Boltzmann collision terms

The first term in this expansion describes the binary collision and leads, as we have shown, to the Boltzmann collision integral. Among the terms of the order n, the first describes the three-particle collision and has, as will be shown, the same structure as the Boltzmann collision term. 

The two-particle Boltzmann collision term if and the three-particle contribution for k = 0 were considered in Section II. It was possible to express those collision integrals in terms of the two- and three-particle scattering matrices. It is also possible to introduce the T matrix in if for the channels k = 1, 2,3, that is, in those cases where three are asymptotically bound states. Here we use the multichannel scattering theory, as outlined in Refs. 9 and 26. 

Since /S Tj0) = , its integral over the collision term is zero (conservation of momentum in a collision). Thus the result of multiplying the Boltzmann equation by and integrating is ... 

Consider electrons of mass m and velocity v, and atoms of mass M and velocity V we have mjM 1. The distribution function for the electrons will be denoted by /(v,<) (we assume no space dependence) that for the atoms, F( V), assumed Maxwellian as usual, in the collision integral, unprimed quantities refer to values before collision, while primed quantities are the values after collision. In general, we would have three Boltzmann equations (one each for the electrons, ions, and neutrals), each containing three collision terms (one for self-collisions, and one each for collisions with the other two species). We are interested only in the equation for the electron distribution function by the assumption of slight ionization, we neglect the electron-electron... 

The most recent effort in this direction is the work of Cohen,8 who established a systematic generalization of the Boltzmann equation. This author obtained the explicit forms of the two-, three-, and four-particle collision terms. His approach is formally very similar to the cluster expansion of Mayer in the equilibrium case. 

Choh and Uhlenbeck6 developed Bogolubov s ideas and extended his formal results. They established a generalized Boltzmann equation which takes account of three-particle collisions. The extension of their results to higher orders in the concentration poses no problem in principle, but it appears difficult, in this formalism, to write a priori the collision term with an arbitrary number of particles. 

The collision term for four particles in Cohen s version and the general Boltzmann operator for n = 4 in the Prigogine formalism (Section VC). 

The dominant term in the Boltzmann Equation (84) is assumed to be the collision term, i.e., the second term on the right-hand side of (84). This then implies that, as t —> 00, solutions to Equation (82), denoted 11l0 (r. v p. u, e) represent a good approximation to the asymptotic solution to (84). Consequently, we choose... 

The osmotic pressure between two flat surfaces can be derived within the Poisson-Boltzmann approximation. The PB equation was originally developed to describe ion distributions outside a large charged surface. However, there are extended PB equations where polymers have been included [32]. The expression for the osmotic pressure given below is valid in the absence of polymers. In the PB equation the correlations between ions are neglected, which means that Pei is identically zero. Furthermore, the ions are normally treated as point particles which means that the collision term disappears. Thus for symmetric systems only two terms remain, the kinetic pressure and the bulk pressure. The net pressure can be written as... 

However, the intermolecular force laws play a central role in the model determining the molecular interaction terms (i.e., related to the collision term on the RHS of the Boltzmann equation). Classical kinetic theory proceeds on the assumption that this law has been separately established, either empirically or from quantum theory. The force of interaction between two molecules is related to the potential energy as expressed by... 

Therefore, in the limit of no molecular interactions, for which the collision term ( )coiiision vanishes, the Boltzmann equation yields... 

This relation enable us to simplify the formulation of the general equation of change considerably. Fortunately, the fundamental fluid dynamic conservation equations of continuity, momentum, and energy are thus derived from the Boltzmann equation without actually determining the form of either the collision term or the distribution function /. 

The conservation equations (2.202), (2.207) and (2.213) are rigorous (i.e., for mono-atomic gases) consequences of the Boltzmann equation (2.185). It is important to note that we have derived the governing conservation equations without knowing the exact form of the collision term, the only requirement is that we are considering summation invariant properties of mono-atomic gases. That is, we are considering properties that are conserved in molecular collisions. 

The flrst step in the Enskog expansion is to introduce a perturbation parameter e into the Boltzmann equation to enforce a state of equilibrium flow as the gas is dominated by a large collision term ... 

The starting point for the kinetic theory of dilute mono-atomic gases is the Boltzmann equation determining the evolution of the distribution function in time and space. The formulation of the collision term is restricted to gases that are sufficiently dilute so that only binary collisions need to be taken into account. It is also required that the molecular dimensions are small in comparison with the mean distance between the molecules, hence the transfer of molecular properties is solely regarded as a consequence of the free motion of molecules between collisions. 

T source term in generalized Boltzmann type of equation representing the effects of particle coalescence, breakage and collisions J c)) collision term in the Boltzmann equation... 

Hydrodynamic models are derived from the mesoscale model (e.g. the Boltzmann equation) using a Chapman-Enskog expansion in powers of the Knudsen number (Bardos et al., 1991 Cercignani et al, 1994 Chapman Cowling, 1961 Ferziger Kaper, 1972 Jenkins Mancini, 1989). The basic idea is that the collision term will drive the velocity distribution n towards an equilibrium function eq (i-e. the solution to C( eq) = 0), and thus the deviation from equilibrium can be approximated by n -i- Knui. From the... 

