Binary collision operators

The pseudo-Liouville operator for hard-sphere interactions can be written directly in terms of the binary collision operator and this leads to considerable simplifications in the formalism. It is not difficult to treat more general interactions, but a considerable number of manipulations must be carried out to express the results in terms of generalized binary collision operators. To avoid these difficulties, we make use of hard-sphere interactions whenever no violence is done to the qualitative features of the effects we are studying. 

For themial unimolecular reactions with bimolecular collisional activation steps and for bimolecular reactions, more specifically one takes the limit of tire time evolution operator for - co and t —> + co to describe isolated binary collision events. The corresponding matrix representation of f)is called the scattering matrix or S-matrix with matrix elements... 

In a similar spirit, Inoue et al. [120] and Hashimoto et al. [121] generalized MPC dynamics so that the collision operator reflects the species compositions in the neighborhood of a chosen cell. More specifically, consider a binary mixture of particles with different colors. The color of particle i is denoted by c,-. The color flux of particles with color c in cell E, is defined as... 

B. Binary Density Operator in Three-Particle Collision Approximation— Boltzmann Equation for Nonideal Gases... 

B. Binary Collision Approximation for the Two-Particle Density Operator— Kinetic Equations for Free Particles and Atoms... 

As mentioned in Section 2.1, the usual Boltzmann equation conserves the kinetic energy only. In this sense the Boltzmann equation is referred to as an equation for ideal systems. For nonideal systems we will show that the binary density operator, in the three-particle collision approximation, provides for an energy conservation up to the next-higher order in the density (second virial coefficient). For this reason we consider the time derivative of the mean value of the kinetic energy,12 16 17... 

Now we want to generalize the kinetic equation for free (unbound) particles that is, we want to derive a kinetic equation for free particles that takes into account collisions between free and bound particles as well. For this purpose it is necessary to determine the binary density operator, occurring in the collision integral of the single-particle kinetic equation, at least in the three-particle collision approximation. An approximation of such type was given in Section II.2 for systems without bound states. Thus we have to generalize, for example, the approximation for/12 given by Eq. (2.40), to systems with bound states. 

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. 

The contributions to the collision operator A (l z) describe the following types of dynamic event the A operators are Enskog collision operators and describe uncorrelated binary collision events describes uncorrelated elastic collisions of A with solvent molecules... 

Fig. 24. Some of the four successive binary collision events taking place among four particles (four-body ring events) that are responsible for the divergence of the four-body collision operator.

Since the leading divergences in the generalized Boltzmann equation are associated with sequences of binary collisions, the resummations are usually carried out by first expressing the 5-particle streaming operators 5, (xi,..., x ) in Eq. (208) in terms of sequences of binary collisions that take place between the s particles. This is accomplished by means of the binary collision expansion, which has proven to be one of the most useful tools in the kinetic theory of gases. " ... 

Effective collision cross sections are related to the reduced matrix elements of the linearized collision operator It and incorporate all of the information about the binary molecular interactions, and therefore, about the intermolecular potential. Effective collision cross sections represent the collisional coupling between microscopic tensor polarizations which depend in general upon the reduced peculiar velocity C and the rotational angular momentum j. The meaning of the indices p, p q, q s, s and t, t is the same as already introduced for the basis tensors In the two-flux approach only cross sections of equal rank in velocity (p = p ) and zero rank in angular momentum (q = q = 0) enter die description of the traditional transport properties. Such cross sections are defined by... 

The effective cross section 6(p0j/ 9) exactly the same as the cross section (pOst) introduced in Section 4.2 for a pure gas. All of the other cross sections introduced depend upon the binary interaction of two molecular species alone, as is made clear by the definition of the Wang Chang-Uhlenbeck or Boltzmann collision operators. From the point of view of the present volume this is a very important result because it means that if the collision cross sections can be determined for each binary interaction, then it becomes possible to evaluate the contribution of that interaction to the transport properties of a multicomponent mixture immediately. If repeated for all possible binary interactions in the mixture then the prediction of the transport properties of any multicomponent mixture containing them is possible. Furthermore, if the effective cross sections for the pure components of a binary mixture are known then those characteristic of the single, unlike binary interaction may be deduced from the properties of a mixture. It is worthy of note that this result remains valid even if higher orders of kinetic theory approximation are invoked. 

Following the approach of Jenkins and Savage [68], inserting ci, r + di2k, C2, t) from (4.19) into (4.15), adding the intermediate result to (4.16) and divide by 2, Lun et al. [102] obtained a symmetric approximation of the binary solid particle collision operator ... 

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