  All liquids are topologically disordered materials; however, the degree of
disorder can vary as a result of internal fluctuations in structure and
topology. These fluctuations depend on both the composition and temperature of
the system. Most prior work has considered the mean values of liquid or glass
properties, such as the average number of topological degrees of freedom per
atom; however, the localized fluctuations in properties also play a key role in
governing the macroscopic characteristics of any glass-forming system. This
paper proposes a generalized approach for modeling topological fluctuations in
glass-forming liquids by linking the statistical mechanics of the disordered
structure to topological constraint theory. In doing so we introduce the
contributions of localized fluctuations into the calculation of the topological
degrees of freedoms in the network. With this approach the full distribution of
properties in the disordered network can be calculated as an arbitrary function
of composition, temperature, and thermal history (for the nonequilibrium glassy
state). The scope of this current investigation focuses on describing
topological fluctuations in liquids, concentrating on composition and
temperature effects.
