Superdiffusion in self-reinforcing run-and-tumble model with rests

Run-and-tumble like motion of a camphor-infused Marangoni swimmer

'Run-and-tumble' (RT) motion has been a subject of intense research for several decades. Many organisms, such as bacteria, perform such motion in the presence or absence of local chemical concentration gradients and it is found to be advantageous in search processes. Although there are previous reports involving the successful design of non-living self-propelled particles exhibiting such motion in the presence of external stimuli (chemical/mechanical), RT motion with 'rest' has not yet been observed for autonomous non-living active particles. We have designed a swimmer that performs motion using a combination of 'run', 'tumble', and 'rest' states with stochastic transitions. In the present scenario, it arises solely due to self-generated local surface tension gradients. We quantify the residence time statistics by analyzing the swimmer trajectories from the experimental data, which suggests that the 'rest' and 'tumble' states are more frequent than 'run'. Then, we quantify the motion properties by computing the mean squared displacement, which shows that the swimmer performs ballistic motion on a short time scale and then slows down due to tumbling and resting. To validate the observed transport properties, we introduce a minimal model of a chiral active Brownian particle, stochastically switching between three internal states. The model parameters were extracted from the experiments, which rendered a good agreement between the experiments and simulations.

Population heterogeneity in the fractional master equation, ensemble self-reinforcement, and strong memory effects

We formulate a fractional master equation in continuous time with random transition probabilities across the population of random walkers such that the effective underlying random walk exhibits ensemble self-reinforcement. The population heterogeneity generates a random walk with conditional transition probabilities that increase with the number of steps taken previously (self-reinforcement). Through this, we establish the connection between random walks with a heterogeneous ensemble and those with strong memory where the transition probability depends on the entire history of steps. We find the ensemble-averaged solution of the fractional master equation through subordination involving the fractional Poisson process counting the number of steps at a given time and the underlying discrete random walk with self-reinforcement. We also find the exact solution for the variance which exhibits superdiffusion even as the fractional exponent tends to 1.