The rest of this chapter is organized as follows. First, in Section 6.1, we consider the collision term for monodisperse hard-sphere collisions both for elastic and for inelastic particles. We introduce the kinetic closures due to Boltzmann (1872) and Enksog (1921) for the pair correlation function, and then derive the exact source terms for the velocity moments of arbitrary order and then for integer moments. Second, in Section 6.2, we consider the exact source terms for polydisperse hard-sphere collisions, deriving exact expressions for arbitrary and integer-order moments. Next, in Section 6.3, we consider simplified kinetic models for monodisperse and polydisperse systems that are derived from the exact collision source terms, and discuss their properties vis-d-vis the hard-sphere collision models. In Section 6.4, we discuss properties of the moment-transport equations derived from Eq. (6.1) with the hard-sphere collision models. Finally, in Section 6.5 we briefly describe how quadrature-based moment methods are applied to close the collision source terms for the velocity moments. 

Note that the transport term on the left-hand side of Eq. (6.1) can be larger or smaller in magnitude than the collision term. For cases in which the collision term is much more important than the transport term, the solution to Eq. (6.1) with the Boltzmann collision model is a local Maxwellian wherein ap. Up, and p depend on space and time but / is well approximated by Eq. (6.10). In this limit, the particles behave as an ideal gas and the mean velocity obeys the Euler equation. 

For elastic collisions, several different kinetic models have been proposed in order to close the Boltzmann hard-sphere collision term (Eq. 6.9). For inelastic collisions (e < 1), one must correctly account for the dependence of the dissipation of granular energy on the value of e. One method for accomplishing this task is to start from the exact (unclosed) collision integral in Eq. (6.68). From the definition of if given in Eq. (6.60), it can be... 

In LBM, a fluid is modeled as fictitious particles moving in a lattice domain at discrete time steps. The major variable in LBM is the density distribution fi x, t), indicating the quantity of particles moving along the /-th lattice direction at position X and time t. The time evolution of density distributions is governed by the so-called lattice Boltzmann equation with a BGK collision term [1, 2] ... 

The simplified Boltzmann equation can be solved using the Lattice Boltzmann Method (LBM) for the distributed function on a regular lattice. LBM considers each lattice structure as a volume element that consists of a collection of particles in the fluid. This simplified Boltzmann equation approximates the collision term, Q f, / ), in Eq. 37 using a relaxation time, t, providing a linear correlation. The most well-known form of the LBM is the BGKLBM, where the relaxation time is a constant. 

Any solution about Boltzmann equation needs an expression for collision term Q(f). The complexity of it, carry the search of simple models of collision processes, it will permit to make easy the mathematical analysis. Perhaps collision model more known was suggested simultaneously by Bhatnagar, Gross and Krook (Bhatnagar et al., 1954) and it is known like BGK ... 

Where i) is an adjusted parameter and f (r,v,t) is the local thermodynamic equilibrium of distribution. This simplification is called the "single-time-relaxation" approximation, already that the absence of space dependence, is implicating that f (t)-i f (v) the exponential manner in time like exp (-vt). This model keeps important properties of collision term in Boltzmann equation. For example, satisfies theorem H and obeys the laws of mass. 

The ordinary kinetics theory of neuter gas, the Boltzmann equation is considered with collision term for binary collisions and is despised the body s force F . This simplified Boltzmann equations is an integro - differential non lineal equation, and its solution is very complicated for solve practical problems of fluids. However, Boltzmann equation is used in two important aspects of dynamic fluids. First the fundamental mechanic fluids equation of point of view microscopic can be derivate of Boltzmann equation. By a first approximation could obtain the Navier-Stokes equations starting from Boltzmann equation. The second the Boltzmann equation can bring information about transport coefficient, like viscosity, diffusion and thermal conductivity coefficients (Pai, 1981 Maxwell, 1997). 

The equation (1) with the collision term described for binary collisions, it isn t lineal, is for that reason that the solution is very difficult. Nevertheless, exists a solution for Boltzmann equation, it isn t trivial and is very important and is known like distribution function Maxwellian. For this case the Boltzmann equation presents a non reversible behavior and distribution function lays to distribution Maxwellian, this represent the situation of an uniform gas in stationary state. 

For derivate the distribution fuction Maxwellian, supposes the absence of external forces F emd uniform gas, the distribution function isf (r,v,t) independent of sp>ace coordinates , i.e, f = f (v,t). Whereas these conditions, Boltzmann equation (1) with binary collision term is... 

In the equation (14), f = f(x,, t)is distribution fimction of only particle, is microscopic velocity, X is the relaxation time due to collision, and g is the Boltzmann-Maxwellian distribution function (fM), is important mention that collision term has been transforming in accordance with equation (2). 

The Boltzmann equation must be solved with appropriate boundary conditions to obtain f and f. The full Boltzmann equation has not been solved analytically or numerically. Current approximate methods for extracting the desired information from the Boltzmann equation are covered in detail in a recent reference [2.84]. In view of this review, discussion of these methods will not be given. It is sufficient to indicate some of the principal methods which have been employed. These are moment or integral methods for specific molecular scattering laws, the use of "models" (of which the BGK model is the simplest) for the collisions term J(fgfg), and direct simulation by Monte Carlo or molecular-dynamics techniques. 

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