Spatial structures and giant number fluctuations in models of active matter

The large scale fluctuations of the ordered state in active matter systems are usually characterized by studying the "giant number fluctuations" of particles in any finite volume, as compared to the expectations from the central limit theorem. However, in ordering systems, the fluctuations in density ordering are often captured through their structure functions deviating from Porod's law. In this Letter we study the relationship between giant number fluctuations and structure functions for different models of active matter as well as other nonequilibrium systems. A unified picture emerges, with different models falling in four distinct classes depending on the nature of their structure functions. For one class, we show that experimentalists may find Porod's law violation, by measuring subleading corrections to the number fluctuations.