The Boltzmann-Enskog Theory of Thermal Fluctuations

The approximation wherein one retains the first term in the memory function expression (105) is of special interest because it leads to a kinetic equation that is closely related to the Boltzmann-Enskog equation in transport theory. In this section we will investigate this particular approximation in some detail not only from the standpoint of further analytical analysis but also from the standpoint of practical calculations. We will see that within certain limitations the approximation results in a reasonably realistic description of dense gases and liquids, and in this sense represents the first step in a systematic microscopic calculation. 

Since the static memory function affects the sound speed the two kinetic equations do not give the same hydrodynamic limit. On the other hand, it is known that transport coefficients calculated in the two descriptions are the same.  

The kinetic model solutions are particularly useful for analyzing fluctuation spectra at finite wavenumbers. The solutions do not involve any expansion in ky and since they are constrained to give the correct limiting behavior at large and small k, the models prove to be quite successful means of interpolating between well-known limits. 

It is sufficient to illustrate the essential techniques of kinetic model solution by considering a triple relaxation time model. This is the simplest kinetic equation capable of describing quantitatively the thermal fluctuations at long wavelengths. We will first summarize the kinetic model method and formulate the triple relaxation time model explicitly. Then we will examine some of the calculations in the context of experimental data on S(k, a ). 

The basic ingredients in the formulation of kinetic models are the matrix elements of the memory function calculated using an appropriate set of momentum basis functions. In the analysis of the Boltzmann collision operator it has been found convenient to use the Sonine polynomials. However, the k r dependence in (124) destroys the rotational symmetry present in the Boltzmann collision operator for a spherically symmetric potential therefore it is equally appropriate to choose the function 

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