Local volume fraction fluctuations in random media

Although the volume fraction is a constant for a statistically homogeneous random medium, on a spatially local level it fluctuates. We study the full distribution of volume fraction within an observation window of finite size for models of random media. A formula due to Lu and Torquato for the standard deviation or "coarseness" associated with the local volume fraction ξ is extended for the nth moment of ξ for any n. The distribution function F_L of the local volume fraction of five different model microstructures is evaluated using analytical and computer-simulation methods for a wide range of window sizes and overall volume fractions. On the line, we examine a system of fully penetrable rods and a system of totally impenetrable rods formed by random sequential addition (RSA). In the plane, we study RSA totally impenetrable disks and fully penetrable aligned squares. In three dimensions, we study fully penetrable aligned cubes. In the case of fully penetrable rods, we will also simplify and numerically invert a prior analytical result for the Laplace transform of F_L. In all of these models, we show that, for sufficiently large window sizes, F_L can be reasonably approximated by the normal distribution.