On the computation of ensemble averages for spatially non-uniform particle systems

When there is no clear separation between microand macro-scales, ergodicity cannot be invoked to transform ensemble into volume averages. In such cases it is necessary to use ensemble averaging directly. The calculation of such averages, however, converges slowly and therefore requires a large number of realizations of the system. This paper describes a much more efficient method based on the use of a Fourier expansion of the quantity to be averaged. The advantages of the Fourier approach are demonstrated explicitly with several examples for the specific problem of equal spheres in a viscous fluid, but the analytical estimates given suggest that similar results can be expected for other situations as well. It is shown both analytically and numerically that the variance of the Fourier coefficients is in many cases significantly smaller than that of the direct method, which leads to a much faster convergence of the former. The paper also describes a method by which the probability distribution of a uniform ensemble can be biased so as to mimic that of a non-uniform one with prescribed properties.