Boltzmann collision equation

In its more advanced aspects, kinetic theory is based upon a description of the gas in terms of the probability of a particle having certain values of coordinates and velocity, at a given time. Particle interactions are developed by the ordinary laws of mechanics, and the results of these are averaged over the probability distribution. The probability distribution function that is used for a given macroscopic physical situation is determined by means of an equation, the Boltzmann transport equation, which describes the space, velocity, and time changes of the distribution function in terms of collisions between particles. This equation is usually solved to give the distribution function in terms of certain macroscopic functions thus, the macroscopic conditions imposed upon the gas are taken into account in the probability function description of the microscopic situation. 

Twelve years later, Choh and Uhlenbeck8 published the first explicit generalization of the Boltzmann equation involving triple collisions. Their work rests on Bogolubov s ideas and formal results. Green11 and Rice, Kirkwood, and Harris26 also obtained the triple collision equation by other methods. 

In other words, in approximate accordance with the original paper by Boltzmann [6], we assume that in a given volume element the expected number of collisions between molecules that belong to different velocity ranges can be computed statistically. This assumption is referred to as the Boltzmann Stosszahlansatz (German for Collision number assumption). A result of the Boltzmann H-theorem analysis is that the latter statistical assumption makes Boltzmann s equation irreversible in time (e.g., [28], sect. 4.2). 

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. 

In this section we will survey both the informal and formal versions of the kinetic theory of gases, starting with the simpler informal version. Here the basic idea is to combine both probabilistic and mechanical arguments to calculate quantities such as the equilibrium pressure of a gas, the mean free distance between collisions for a typical gas particle, and the transport properties of the gas, such as its viscosity and thermal conductivity. The formal version again uses both probabilistic and mechanical arguments to obtain an equation, the Boltzmann transport equation, that determines the distribution function,/(r, v, t), that describes the number of gas particles in a small spatial region, 5r, about a point r, and in a small region of velocities,... 

The foregoing treatment of ICR spectra and line shapes does not cover the effects of collisions or bimolecular reactions. Hence the equations derived are only valid in the low-pressure limit. Wobschall et have treated the effect of nonreactive collisions on power absorption, and Beauchamp has treated the problem of nonreactive collisions and charge exchange somewhat more formally by using the Boltzmann transport equation as applied to the properties of slightly ionized gases. 

Finally, we note an important property of the conservation equations, Eqs. (3.30) and (3.32), in the context of the Boltzmann collision integral. Letting primes denote postcollisional values and unprimed denote the precollisional values, as in the Boltzmann collision integral, the conservation equations, when applied to a collision, lead to... 

In (1), the lattice velocity vector is denoted by e,. The distribution functions satisfy a linear algebraic evolution equation that involves a collision term. The most popular version of the LBE is called the BGK formulation and it is based on the simplified model of the Boltzmann kinetic equation that was devised by Bhamagar et al. (1954). In the BGK formulation, the LBE takes the following form ... 

Because the Sigmund-Thompson relation is derived from a linearized Boltzmann transport equation (a statistical approach), it can only approximate macroscopic properties arising from an isotropic collision cascade in nonstructured solids,... 

The functions and depend on the collision function model using gas density and temperature, and should satisfy the moment equation. The above-mentioned equation is substituted for the Boltzmann s equation, and a set of inhomogeneous linear equations is obtained by equating terms of equal order. The use of distribution functionsand so on leads to the determination of transport terms needed to close the continuum equations appropriate to the particular level of approximation. The continuum stress tensor and heat flux vector can be written in terms of the distribution function (f >). This can be further simplified in terms of macroscopic velocity and temperature derivatives. 

The Boltzmann equation is considered valid as long as the density of the gas is sufficiently low and the gas properties are sufficiently uniform in space. Although an exact solution is only achieved for a gas at equilibrium for which the Maxwell velocity distribution is supposed to be valid, one can still obtain approximate solutions for gases near equilibrium states. However, it is evident that the range of densities for which a formal mathematical theory of transport processes can be deduced from Boltzmann s equation is limited to dilute gases, since this relation is reflecting an asymptotic formulation valid in the limit of no coUisional transfer fluxes and restricted to binary collisions only. Hence, this theory cannot without ad hoc modifications be applied to dense gases and liquids. 

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