Solutions of the Boltzmann Equation

The Boltzmann equation is a nonlinear, integrodifferential equation. As such it is extremely difficult to solve and, in fact, almost no exact solutions are known, apart from the Maxwell-Boltzmann equilibrium solution. Furthermore, only a few existence theorems are known notable are the theorems of Carleman, later extended by Wild and by Morgenstern, proving the existence of a solution of the nonlinear Boltzmann equation for special intermolecular potentials in the case that the system is spatially uniform, i.e., that the distribution function does not depend on r. However, there are a number of circumstances where the system is close enough to equilibrium that the distribution function may be written 

Before constructing the normal solutions, we find it convenient to rewrite the Boltzmann equation, Eq. (36), as two equations, one of which determines the distribution function for points in the interior of the container and the other can be interpreted as a matching condition for the distribution function at the walls. To obtain these two equations we look for solutions/(r, v, t) that vanish outside the container. That is, we write 

If we examine Eq. (74), we see that the left-hand side cannot contain delta functions evaluated at r = p, since we have assumed that / is continuous at the walls. However, the right-hand side does contain 5(r—p,). Therefore, Eq. (74) can only be satisfied if both the right- and left-hand sides separately vanish. We are thus led to the two equations 

Here we see that/ must satisfy (a) the Boltzmann equation without the gas-wall collision term for points in V, and (b) a condition that relates the distribution function for outgoing molecules at the wall to the distribution function for incoming molecules.  

Our procedure now is to construct the normal solution to Eq. (76a) for points in the interior and then to see if the solution also satisfies the boundary condition Eq. (76b). 

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A3.1.3.2 THE CHAPMAN-ENSKOG NORMAL SOLUTIONS OF THE BOLTZMANN EQUATION... 

In order to calculate the distribution function must be obtained in terms of local gas properties, electric and magnetic fields, etc, by direct solution of the Boltzmann equation. One such Boltzmann equation exists for each species in the gas, resulting in the need to solve many Boltzmann equations with as many unknowns. This is not possible in practice. Instead, a number of expressions are derived, using different simplifying assumptions and with varying degrees of vaUdity. A more complete discussion can be found in Reference 34. 

Boltzmann s H-Theorem. —One of the most striking features of transport theory is seen from the result that, although collisions are completely reversible phenomena (since they are based upon the reversible laws of mechanics), the solutions of the Boltzmann equation depict irreversible phenomena. This effect is most clearly seen from a consideration of Boltzmann s IZ-function, which will be discussed here for a gas in a uniform state (no dependence of the distribution function on position and no external forces) for simplicity. 

Hydrodynamic Equations.—Before deriving the hydro-dynamic equations, some integral theorems that are useful in the solution of the Boltzmann equation will be proved. Consider a function of velocity, G(Vx), which may also be a function of position and time let... 

The Burnett Expansion.—The Chapman-Enskog solution of the Boltzmann equation can be most easily developed through an expansion procedure due to Burnett.15 For the distribution function of a system that is close to equilibrium, we may use as a zeroth approximation a local equilibrium distribution function given by the maxwellian form ... 

Chapman-Enskog Solution.—The solution of the Boltzmann equation obtained by Chapman and Enskog involves the assumption... 

Since the middle of the 1990s, another computation method, direct simulation Monte Carlo (DSMC), has been employed in analysis of ultra-thin film gas lubrication problems [13-15]. DSMC is a particle-based simulation scheme suitable to treat rarefied gas flow problems. It was introduced by Bird [16] in the 1970s. It has been proven that a DSMC solution is an equivalent solution of the Boltzmann equation, and the method has been effectively used to solve gas flow problems in aerospace engineering. However, a disadvantageous feature of DSMC is heavy time consumption in computing, compared with the approach by solving the slip-flow or F-K models. This limits its application to two- or three-dimensional gas flow problems in microscale. In the... 

The early theories for the transport coefficients were based on the concept of the mean free path. Excellent summaries of these older theories and their later modifications are to be found in standard text books on kinetic theory (J2, K2). The mean-free-path theories, while still very useful from a pedagogical standpoint, have to a large extent been supplanted by the rigorous mathematical theory of nonuniform gases, which is based on the solution of the Boltzmann equation. This theory is... 

Consider a mixture of acoustic-mode (rL) and ionized-impurity (r,) scattering. For tL t, we would expect r 0 = 1.18 and for r, tl, rn0 = 1.93. But for intermediate mixtures, r 0 goes through a minimum value, dropping to about 1.05 at 15% ionized-impurity scattering (Nam, 1980). For this special case (sL = i, s, = — f), the integrals can be evaluated in terms of tabulated functions (Bube, 1974). For optical-mode scattering the relaxation-time approach is not valid, at least below the Debye temperature, but rn may still be obtained by such theoretical methods as a variational calculation (Ehrenreich, 1960 Nag, 1980) or an iterative solution of the Boltzmann equation (Rode, 1970), and typically varies between 1.0 and 1.4 as a function of temperature (Stillman et al., 1970 Debney and Jay, 1980). 

