Thermal Fluctuations in a Hard-Sphere Gas

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Abstract
Density fluctuations in a moderately dilute gas of hard spheres are analyzed using a kinetic equation which is the generalization of the linearized Boltzmann equation to arbitrary frequencies and wave numbers. The phase-space memory function derived recently by Mazenko is evaluated for the hard-sphere interaction, and the result is shown to be explicitly frequency independent. In the case of the self-correlation function, the memory function is found to be identical to the collision operator in the appropriate Boltzmann equation, thus implying that in this special case the Boltzmann-equation description is valid at all frequencies and wavelengths. Wave-number-dependent matrix elements of the memory functions are calculated using the Hermite function as a basis, and a number of interesting features are observed. The matrix elements are utilized in a kinetic-model formulation of the initial-value problem of evaluating the spectral densities of thermal fluctuations. Numerically converged results are presented and compared with previous Boltzmann-equation calculations. While the qualitative behavior of the spectral distribution still can be characterized by the ratio of wavelength to collision mean free path, details of the line shape now depend on density and wave number. At reasonable density and wave-number values, a narrowing effect is observed. This effect, which is most pronounced at small and moderate wavelength-to-mean-free-path ratios, is sufficiently significant that further investigation is warranted.