Inverse Statistics of Active Matter Trajectories to Distinguish Interaction Kernel Anisotropy from Emergent Correlations

High-resolution imaging provides dense trajectories of migrating cells, flocking animals, and synthetic active particles, from which interaction laws can be determined with a wide variety of methods. Yet, distinguishing whether front-back or lateral biases seen in such data reflect intrinsic anisotropy in the interaction kernel or emergent correlations that are nevertheless produced by isotropic pairwise interaction forces remains an open challenge. We resolve this ambiguity by deriving a linear partial differential equation that connects measurable two-point velocity correlations to an unknown, distance- and angle-dependent interaction kernel. Turing-like instabilities can occur which allows for dipolar or quadrupolar patterns to arise even when agents interact according to an underlying attraction-repulsion law that is angle-independent. We then show that incorporating a weak velocity-alignment force can interfere with anisotropic pattern formation by suppressing dipolar patterns. We validate these predictions with agent-based simulations and provide design guidance for experiments that seek to discriminate intrinsic anisotropy from emergent effects.