Kinetic Theory Boltzmans Equation

Let us now shift our focus and consider the bottom-up , or kinetic theory, approach to fluid dynamics. Kinetic theory describes fluids by assuming that they are made up of a large number of individual atoms or molecules, each subject to the laws of 

Newtonian inochaiiics. Since it is impossible to completely specify the initial state of such a system, kinetic theory (contents itself with describing a smoothed version of the system. The smoothed version is simply one whore all exact state-information below some characteristic length and time is replaced by a probabilistic description. 

To be more precise, let us assume, as Boltzman first did in 1872 [boltz72], that we have N perfectly elastic billiard balls, or hard-spheres, inside a volume V, and that a complete statistical description of our system (be it a gas or fluid) at, or near, its equilibrium state is contained in the one-particle phase-space distribution function f x,v,t)  

Integrating ovtir all of phase space must give us back the total number of hard-spheres in our system J f x, v, t) (Px (Pv = N. 

The goal of kinetic theory is to find the time-evolution equation for f x,v,t), with the ultimate aim being to obtain the equilibrium properties of the system through the limiting form of f x, v,t) as t cx . 

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This balance equation can also be derived from kinetic theory [101], In the Maxwellian average Boltzman equation for the species s type of molecules, the collision operator does not vanish because the momentum mgCs is not an invariant quantity. Rigorous determination of the collision operator in this balance equation is hardly possible, thus an appropriate model closure for the diffusive force is required. Maxwell [65] proposed a model for the diffusive force based on the principles of kinetic theory of dilute gases. The dilute gas kinetic theory result of Maxwell [65] is generally assumed to be an acceptable form for dense gases and liquids as well, although for these mixtures the binary diffusion coefficient is a concentration dependent, experimentally determined empirical parameter. 

In spite of the difference in the underlying concepts and the forms of equations, Eqs. (3.3) and (3.4), both descriptions reflect the statistical sense of the rate constant. The latter statement is crucially important for better understanding of the problem existing in heterogeneous kinetics. Indeed, the above-mentioned theories are based on gas statistics and the given equations assume an equilibrium Maxwell-Boltzman distribution for gas species, which in the absence of reaction interact only via elastic collisions. If this can be considered as a satisfactory approximation for gas reactions at moderate temperatures and pressures discussed here (with some exceptions—see Section III.D), its applicability to the processes involving surface sites (i.e., elements of solid lattice) or adsorbed species is not so obvious. 

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