Equation, Boltzmann, generalized Liouville

Several attempts were made to derive the Boltzmann equation from the Liouville equation, but it was not until the work of Bogoliubov in 1947 and the later work of Cohen and of Green that a satisfactory derivation was given. The approach to the Boltzmann equation from the Liouville equation has proved to be important for two reasons. First, it makes it possible to replace the Stosszahlansatz by a more fundamental assumption about the statistical ensemble that is sampled when an experiment is performed. Second, it allows one to generalize the Boltzmann equation to dense gases. 

As a result of the secular growth of the /-body collision integrals with time, we are compelled to conclude that, although the cluster expansion method can be used successfully to derive the Boltzmann equation from the liouville equation and to obtain corrections to the Boltzmann equation, there are serious difficulties in trying to represent these corrections as a power series in the density. An example of the difficulties that appear if one attempts to apply the generalized Boltzmann equation as it stands now to a problem of some interest is provided by the calculation of the density expansion of the coefficient of shear viscosity. By constructing normal solutions to the generalized Boltzmann equation, one finds that the viscosity 17 has the expansion of the form mentioned in Eq. (224),... 

Before leaving this chapter, we briefly look at an important quantity known as Boltzmann s entropy, and we will examine reduced forms of the Liouville equation in generalized coordinates. 

The general equations of change given in the previous chapter show that the property flux vectors P, q, and s depend on the nonequi-lihrium behavior of the lower-order distribution functions g(r, R, t), f2(r, rf, p, p, t), and fi(r, P, t). These functions are, in turn, obtained from solutions to the reduced Liouville equation (RLE) given in Chap. 3. Unfortunately, this equation is difficult to solve without a significant number of approximations. On the other hand, these approximate solutions have led to the theoretical basis of the so-called phenomenological laws, such as Newton s law of viscosity, Fourier s law of heat conduction, and Boltzmann s entropy generation, and have consequently provided a firm molecular, theoretical basis for such well-known equations as the Navier-Stokes equation in fluid mechanics, Laplace s equation in heat transfer, and the second law of thermodynamics, respectively. Furthermore, theoretical expressions to quantitatively predict fluid transport properties, such as the coefficient of viscosity and thermal... 

Let be a function of coordinates only, so a point in is specified by [3(A(ab - 1) - 1] coordinates and 3(A(ab - ) momenta. Identify the missing coordinate as the reaction coordinate s (so s becomes a coordinate normal to the hypersurface), and identify the momentum conjugate to s as p. Let C denote the [6(A(ab — 1) - 2]-dimensional hyperface in in which ps = 0. Assume that the % region of phase space is populated according to a Boltzmann equilibrium distribution then Liouville s theorem of classical statistical mechanics shows it will evolve into a Boltzmann equilibrium distribution at and hence also at C. Consider the one-way flux of this equilibrium ensemble of phase points through in the 5 —> P direction. This flux may be calculated quite generally, and using this calculation plus equation (2) yields... 

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