Solution of the Boltzmann equation gives the velocity distribution function throughout the gas as it evolves through time, for example, due to velocity, temperature, or concentration gradients. A practical solution to the Boltzmann equation was found by Enskog [114], which is discussed in the next section. This approach is used to calculate rigorous expressions for gas transport coefficients. 

Plasmas typical of C02 laser discharges operate over a pressure range from 1 Torr to several atmospheres with degrees of ionization, that is, nJN (the ratio of electron density to neutral density) in the range from 10-8 to 10-8. Under these conditions the electron energy distribution function is highly non-Maxwellian. As a consequence it is necessary to solve the Boltzmann transport equation based on a detailed knowledge of the electron collisional channels in order to establish the electron distribution function as a function of the ratio of the electric field to the neutral gas density, E/N, and species concentration. Development of the fundamental techniques for solution of the Boltzmann equation are presented in detail by Shkarofsky, Johnston, and Bachynski [44] and Holstein [45]. 

The equations for conservation of mass, momentum, and energy for a one-component continuum are well known and are derived in standard treatises on fluid mechanics [l]-[3]. On the other hand, the conservation equations for reacting, multicomponent gas mixtures are generally obtained as the equations of change for the summational invariants arising in the solution of the Boltzmann equation (see Appendix D and [4] and [5]), One of several exceptions to the last statement is the analysis of von Karman [6], whose results are quoted in [7] and are extended in a more recent publication [8] to a point where the equivalence of the continuum-theory and kinetic-theory results becomes apparent [9]. This appendix is based on material in [8]. 

Muckenfuss, C., Stefan-Maxwell Relations for Multicomponent Diffusion and the Chapman Enskog Solution of the Boltzmann Equations, J. Chem. Phys., 59, 1747-1752 (1973). 

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. 

The rigorous Fickian multicomponent mass diffusion flux formulation is derived from kinetic theory of dilute gases adopting the Enskog solution of the Boltzmann equation (e.g., [17] [18] [19] [89] [5]). This mass flux is defined by the relation given in the last line of (2.281) ... 

That is, instead of determining the transport properties from the rather theoretical Enskog solution of the Boltzmann equation, for practical applications we may often resort to the much simpler but still fairly accurate mean free path approach (e.g., [12], section 5.1 [87], chap. 20 [34], section 9.6). Actually, the form of the relations resulting from the mean free path concept are about the same as those obtained from the much more complex theories, and even the values of the prefactors are considered sufficiently accurate for many reactor modeling applications. 

The kd (v) rate coefficients have been obtained by using the cross sections of Fig. 12 and the non-Maxwellian electron distribution functions of Fig. 13. The edfs have been obtained by a numerical solution of the Boltzmann equation (BE) which includes the superelastic vibrational collisions involving the first three vibrational levels, and the dissociation process from all vibrational levels (see Ref.9) for details). The vibrational population densities inserted in the BE are self-consistent with the quasi-stationary values reported in Figs. 8 and 10. It should be noted that the DEM rates (Fig. 14) depend on E/N as well as on the vibrational non equilibrium present in the discharge, which affects the electron distribution functions, as discussed in Sect. 2.1. 

It can be shown that if one assumes that only elastic collisions occur one can obtain an approximate solution of the Boltzmann equation for the distribution of speeds. The result is not the Maxwell-Boltzmann dis-... 

The extension of the kinetic theory approach to include large values of a (and hence large deviations from equilibrium) requires higher order perturbations for the solution of the Boltzmann equation. It is probably unprofitable to proceed in this difficult and laborious direction until one understands the detailed analytical dependence of the transition probability a on the mechanism of molecular energy exchange and redistribution on collision. Currently available information on intermolecular forces is insufficient to establish this dependence. 

If the exciton-phonon coupling is sufficiently weak, the solution of the equation for the correlation function (B (t)Bm(t)B, (0)Bm (0)) is equivalent to the solution of the Boltzmann equation (11). In this coherent limit, the exciton states of... 

The Chapman-Enskog solution of the Boltzmann equation [112] leads to the following expressions for the transport coefficients. The viscosity of a pure, monatomic gas can be written as... 

Taking the natural logarithm of (A3.1.54), we see that In/j + In has to be conserved for an equilibrium solution of the Boltzmann equation. Therefore, In/j can generally be expressed as a linear combination with constant coefficients... 

Sitarski, M., and Nowakowski, B. (1979) Condensation rate of trace vapor on Kundsen aerosols from solution of the Boltzmann equation, J. Colloid Interface Sci. 72, 113-122. 

It is known that an exact description of transfer processes in the aerosol particles-gas phase system with chemical or phase transformations on the particle surface for arbitrary particle sizes (and correspondingly for arbitrary Knudsen numbers) can be found only by solving the Boltzmann kinetic equation. However, the mathematical difficulties associated with the solution of the given equation lead to the necessity of obtaining rather simple expressions for mass and energy fluxes either on the basis of an approximate solution of the Boltzmann equation or with the use of simpler models. In particular, it is known that the use of the diffusion equation with appropriate boundary conditions on the particle surface leads to the equation that gives correct limiting cases with respect to the Knudsen number [2]. 

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