Boltzmann kinetics

An essential feature of this equation is that the Boltzmann kinetic equation is only valid for ideal systems. Indeed, as can be shown, this kinetic equation does not take into account the contributions of the interaction to the thermodynamic functions. For example, the conservation of the energy following from the Boltzmann equation has the form... 

Abdallah Jr. J, Csanak G, Fukuda Y, Yamakawa K, Akahane Y, Aoyama M, Inoue N, Ueda H, Faenov AYa, Magunov AI, Pikuz TA, Skobelev IYu (2003) Time-dependent Boltzmann kinetic model of X-rays produced by ultrashort-pulse laser irradiation of argon clusters. Phys. Rev. A 68 0632011-0632018... 

The kinetic theory leads to the definitions of the temperature, pressure, internal energy, heat flow density, diffusion flows, entropy flow, and entropy source in terms of definite integrals of the distribution function with respect to the molecular velocities. The classical phenomenological expressions for the entropy flow and entropy source (the product of flows and forces) follow from the approximate solution of the Boltzmann kinetic equation. This corresponds to the linear nonequilibrium thermodynamics approach of irreversible processes, and to Onsager s symmetry relations with the assumption of local equilibrium. 

Fig. 9. Kinetic energy distributions for physically sputtered products obtained from the simulation (symbols). Data taken from all simulated layers with F/Si > 0.39 for the chemically sputtering results, represented here as the dashed, dot-dash, and dotted lines, assuming a Maxwell-Boltzmann kinetic energy distribution and 300 K. Solid-line fit to the symbols is collision cascade model with a fitted value of Uq. (From Barone and Graves, 1995a.)...

In the literature the GPBE is known by many different names, depending on the research field. For example, in statistical physics, when written for the velocity NDF, it is called the Boltzmann kinetic equation and in the combustion literature, when written for droplets, it is known as the spray or Williams equation. 

By using a very similar approach to the one outlined above for the PBE, it is possible to derive a GPBE for an NDF that includes particle velocity as an internal variable. We will denote this general NDF as n(t,x,, ) (i.e. without subscripts on n). The simplest GPBE (i.e. velocity without other internal coordinates) is known as the Boltzmann kinetic equation and was first derived in the context of gas theory (Chapman Cowling, 1961). The final form of the GPBE is... 

Readers familiar with the kinetic theory of gas dynamics should find many similarities with the derivation of the Boltzmann kinetic equation. 

It is known that an exact description of transfer processes in the aerosol particles-gas phase system with chemical or phase transformations on the particle surface for arbitrary particle sizes (and correspondingly for arbitrary Knudsen numbers) can be found only by solving the Boltzmann kinetic equation. However, the mathematical difficulties associated with the solution of the given equation lead to the necessity of obtaining rather simple expressions for mass and energy fluxes either on the basis of an approximate solution of the Boltzmann equation or with the use of simpler models. In particular, it is known that the use of the diffusion equation with appropriate boundary conditions on the particle surface leads to the equation that gives correct limiting cases with respect to the Knudsen number [2]. 

Tcherimissine FG (2004) Solution of the Boltzmann kinetic equation for high speed flows of a rarefied gas. In Proceedings of the 24th international symposium on rarefied gas dynamics, Bari, 10-16 July 2004... 

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 ... 

The most complete mathematical description of gas-particle jet interaction is based on the Boltzmann kinetic equation [1]. Due to computational difficulties it is expedient to apply approximate methods of modeling. For example, Babukha et al [2] developed the effective simulation methods of flows in nozzles and Berman et al [3] in intersecting jets. These models take into consideration particle momentum exchanges in polydispersed flows. 

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