  This work presents an analytical investigation of anomalous diffusion and
turbulence in a dusty plasma monolayer, where energy transport across scales
leads to the spontaneous formation of spatially disordered patterns. Many-body
simulations of 10,000-particle dusty plasma monolayers are used to demonstrate
how the global dynamics depend on the statistical properties of the dust
assembly for realistic laboratory conditions. We find that disorder due to
variations in the dust size distribution and charge-driven nonlocal
interactions resulting in anomalous dust diffusion are key factors for the
onset of instabilities. The resulting dynamics exhibit features of inertial
turbulence over slightly more than half a decade of scales proportional or
smaller than the Debye shielding length. These processes are examined
analytically using a recently developed Fractional Laplacian Spectral (FLS)
technique, which identifies the active energy channels as a function of scale,
disorder concentration, and features of the nonlocal interactions. The
predictions from the theoretical (spectral) analysis demonstrate agreement with
the results from the many-body (kinetic) simulations, thus providing a powerful
tool for the study of active turbulence.
