calculated by Burnett. In general circumstances, however, these higher-order corrections may not give reliable improvements to the first-order theory, due to the fact that the Chapman–Enskog expansion does not always converge. (On the other hand, the expansion is thought to be at least asymptotic to solutions of the Boltzmann equation, in which case truncating at low order still gives accurate results.) Even if the higher order corrections do afford improvement in a given system, the interpretation of the corresponding hydrodynamical equations is still debated. References . The classic monograph on the topic: Contains a technical introduction to normal solutions of the Boltzmann equation: