Kinetic equation collision integrals

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. 

We will carry out our program in two steps. In this section we will derive the two-particle density operator Fn in a three-particle collision approximation for the application in the collision integral of Fl. As compared with Section II.2, the main difference will be the occurrence of bound states and, especially, the generalization of the asymptotic condition, which now has to account for bound states too. For the purpose of the application in the kinetic equation of the atoms (bound states) we need an approximation of the next-higher-order density matrix, that is, F 23 This quantity will be determined under inclusion of certain four-particle interaction. 

In order to construct a collision integral for a bound-state kinetic equation (kinetic equation for atoms, consisting of elementary particles), which accounts for the scattering between atoms and between atoms and free particles, it is necessary to determine the three-particle density operator in four-particle approximation. Four-particle collision approximation means that in the formal solution, for example, (1.30), for F 234 the integral term is neglected. Then we obtain the expression... 

Similarly, we may discuss the properties of the kinetic equations for atoms. For example, the collision integral hnab of Eq. (3.68) contains the following contributions for k = 3 ... 

The collision integrals of the kinetic equations (3.68) and (3.69) are essentially given by the scattering matrix ( T ). For this reason, the determination of such quantities from the Lippmann-Schwinger equations... 

Let us now consider the atomic kinetic equation (4.70). There are also two contributions to the collision integral I (Pt), the first one leaves the incident atom unchanged. The relevant processes are... 

According to (4.38), the correlation function (5.4) determines the collision integral of the relevant kinetic equation for the distribution of pairs. 

As a consequence of such additional correlations, we have an additional contribution to the collision integral, la, for the kinetic equation for fa. Now, the kinetic equation has the structure... 

The basis for the semiclassical description of kinetics is the existence of two well separated time scales, one of which describes a slow classical evolution of the system and the other describes fast quantum processes. For example, the collision integral in the Boltzmann equation may be written as local in time because quantum-mechanical scattering is assumed to be fast as compared to the evolution of the distribution function. 

The term isothermal was previously used in terms of the model of kinetic equations applied to free motion of the particles between strong collisions [18, p. 126 65], In this particular case the collision integral St(/) of the Bhatnagar-Gross-Krook (BGK) model is found for T (q,t) 7 const. 

Equation (7.54) holds at the equilibrium and when the initial NDF is isotropic. In order to illustrate how the discretization for the kinetic equation is typically carried out, some additional simplifications must be introduced. It is convenient to write the collision integral as... 

The kinetic equation for homogeneous systems is given by Eq. (7.47). The evolution equation for the zeroth-order moment of the NDF is null, which is due to the fact that the collision integral does not change the number of particles, or, more explicitly, f Cdf = 0. If the rate of change of the particle velocity (i.e. particle acceleration) is a linear function of the particle velocity (i.e. f = a + b ), then the evolution equation for the first-order moments are... 

In deriving the equation system, it is commonly assumed with respect to the collision integrals that the atoms or molecules are at rest before the collision events. Furthermore, each collision integral is additionally expanded with respect to the mass ratio nig/M, and only the leading term with regard to m. /M of each collision integral has been taken into account in each coefficient of the Legendre polynomial expansion of the kinetic equation. 

Consider a polydisperse emulsion, assuming a spatially homogeneous case, with low volume concentrations of the disperse phase. Assume further that it is possible to limit ourselves to consideration of pair interactions of drops. The dynamics of enlargement (integration) of drops due to their collision and coalescence is then described by the following kinetic equation... 

Usually, the so-called model kinetic equations are applied in practical calculations. They maintain the main properties of the exact collision integral and, at the same time, they reduce significantly the computational efforts. The most usual model kinetic equation was proposed by Bhatnagar, Gross, and Krook (BGK) which reads [4]... 

Model kinetic equation is the Boltzmann equation with a simplified form of the collision integral. 

By way of illustration, we note that in the recombination problem mentioned above the energies of the electrons are the variables in the diffusion equation, the bottleneck is the region of energies near the boundary of the continuous spectrum, and the slowness of the process is related to the small amount of energy transfer from an electron to a heavy particle in one collision. In the problem of escaping electrons in a plasma, slowness is ensured by the weakness of the electric field, the independent variable in the diffusion equation is the momentum component along the field, and the bottleneck is determined, as in the kinetics of new phase formation, by the saddle point of the integral. 

Por the computation we have used the integral method using cubic spline and the combined gradient method of Levenberg-Marquardt [57, 58]. The kinetic models chosen describe well the hydrogenation kinetics. In the formulas presented in Table 3.1 k is the kinetic parameter of the reaction and Q takes into account the coordination (adsorption) of the product (LN) and substrate (DHL) with the catalyst (the ratio of the adsorption-desoprtion equilibrium constants for LN and DHL). Parameters of the Arrhenius equation, apparent activation energy kj mol , and frequency factor k, have been determined from the data on activities at different temperatures. The frequency factor is derived from the ordinate intercept of the Arrhenius dependence and provides a measure of the number of collisions or active centers on the surface of catalytic nanoparticles. 

This method involves the numerical integration of the equations of motion (F = ma) for each of the molecules, subject to intermolecular forces, in time. The molecules are positioned arbitrarily in a simulation cell, that is, a three-dimensional cube, with initial velocities also specified arbitrarily. Subsequently, the velocities are scaled so that the summation of the kinetic energies of the molecules, 3NkTI2, gives the specified temperature, T, where W is the number of molecules and k is the Boltzmann constant. Note that after many collisions with the walls and the other molecules, the relative positions and velocities of the molecules arc independent of the initial conditions. 

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