Near-Maxwellian particle velocity distribution

In order to evaluate the collisional integrals Pc, qc, and y, explicitly, it is important to know the specific form of the pair distribution function /(2)(vi, ri V2, ry. t). The pair distribution function /(2) may be related to the single-particle velocity distribution function f by introducing a configurational pair-correlation function g(ri, r2). In the following, we first introduce the distribution functions and then derive the expression of /(2) in terms of f by assuming /(1) is Maxwellian and particles are nearly elastic (i.e., 1 — e 1). 

To solve eqn. (294) for the doublet density, the hierarchy of the equation must be broken in a manner analogous to the super-position approximation of Kirkwood or that of Felderhof and Deutch [25], which was presented in Chap. 9, Sect. 5. Furthermore, it is not unreasonable to assume that the system is quite near to thermal equilibrium. Were the system at thermal equilibrium, then collisions would not change the velocity distribution of the particles and the equilibrium distribution would be of the usual Maxwellian form, 0 (v,), etc. These are the solutions of the psuedo-Liouville equation... 

For a chute flow with relatively massive particles, the velocity distribution function of particles,/ should be nearly Maxwellian. Therefore the volume-averaged drag force based on an element volume of particles can be expressed as... 

These probability distributions are constructed so that the probability distribution for particles near the wall remains Maxwellian. The probability distribution, px. for the tangential components of the velocity is Maxwellian, and both positive and negative values are permitted. The normal component must be positive, since after scattering at the surface, the particle must move away from the wall. The form of Pn is a reflection of the fact that more particles with large vn hit the wall per unit time than with small vn [78]. 

In a hybrid method, molecules are displaced in time according to conventional molecular dynamics (MD) algorithms, specifically, by integrating Newton s equations of motion for the system of interest. Once the initial coordinates and momenta of the particles are specified, motion is deterministic (i.e., one can determine with machine precision where the system will be in the near future). In the context of Eq. (2.1), the probability of proposing a transition from a state 0 to a state 1 is determined by the probability with which the initial velocities of the particles are assigned from that point on, motion is deterministic (it occurs with probability one). If initial velocities are sampled at random from a Maxwellian distribution at the temperature of interest, then the transition probability function required by Eq. 

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