  In a previous paper [Jiao, Stillinger and Torquato, PRE 81, 011105 (2010)],
we considered the geometrical ambiguity of pair statistics associated with
point configurations. Here we focus on the analogous problem for heterogeneous
media (materials). The complex structures of heterogeneous media are usually
characterized via statistical descriptors, such as the $n$-point correlation
function $S_n$. An intricate inverse problem of practical importance is to what
extent a medium can be reconstructed from the two-point correlation function
$S_2$ of a target medium. Recently, general claims of the uniqueness of
reconstructions using $S_2$ have been made based on numerical studies. Here, we
provide a systematic approach to characterize the geometrical ambiguity of
$S_2$ for both continuous two-phase heterogeneous media and their digitized
representations in a mathematically precise way. In particular, we derive the
exact conditions for the case where two distinct media possess identical $S_2$.
The degeneracy conditions are given in terms of integral and algebraic
equations for continuous media and their digitized representations,
respectively. By examining these equations and constructing their rigorous
solutions for specific examples, we conclusively show that in general $S_2$ is
indeed not sufficient information to uniquely determine the structure of the
medium, which is consistent with the results of our recent study on
heterogeneous media reconstruction [Jiao, Stillinger and Torquato, PNAS 106,
17634 (2009)]. The uniqueness issue of multiphase media reconstructions and
additional structural information required to characterize heterogeneous media
are discussed.
