  The nature of emergent collective behaviors of moving physical agents
interacting with their neighborhood is a long-standing open issue in physical
and biological systems alike. This calls for studies on the control of
synchronization and the degree of order in a collection of diffusively moving
noisy oscillators. We address this by constructing a generic hydrodynamic
theory for active phase fluctuations in a collection of large number of nearly
phase-coherent moving oscillators in two dimensions. Our theory describes the
general situation where phase fluctuations and oscillator mobility mutually
affect each other. We show that the interplay between the active effects and
the mobility of the oscillators leads to a variety of phenomena, ranging from
synchronization with long range, nearly long range and quasi long range orders
to instabilities and desynchronization with short range order of the oscillator
phases. We highlight the complex dependences of synchronization on the active
effects. These should be testable in wide ranging systems, e.g., oscillating
chemical reactions in the presence of different reaction
inhibitors/facilitators, live oriented cytoskeletal extracts, or vertebrate
segmentation clocks.
