Boltzmann equation first-order approximation

This set of four partial differential equations, then, are the first-order approximations to the Boltzmann equation which satisfy the requirements (7.106), (7.111), (7.137), and (7.146). 

In the small deformation approximation, it is assumed that the deformations undergone by the material are small, at least in the recent past. Approximations of different orders can be developed. The approximation of first order for an incompressible fluid is given by Boltzmann s equation of linear viscoelasticity,... 

The Enskog [24] expansion method for the solution of the Boltzmann equation provides a series approximation to the distribution function. In the zero order approximation the distribution function is locally Maxwellian giving rise to the Euler equations of change. The first order perturbation results in the Navier-Stokes equations, while the second order expansion gives the so-called Burnett equations. The higher order approximations provide corrections for the larger gradients in the physical properties like p, T and v. 

Falkenhagen considered a modified distribution function which took account of the finite size of the ions by recognising that the total space available to the ions is less than the total volume of the solution. This implicitly means that (1 - - Ka) is not approximated to unity. A modified Poisson-Boltzmann equation was thus used in the derivation of the relaxation effect, but the solution of this modified Poisson-Boltzmann equation was approximated to the first two terms. These modifications gave higher order terms in Cacmai of the type which had been empirically observed. 

The Poisson-Boltzmann equation (23.5) is a nonlinear second-order differential equation from which you can compute ip if you know the charge density on P and the bulk salt concentration, rioo- This equation can be solved numerically by a computer. However, a linear approximation, which is easy to solve without a computer, applies when the electrostatic potential is small. For small potentials, zeip/kT 1, you can use the approximation sinh(x) s [(I + x) - (I - x)] 12 = X (which is the first term of the Taylor series expansion for the two exponentials in sinh(x) (see Appendix C, Equation (C.l)). Then Equation (23.5) becomes... 

The development by Ziman (1961,1964) of a pseudopotential theory of the electronic properties of liquid metals provided a substantial impetus to this field. Within this formulation, based on first-order Born approximation and the Boltzmann equation, the electrical resistivity, p, can be cast as... 

Alternatively, if we insert / = / (1 -(- ) in the modified Boltzmann equation 97 and retain only up to the zeroth order in left-hand side and the first order in the right-hand side, the first approximation equation of Chapman and Enskog for a dense gas is obtained ... 

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