Effective collision definition

Politano et al [88] adopted the kernel functionality of Luo and Svendsen but modified the definition of the effective collision cross-sectional area, aAT = in accordance with other work in nuclear engineering (e.g., Kolev [46], p 168). 

All of the effective collision cross sections introduced above can, in principle, be evaluated from a knowledge of the intermolecular pair potential by means of equation (4.17) and the definition of the collision operator (4.5). This process would then make it possible to predict the transport properties of dilute gases from first principles. However, such a procedure would require that it is possible to evaluate the inelastic differential scattering cross section ajj or an equivalent to it which enters (4.5). Until very recently this could only be accomplished for dilute monatomic gases. There were two reasons for this first, only for such systems are accurate intermolecular potentials available (Aziz 1984 Maitland et al. 1987 van der Avoird 1992) second, only for such systems was the... 

This is an indication of the collective nature of the effect. Although collisions between hard spheres are instantaneous the model itself is not binary. Very careful analysis of the free-path distribution has been undertaken in an excellent old work [74], It showed quite definite although small deviations from Poissonian statistics not only in solids, but also in a liquid hard-sphere system. The mean free-path X is used as a scaling length to make a dimensionless free-path distribution, Xp, as a function of a free-path length r/X. In the zero-density limit this is an ideal exponential function (Ap)o- In a one-dimensional system this is an exact result, i.e., Xp/(Xp)0 = 1 at any density. In two dimensions the dense-fluid scaled free-path distributions agree quite well with each other, but not so well with the zero-density scaled distribution, which is represented by a horizontal line (Fig. 1.21(a)). The maximum deviation is about... 

For a flying height around 2 nm, collisions between the molecules and boundary have a strong influence on the gas behavior and lead to an invalidity of the customary definition of the gas mean free path. This influence is called a "nanoscale effect" [46] and will be discussed more specifically in Chapter 6. 

On the contrary, the definition of the collision process, Eq. (429), is such that through a sequence of such events, the perturbation caused by the external force may be propagated at long distances. For instance, in the diagram of Fig. 21, corresponding to a typical term of the iterative solution of Eq. (428), the T operators are not localized around the B-particle. This allows long-range hydrodynamical effects. 

The physical meaning of the constant s in equation (15) is less definite than the meaning of E. In bimolecular reactions, which demand a collision between two molecules before reaction can occur, s refers to the number of molecules colliding, and, as just explained, e E,RT is the fraction of molecules which are activated. The rate constant, k, when expressed in the proper units, is then equal to the number of activated molecules per unit volume colliding. This assumption that every collision of an activated molecule leads to reaction is valid only in a few cases and it is necessary to put another factor, a, into the equation to allow for a steric effect of some kind. Perhaps the activated molecules have to be orientated in a definite way when they collide, in order that reaction may occur. Equation (15) then becomes... 

The Effect of Collisions on the Entropy.—Equation (1.6) represents the first part of our derivation of the effect of collisions in producing an irreversible change in the distribution and hence in increasing the entropy. Now we must go back to the definition of the entropy in Eq. (2.12) in Chap. V, and find how much S changes per unit time on account of the collisions. Differentiating that equation with respect to the time, we have at once... 

Temperature has a definite effect on reaction rate, but the reasons for the changes are not completely understood. The two theories that describe this relationship are the collision... 

This equation appears to predict that Z i2 will be a function of the composition of the gas, although experimentally the diffusion constant is almost independent of composition. However, we must be careful in our definition of Aj and A2. We must take account of the fact that collisions of molecules of one species with one another can have no significant effect on the diffusion such collisions do not affect the total momentum possessed by all the molecules of that species and thus do not affect the mean mass velocity of the species in its diffusion. Thus the total number of molecules of that species crossing the reference plane in a given period of time is not affected by such collisions and is the same as if such... 

It is easily seen by inspection that the biorthogonal basis set definition (3.55) cmnddes with the definifion (3.18) ven in the discussion of the Lanczos method. We recall that the dynamics of operators (liouville equations) or probabilities (Fokker-Planck equations) have a mathematical structure similar to Eq. (3.29) and can thus be treated with the same techniques (see, e.g., Chapter 1) once an appropriate generalization of a scalar product is performed. For instance, this same formalism has been successfully adopted to model phonon thermal baths and to include, in principle, anharmonicity effects in the interesting aspects of lattice dynamics and atom-solid collisions. ... 

From a very general point of view every ion-atom collision system has to be treated as a correlated many-body time-dependent quantum system. To solve this from an ab initio point of view is still impossible. So, one has to rely on various approximations. Nowadays the best method which can be applied to realistic collision systems (which we discuss here) is on the level of the non-selfconsistent time-dependent Hartree-Fock-Slater or, in the relativistic case, the Dirac-Fock-Slater method. Up-to-now no correlation beyond this approximation can be taken into account in the case of 3 or more electrons. (This is in accordance with the definition of correlation given by Lowdin [1] in 1956) In addition no QED contributions, i.e. no correction to the 1/r Coulomb interaction between the electrons, ever have been taken into account, although in very heavy collision systems this effect may become important. This will be discussed in section 5. A short survey of the theory used is followed by our results on impact parameter dependent electron transfer and excitation calculations of ion-atom and ion-solid collisions as well as first results of an ab initio calculation of MO X-rays in such complicated many particle scattering systems. 

Integration of Eq. (10.28) along the cross-section of the hydrodynamic layer allows us to check whether within its limits the radial velocity component is proportional to the tangential derivative of the velocity distribution along the bubble surface, which differs slightly from the potential distribution. The effect of a boundary layer on the normal velocity component and on inertia-free deposition of particles should be therefore very small. The formula for the collision efficiency given by Mileva as an inertia-free approximation is thus VRc times less than the collision efficiency according to Sutherland, which is definitely erroneous. 

The correlated collision term contains several effects, which are exposed by using the definition of / Aas (7.34)],... 

First, there are terms of the form <5S(12)[z —0L (12 z)] 5S(12)>o. From its definition in (9.12), we see that 55(12) is the deviation of the reactive operator from its velocity average. The correlation function above characterizes the time evolution (Laplace transformed) of these fluctuations. If the chemical reaction is slow, we expect that perturbations of the velocity distribution induced by the reaction will be small hence such contributions may be safely neglected in this limit. This argument may be made more formal using limiting procedures analogous to those described in Section V. In principle, one may also use this term to introduce a modification to in S(/- 2) due to velocity relaxation effects. This will lead to some effective reactive collision frequency in place of k p. 

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