Particle number fluctuations $N\left(t\right)$, measured in virtual observation boxes of an image or a simulation, offer a way to quantify particle dynamics when particle tracking is impractical, such as in high-density systems. While traditionally limited to equilibrium diffusive systems, we extend this approach—named “Countoscope”—to out-of-equilibrium self-propelled particles: Active Brownian (ABPs), run and tumble (RTPs), and active Ornstein-Uhlenbeck particles (AOUPs). For AOUPs, we leverage their Gaussian statistics to derive a general formula applicable to any Gaussian system. For ABPs and RTPs, we derive the self-intermediate scattering function (ISF)—and thus the correlations of $N\left(t\right)$—using an exact perturbative expansion over the probability density fields, revealing key physical features of the ISF and of the number correlations. Our theoretical predictions for the mean-squared number difference $\langle \mathrm{\Delta }{N}^2\left(t\right)\rangle =\langle {(N\left(t\right)-N\left(0\right))}^2\rangle$ match stochastic simulations and exhibit three time-dependent scaling regimes: diffusive, ballistic, and long-time enhanced diffusive, reflecting the regimes of the mean-squared particle displacement. We further uncover limiting laws in each of these regimes that are useful to quantify self-propulsion properties.

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• Active Brownian particles
• Dry active matter
• Living matter & active matter
• Theories of collective dynamics & active matter