Integrations of the Boltzmann equation lead to conservation equations for the local fluid densities. The mean free path, l; of the gas particles is assumed to be small compared to the size, L of the container. When the gas is close to local equilibrium, solutions to the Boltzmann equation are obtained by expressing conserved currents as series in powers of the gradients of the local variables starting with the local Maxwell-Boltzmann distribution. Higher terms are obtained by inserting the series in the Boltzmann equation and collecting terms of equal order. This, together with the conservation equations, leads to hydrodynamic equations. The ideal fluid equations are obtained to lowest order, and the next order is the Navier-Stokes equations, with explicit expressions for the transport coefficients, which are compared with experimental results for different model potentials. To second order the rate of entropy production agrees with predictions of non-equilibrium thermodynamics.

The Enskog equation was the first extension of the Boltzmann transport equation to higher densities. It applies only to hard sphere systems and takes into account excluded volume and collisional transport effects. While useful for one component gases, it has serious shortcomings, in particular, for mixtures it leads to expressions for transport coefficients that are inconsistent with the general Onsager reciprocal relations and it has no H-theorem. The Revised Enskog equation is presented and shown to satisfy an H-theorem, and, for mixtures, to have transport coefficients that satisfy the Onsager relations. The revised equation describes spatio-temporal fluctuations in a hard sphere fluid about equilibrium. It 8 is possible to extend the Revised Enskog equation to high densities where hard sphere fluids form a crystal, and to show that this solid has transport properties appropriate for an elastic solid. Explicit expressions for the appropriate transport coefficients are given.