Practical fitting framework for applying Countoscope expressions to experimental data

Develop a precise procedure for using the derived Countoscope expressions—specifically the formulas for the number correlation function C_N(t) and the number mean-squared difference obtained via integrated intermediate scattering functions for Active Brownian Particles, Run-and-Tumble Particles, and Active Ornstein–Uhlenbeck Particles—at different truncation orders to fit experimental measurements and thereby enable practical parameter inference from number-fluctuation data.

Background

The paper derives theoretical expressions for number fluctuations of self-propelled particles by linking number correlations to intermediate scattering functions (ISFs). For Active Ornstein–Uhlenbeck Particles (AOUPs), a closed-form Gaussian expression is obtained, while for Active Brownian Particles (ABPs) and Run-and-Tumble Particles (RTPs), the authors develop an exact continued-fraction framework (exact for RTPs; controlled truncations for ABPs) that yields ISFs and, by integration, predictions for number correlations and the number mean-squared difference (NMSD). These predictions are validated against simulations and analyzed across diffusive, advective, and enhanced-diffusion regimes.

While the work provides limiting laws and guidance about truncation order accuracy as a function of Péclet number, it does not supply a concrete, end-to-end methodology for fitting experimental Countoscope data using the derived expressions. The authors explicitly defer a precise discussion of how to use the expressions at different truncation orders to fit experimental data, indicating the need for a practical fitting framework that operationalizes their theoretical results for parameter extraction.