Linear Hard-Sphere Gas: Variational Eigenvalue Spectra of the Energy Kernel

The relaxation problem for the linear hard-sphere gas is considered using a Rayleigh–Ritz expansion of the energy kernel for transitions in a test-particle/heat-bath system at variable mass ratio γ. The basis sets used are the “exact” eigenfunctions for the problem in the Rayleigh limit γ → 0 and the required expansion matrices can be obtained to order at least 20 × 20 by algebraic procedures. Good, converged eigenvalues λk are obtained in the discrete region 0 < λk < λ* so long as these are not within about 10% of the continuum threshold λ*. Where possible, unconverged “pseudoeigenvalues” in the continuum region λ > λ* are also determined and their use to “represent” the continuum in the initial-value problem is discussed. A number of applications are considered. We obtain the relaxation of distribution functions P(x,t) and mean energies 〈x(t)〉 and investigate the energy-autocorrelation function S(t) for equilibrium fluctuations, as a function of mass ratio. The latter proves to be very nearly Gaussian for all γ in the range 2−9 < γ < 29. The results also allow a detailed evaluation of the range of validity of the Rayleigh–Fokker–Planck (RFP) equation in energy space. For processes dominated by λ1 the RFP approximation is adequate from about γ < 1 / 20, but with progressive deterioration of the predicted higher eigenvalues. Where comparable the results agree with those of Shapiro and Corngold [Phys. Rev. 137, A1686 (1965)] obtained for the neutron thermalization problem with somewhat different methods and scope.