Collision operators equation

Multiparticle collision dynamics basic principles, 92-93 collision operators and evolution equations, 97-99... 

In the LB technique, the fluid to be simulated consists of a large set of fictitious particles. Essentially, the LB technique boils down to tracking a collection of these fictitious particles residing on a regular lattice. A typical lattice that is commonly used for the effective simulation of the NS equations (Somers, 1993) is a 3-D projection of a 4-D face-centred hypercube. This projected lattice has 18 velocity directions. Every time step, the particles move synchronously along these directions to neighboring lattice sites where they collide. The collisions at the lattice sites have to conserve mass and momentum and obey the so-called collision operator comprising a set of collision rules. The characteristic features of the LB technique are the distribution of particle densities over the various directions, the lattice velocities, and the collision rules. 

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... 

This equation is readily transformed to an integral equation for different from i and in <— k,- Y(z] — k )) never appear in two successive collision operators because otherwise we would get a negligible contribution in the limit of an infinite system moreover as these dummy particles have zero wave vectors in the initial state, they have a Maxwellian distribution of velocities (see Eq. (418)). This allows us to write Eq. (A.74) in the compact form ... 

Having obtained two simultaneous equations for the singlet and doublet correlation functions, X and, these have to be solved. Furthermore, Kapral has pointed out that these correlations do not contain any spatial dependence at equilibrium because the direct and indirect correlations of position in an equilibrium fluid (static structures) have not been included into the psuedo-Liouville collision operators, T, [285]. Ignoring this point, Kapral then transformed the equation for the singlet density, by means of a Laplace transformation, which removes the time derivative from the equation. Using z as the Laplace transform parameter to avoid confusion with S as the solvent index, gives... 

In this equation, G(t) is a generalized collision operator defined formally in terms of all irreducible transitions from vacuum of correlations to vacuum of correlations. A fundamental role is played by the Laplace transform /(z) of G(t). 

We would like to emphasize a third aspect. Each term in the collision operator now contains the correct 8-function expressing conservation of energy. It is, therefore, in terms of these equations that we may describe in a simple way the production and destruction processes involving unstable particles or excited states. It is interesting to note that while in the usual dynamical picture excited states are broad ( uncertainty principle ... 

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

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 main challenge in formulating these equations is related to the definition of the collision operator. So far this approach has been restricted to the formulation of the population balance equation. That is, in most cases a general transport equation which is complemented with postulated source term formulations for the particle behavior is used. Randolph [80] and Randolph and Larson [81] used this approach deriving a microscopic population balance equation for the purpose of describing the behavior of particulate systems. Ramkrishna [79] provides further details on this approach considering also fluid particle systems. 

Note that left-hand side of this expression is, in fact, a continuity equation for which states that the multi-particle joint PDF is constant along trajectories in phase space. The term on the right-hand side of Fq. (4.32) has a contribution due to the Alp-particle collision operator, which generates discontinuous changes in particle velocities Up" and internal coordinates p", and to particle nucleation or evaporation. The first term on the left-hand side is accumulation of The remaining terms on the left-hand side represent... 

The evolution operator is the composition of the propagation and collision operators E - C S. The entire updating of the system can be described by the following equation ... 

The form of the Boltzmann-Enskog collision operator is thus specified out task is to find its generalization. We denote the general collision operator by /l (1, 1 t ), where we have allowed for the possibility that it may be nonlocal in time as well as space. The general kinetic equation may then be written as... 

Analogous BGK models can be constructed at the pair kinetic equation level. The simplest model of this type would approximate the collision operator in (7.32) as... 

Simpler BGK kinetic theory models have, however, been applied to the study of isomerization dynamics. The solutions to the kinetic equation have been carried out either by expansions in eigenfunctions of the BGK collision operator (these are similar in spirit to the discussion in Section IX.B) or by stochastic simulation of the kinetic equation. The stochastic trajectory simulation of the BGK kinetic equation involves the calculation of the trajectories of an ensemble of particles as in the Brownian dynamics method described earlier. 

An external force F can be included in the above LBM algorithm by adding an extra term to the collision operator (RHS of Eq. 1) and the lattice Boltzmann equation becomes... 

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. 

We observe that the left-hand side of this equation is, roughly speaking, characterized by the time scale t, and length scale l, over which the quantities n, u, and T change with time and position and the external potential changes with r. The collision operator appearing on the right-hand side of Eq. (85) is characterized, on the other hand, by the collision scales tc and Ic. Thus, the... 

To avoid the difficulties in solving Eq. (138b) due to the complicated spectrum of L, it has been suggested that L in the equation be replaced by a model integral operator that has most of the properties of the actual collision operator L but that has a simple spectrum. The most commonly used model for L is the BGK rnodel given [when V(r) = 0] by... 

We would like to prove that the initial state is forgotten, and that the collision operators reach an asymptotic form for times long compared to some microscopic time. If this were true, then the normal solution method, when applied to the generalized Boltzmann equation, would lead to expressions for the transport coefficients for a dense gas that would (a) be independent of the precise initial state of the gas, (b) be independent of the time elapsed since the initial state of the gas, and (c) have a density expansion of the form... 

We consider first Oxi tixux x-, the three-particle collision operator. H Because of the presence of the 12 operator, we are again only interested, for the generalized Boltzmann equation, in phases Xx and X2 of particles 1 and 2, where ri2 <. For these phases of particles 1 and 2, a dynamical analysis of the operator t(xi, X2IX3) similar to that given for tixxi X2) shows that the following dynamical events contribute to this collision operator. ... 

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