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Microscopy can now resolve complex systems at the single particle level, fromcellular organisms to individual ions. The scientific community thus shares acommon interest in quantifying particle dynamics from images. Yet, availableanalysis techniques face challenges reconstructing trajectories in dense andheterogeneous systems where accurately labelling particles is difficult.Furthermore, the inescapable finite field of view of experiments hinders themeasurement of collective effects central to biological or active systems.Inspired by Smoluchowski, we introduce a broadly applicable analysis techniquethat probes dynamics simply by counting particles in finite observation boxes.Using colloidal experiments, simulations, and theory, we first demonstrate thatstatistical properties of fluctuating counts can be used to determineself-diffusion coefficients, including the effect of hydrodynamic interactions,so alleviating the hurdles associated with trajectory reconstruction. Wesubsequently show that by increasing the observation box size, countingnaturally enables the study of collective dynamics. Using our novel analysis ofparticle counts, we uncover a surprising enhancement of collective behaviour,as well as a new length scale associated with hyperuniform-like structure. Ourcounting framework, the Countoscope, thus liberates experiments from finitefields of view and enables efficient collective dynamics measurements, openingthe way to study diverse complex systems.

The Countoscope: Measuring Self and Collective Dynamics without Trajectories