Memory effects in non-Markovian random walks for swimming droplets

Swimming droplets leave a repulsive trail when they dissolve in water. This leads to self-avoidance and memory effects in the active droplet dynamics. Classical polymer physics models would predict a super diffusive scaling of 3/2 for the mean-squared displacement (MSD) in the limit of a self-avoiding random walk with no noise.  Here we find that the ensemble-averaged and time-averaged MSD reveal a discrepancy due to the history dependence of the droplet motion. To capture this behavior, we derive a theoretical Non-Markovian model (NMM), in which concentration gradient of the dissolved oil repels the particle motion. Interestingly, the ensemble-averaged MSD can be equally well fit with the Markovian active Brownian particle (ABP) model and the NMM that takes into account the trail repulsion. We therefore implement a deep-learning method known as convolutional neural network (CNN) to classify experimental trajectories by the two models. The network is trained and tested on trajectories simulated with parameters obtained from fitting both models to ensemble-averaged MSD. Then experimental trajectories are classified using the well-trained network. It turns out that more than 85% of the experimental trajectories are identified as NMM. The success of the NMM in fitting the data allows us to extract the active noise in the system, arising from hydrodynamic effects, which can be interpreted as an effective temperature of ~ 10⁶ k_BT. Our findings indicate that even though the MSD scaling laws do not change because of the memory effect, there are distinguishing features that select NMM over ABP.