Density fluctuations in many-body systems

The characterization of density fluctuations in systems of interacting particles is of fundamental importance in the physical sciences. We present a formalism for studying local density fluctuations in two special subvolumes (centered around either a reference particle or some arbitrary point in the system) termed particle and void regions, respectively. We present formal expressions for the probability, as well as the moments, associated with finding exactly n particles inside of either of these subvolumes. Furthermore, we derive the relationship between the probability functions and closely related quantities of interest, such as the $n\mathrm{th}$ nearest-neighbor distribution functions and the n-particle conditional pair distribution functions associated with each region. We solve for these quantities exactly in the one-dimensional hard-rod system. The methods developed for studying the hard-rod fluid are applicable for studying a wide class of one-dimensional systems.