Detecting active L\'evy particles using differential dynamic microscopy

Detecting L\'evy flights of cells has been a challenging problem in experiments. The challenge lies in accessing data in spatiotemporal scales across orders of magnitude, which is necessary for reliably extracting a power-law scaling. Differential dynamic microscopy has been shown to be a powerful method that allows one to acquire statistics of cell motion across scales, which is a potentially versatile method for detecting L\'evy walks in biological systems. In this article, we extend the differential dynamic microscopy method to self-propelled L\'evy particles, whose run-time distribution has an algebraic tail. We validate our protocol using synthetic imaging data and show that a reliable detection of active L\'evy particles requires accessing length scales of an order of magnitude larger than its persistence length, if the variability in particle speed is moderate. Applying the protocol to experimental data of E. coli and E. gracilis, we find that E. coli does not exhibit a signature of L\'evy walks, while E. gracilis is better described as active L\'evy particles.