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.
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.
This condition ensures that collisions are well-defined events in space and time, and holds if the dimensionless parameter
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.
[4] 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
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
Conceptually, the resulting hydrodynamic equations describe the dynamical interplay between free streaming and interparticle collisions.
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
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.
a tensor, each a solution of a linear inhomogeneous integral equation that can be solved explicitly by a polynomial expansion.
This counterintuitive result traces back to James Clerk Maxwell, who inferred it in 1860 on the basis of more elementary kinetic arguments.
[13] 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
[12] 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.
Obtaining the corrections used to account for transport during a collision for soft molecules (i.e. Lennard-Jones or Mie molecules) is in general non-trivial, but success has been achieved at applying Barker-Henderson perturbation theory to accurately describe these effects up to the critical density of various fluid mixtures.
[17] 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.
[18] (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.
)[19] Even if the higher order corrections do afford improvement in a given system, the interpretation of the corresponding hydrodynamical equations is still debated.
The successful derivation of RET followed several previous attempt at the same, but which gave results that were shown to be inconsistent with irreversible thermodynamics.
An important factor to note here is that in order to obtain results in agreement with irreversible thermodynamics, the
While from classical Chapman–Enskog theory the ideal gas law is recovered, RET developed for rigid elastic spheres yields the pressure equation
, which is consistent with the Carnahan-Starling Equation of State, and reduces to the ideal gas law in the limit of infinite dilution (i.e. when
However, one of the major advantages of RET over classical Chapman–Enskog theory is that the dependence of diffusion coefficients on the thermodynamic factors, i.e. the derivatives of the chemical potentials with respect to composition, is predicted.
, while classical Chapman–Enskog theory predicts that the Soret coefficient, like the viscosity and thermal conductivity, is independent of density.
While classical Chapman–Enskog theory can be applied to arbitrarily complex spherical potentials, given sufficiently accurate and fast integration routines to evaluate the required collision integrals, Revised Enskog Theory, in addition to this, requires knowledge of the contact value of the pair distribution function.
For mixtures of hard spheres, this value can be computed without large difficulties, but for more complex intermolecular potentials it is generally non-trivial to obtain.
However, some success has been achieved at estimating the contact value of the pair distribution function for Mie fluids (which consists of particles interacting through a generalised Lennard-Jones potential) and using these estimates to predict the transport properties of dense gas mixtures and supercritical fluids.
While these are unambiguously defined for hard spheres, there is still no generally agreed upon value that one should use for the contact diameter of soft particles.