Boltzmann equation closure

Equation (12.4.1-11) seems to be adequate for a variety of problems. It is very similar to the Boltzmann equation of kinetic theory, where /represents the distribution function of molecular velocities. Closure models are required for all the conditional terms in (12.4.1-11). They depend on the specific physical phenomena and reactions involved. 

A survey of the kinetic theory required deriving the Boltzmann equation, for a dilute gas constituting identical molecular particles of mass m, is provided in the succeeding subsections. The translational part of the equation in the limit of no collisions is discussed first, before the more complex collision term closure is outlined. 

This balance equation can also be derived from kinetic theory [ 159]. In the Maxwellian average Boltzmann equation for the species s type of molecules, the collision operator does not vanish because the momentum ntsCg 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 Pjr is required. Maxwell [95] 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 [95] 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. 

Simulation results were compared with the predictions of the Ornstein-Zernike (OZ) equation with the hypernetted chain (HNC) closure approximation and the non-linear Poisson-Boltzmann equation, both augmented by pertinent Lifshitz NES potentials. We show in Fig. 1 that there is very good agreement between modified Poisson-Boltzmann theory, MC simulations, and HNC calculations when the counterions and co-ions are monovalent. There is also good agreement between the different approaches with divalent co-ions (not shown here). However, the results from MPBE cannot account for ion correlation effects that occur in Fig. 2 when the counterions are divalent. The reason is simply that 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. 

Owing to the presence of the pair correlation function, the collision model in Eq. (6.2) is unclosed. Thus, in order to close the kinetic equation (Eq. 6.1), we must provide a closure for written in terms of /. The simplest closure is the Boltzmann Stofizahlansatz (Boltzmann, 1872) ... 

The 3D-RISM/HNC equations (4.9) and (4.9) appropriately treat the excluded volume of the solute regarded as a whole. However, the imperfectness persists for solvent molecules since their orientation is reduced. Cortis, Rossky, and Friesner [23] proposed to perform explicit orientational averaging of the Boltzmann factor for a short-range repulsive part of the initial orientationally dependent interaction potential. The 3D-HNC closure thus modified to include multi-site correlations between solvent sites around the solute is written as... 

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