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Anomalous diffusion, manifest as a nonlinear temporal evolution of theposition mean square displacement, and/or non-Gaussian features of the positionstatistics, is prevalent in biological transport processes. Likewise,collective behavior is often observed to emerge spontaneously from the mutualinteractions between constituent motile units in biological systems. Exampleswhere these phenomena can be observed simultaneously have been identified inrecent experiments on bird flocks, fish schools and bacterial swarms. Theseresults pose an intriguing question, which cannot be resolved by existingtheories of active matter: How is the collective motion of these systemsaffected by the anomalous diffusion of the constituent units? Here, we answerthis question for a microscopic model of active L\'evy matter -- a collectionof active particles that perform superdiffusion akin to a L\'evy flight andinteract by promoting polar alignment of their orientations. We present indetails the derivation of the hydrodynamic equations of motion of the model,obtain from these equations the criteria for a disordered or ordered state, andapply linear stability analysis on these states at the onset of collectivemotion. Our analysis reveals that the disorder-order phase transition in activeL\'evy matter is critical, in contrast to ordinary active fluids where thephase transition is, instead, first-order. Correspondingly, we estimate thecritical exponents of the transition by finite size scaling analysis and usethese numerical estimates to relate our findings to known universality classes.These results highlight the novel physics exhibited by active matterintegrating both anomalous diffusive single-particle motility andinter-particle interactions.

Active L\'evy Matter: Anomalous Diffusion, Hydrodynamics and Linear Stability