Linearized collision operator

The quantity 8Ia is the linearized collision operator corresponding to (4.62). The subscript 0 in the second term means that the operator acts only on the function fa and not on the delta distribution. The contribution Aap in (5.14) is determined by /a/3, that is by the large-scale fluctuations. We consider only the approximation that Aa/ is equal to zero. 

A q,co) spectral density of paramagnons Q inverse linear collision operator... 

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... 

Together with Eq. (66), this equation describes exactly the linear response of the system to an external field, with arbitrary initial conditions. Its physical meaning is very simple and may be explained precisely as for Eq. (66) 32 the evolution of the velocity distribution results in two effects (1) the dissipative collisions between the particles which are described by the same non-Markoffian collision operator G0o(T) 35 1 the field-free case and (2) the acceleration of the particles due to the external field. As we are interested in a linear theory, this acceleration only affects the zeroth-order distribution function It is... 

The collision operators C are bi-linear in the distribution functions, and the summation indices run over all charged species / and neutral species m in the plasma, respectively. 

Collision operators of this form keep the neutral particle equations linear, if they are considered separate (decoupled) from the charged particle equations. 

Collision is a local exchange of the linear momentum of the particles. The momentum exchange rules are chosen such that they conserve the total mass and linear momentum, like real elastic collisions. The collision operator may be written as... 

The quantities n, Tr, and ur are chosen to reduce the computational efforts. Particularly, they can be constant and equal to their equilibrium values. The operator L in its exact form has a cumbersome form that is why just the model linearized operators will be given. With the help of Eqs. 1, 11, 12, and 21, the linearized BGK collision operator takes the form... 

To solve numerically the linearized kinetic Eq. 24 with the boundary condition (35), a set of values of the velocity c, is chosen. The collision operator Lh is expressed via the values hi x) = h x,Ci). Thus, Eq. 24 is replaced by a system of differential equations for the functions hi x), which can be solved numerically by a finite difference method. First, some values are assumed for the moments being part of the collision operator. Then, the distribution function moments are calculated in accordance with Eqs. 30-34 using some quadrature. The differential equations are solved again with the new moments. The procedure is repeated up to the convergence. 

Can one determine the spectrum of the linearized Boltzmann collision operator L for the class of interaction potentials illustrated in Fig. 1 ... 

In these formulae, [A, B] (where A, B are arbitrary functions of molecular velocities) denotes a bilinear form depending on the linearized integral collision operator for rapid processes. In the kinetic theory, such bilinear forms are basically called bracket integrals. The bracket integrals in the expressions (40) are introduced in Nagnibeda Kustova (2009) similarly to those defined in Ferziger Kaper (1972) for a non-reacting gas mixture under the conditions of weak deviations from the equilibrium. 

A given matrix element of the collision operator will in general involve cross sections with several values of K, K, and K because the cross section is defined to exploit linear and angular momentum conservation, respectively. In Table 1, we give a listing of some bulk phenomena and the dominant cross sections which are required for their description. The cross sections listed for phenomena other than transport phenomena are characterized by K =K =0 and... 

Without essential limitation of generality it may be assumed that the orientation of the molecule and its angular momentum are changed by collision independently, therefore F(JU Ji+, gt) = f (Jt, Ji+i)ip(gi). At the same time the functions /(/ , Ji+ ) and xp(gi) have common variables. There are two reasons for this. First, it may be due to the fact that the angle between / and u must be conserved for linear rotators for any transformation. Second, a transformation T includes rotation of the reference system by an angle sufficient to combine axis z with vector /. After substitution of (A7.16) and (A7.14) into (A7.13), one has to integrate over those variables from the set g , which are not common with the arguments of the function / (/ , /j+i). As a result, in the MF operator T becomes the same for all i and depends on the moments of tp as parameters. 

Once the Fock operators have been constructed from a set of MSOs, this matrix equation is linear in its unknowns. Its coefficients are dependent on time in a way determined by the forces driving the electrons. These forces are the nuclear Coulomb potentials in molecular collisions or dynamics, but they could also be weak external fields. 

Linear momentum (L) operator, time reversal symmetry and, 243-244 Linear scaling, multiparticle collision dynamics, nonideal fluids, 137 Linear thermodynamics entropy production, 20-23 formalities, 8-11... 

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