Hyper-ballistic diffusion is shown to arise from a simple model of microswimmers moving through a porous media while competing for resources. By using a mean-field model where swimmers interact through the local concentration, we show that a non-linear Fokker-Planck equation arises. The solution exhibits hyper-ballistic superdiffusive motion, with a diffusion exponent of four. A microscopic simulation strategy is proposed, which shows excellent agreement with theoretical analysis.

Stochastic systems that undergo random restarts to their initial state have been widely investigated in recent years, both theoretically and in experiments. Oftentimes, however, resetting to a fixed state is impossible due to thermal noise or other limitations. As a result, the system configuration after a resetting event is random. Here, we consider such a resetting protocol for an overdamped Brownian particle in a confining potential V (x). We assume that the position of the particle is reset at a constant rate to a random location x, drawn from a distribution pR(x). To investigate the thermodynamic cost of resetting, we study the stochastic entropy production STotal. We derive a general expression for the average entropy production for any V(x), and the full distribution P(STotal|t) of the entropy production for V(x) = 0. At late times, we show that this distribution assumes the large-deviation form P(STotal|t) <^> exp{-t2 alpha-1 phi [(STotal - (STotal )/t alpha]}, with 1/2 < alpha 1. We compute the rate function phi(z) and the exponent alpha for exponential and Gaussian resetting distributions pR(x). In the latter case, we find the anomalous exponent alpha = 2/3 and show that phi(z) has a first-order singularity at a critical value of z, corresponding to a real-space condensation transition.

The nonequilibrium steady state emerging from stochastic resetting to a distribution is studied. We show that for a range of processes, the steady-state moments can be expressed as a linear combination of the moments of the distribution of resetting positions. The coefficients of this series are universal in the sense that they do not depend on the resetting distribution, only the underlying dynamics. We consider the case of a Brownian particle and a run-and-tumble particle confined in a harmonic potential, where we derive explicit closed-form expressions for all moments for any resetting distribution. Numerical simulations are used to verify the results, showing excellent agreement.

We study the collective dynamical behavior of active particles with topological interactions and directional reversals. Surprising phenomena are shown to emerge as the interaction relaxation time is varied relative to the reversal rate, such as the spontaneous formation of collective chiral states and collective flow reversals in the presence of confinement. The results have a direct relevance to modeling and understanding collective reversals in active matter. 

We use a first-passage time approach to study the statistics of the trapping times induced by persistent motion of active particles colliding with flat boundaries. The angular first-passage time distribution and mean first-passage time are calculated exactly for active Brownian and run-and-tumble particles, and the results of the two prototypes of active particles are contrasted. Theoretical predictions are in excellent agreement with Langevin simulations of active particle dynamics. Our results shed further light on how active particles with different dynamics may have equivalent statistical properties in the bulk yet behave differently near boundaries or obstacles.