Generalized Boltzmann-Enskog

The term will lead to a generalized Boltzmann-Enskog approximation. This... 

Our expression for [Eq. (105)] is thus far still exact. The first term in (105) is seen to be the contribution from local two-body types of interaction including static correlation effects. This term represents a generalized Boltzmann-Enskog term, which controls the short-time behavior of We note that the first two terms in a power series expansion of (l/z) both come from YG V. The second term in (105) contains recollision and mode-coupling " effects. The T-matrices on the ends have a simple form in the case of hard spheres, and they describe close collisions between two particles. The quantity R describes the motion of these two particles after their collision. 

The generalized Boltzmann-Enskog memory function is given by... 

The form of the Boltzmann-Enskog collision operator is thus specified out task is to find its generalization. We denote the general collision operator by /l (1, 1 t ), where we have allowed for the possibility that it may be nonlocal in time as well as space. The general kinetic equation may then be written as... 

The kinetic theory of dense gases began with the work of Enskog, who in 1922, generalized Boltzmann s derivation of the transport equation to apply it to a dense gas of hard spheres. Enskog showed that for dense gases there is a mechanism for the transport of momentum and energy by means of the intermolecular potential, which is not taken into account by the Boltzmann equation at low densities, and he derived expressions for the transport coeffi-... 

We see that if particles 1 and 2 have suffered earlier collisions with each other, as illustrated in Fig. 22b, or with the same third particle, as in Figs. 22c and 22d, then there will be both spatial and velocity correlations between particles 1 and 2 before the onset of their collision at time t. These dynamical correlations are not taken into account in the Boltzmann equation, since the Stosszahlansatz assumes they do not exist, nor are they taken into account by the Enskog theory for hard spheres, which ignores all velocity correlations. Here we see that dynamical correlations do exist in the gas and that they are accounted for in the generalized Boltzmann equation since it takes the concerted action of three or more particles to produce such correlations. 

To solve the Boltzmann equation Enskog s method of successive approximations is generally used This consists of writing firstly,... 

Up to the early 1970s a kinetic approach to the time-dependent properties of fluids was synonymous with a framework based on the Boltzmann equation and its extension by Enskog, in which a central role is played by those dynamical events referred to as uncorrelated binary collisions [29]. Because of this feature the Boltzmann equation is in general not applicable to dense fluids, where the collisions are so frequent that they are likely to interfere with each other. The uncorrelation ansatz is clearly equivalent to a loss of memory, or to a Markov approximation. As a result, for dense fluids the traditional kinetic approach should be critically revised to allow for the presence of non-Markovian effects. 

In spite of have been proposed many approximated solutions to Boltzmann equation (including the Grad s method of 13 moments, expansions of generalized polynomial, bimodal distributions functions), however the Chapman-Enskog is the most popular outline for generalize hydrodynamic equations starting from kinetics equations kind Boltzmann (James William, 1979 Cercignani, 1988). 

The theoretical basis for the corresponding-states principle is as follows. Although molecules interact with a pair potential U r, that depends on relative orientation angles 2 as well as on separation r, the formal extension of the Boltzmann equation and its subsequent solution by the methods of Chtq>man and Enskog show that the expression for the viscosity r) and the so-called self-diffusion coefficient D have the same general form as for central potentials (Taxman 1958 Wang Chang et al. 1964) ... 

We have shown that the approximate values of the transport coefficients of a dilute gas can be calculated by the variational method. It seems to be desirable to generalize such a principle to the case of a dense gas. A generalization has been done by Murakami for dense gases consisting of rigid sphere molecules without attraction, based upon Enskog s modification of the Boltzmann equation. ... 

If the species s Boltzmann equation (2.289) is multiplied by the molecular property V>i(cj) and thereafter integrated over all molecular velocities, the general Enskog s equation of change is achieved ... 

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