Hyperdiffusion of Poissonian run-and-tumble particles in two dimensions

Abstract

We study non-interacting Poissonian run-and-tumble particles (RTPs) in two dimensions whose velocity orientations are controlled by an arbitrary circular distribution $Q(\phi)$. RTP-type active transport has been reported to undergo localization inside crowded and disordered environments, yet its non-equilibrium dynamics, especially at intermediate times, has not been elucidated analytically. Here, starting from the standard (one-state) RTPs, we formulate the localized (two-state) RTPs by concatenating an overdamped Brownian motion in a Markovian manner. Using the space-time coupling technique in continuous-time random walk theory, we generalize the Montroll-Weiss formula in a decomposable form over the Fourier coefficient $Q_{\nu}$ and reveal that the displacement moment $\left \langle \mathbf{r}^{2\mu}(t) \right \rangle$ depends on finite angular moments $Q_{\nu}$ for $|\nu|\leq \mu$. Based on this finding, we provide (i) the angular distribution of velocity reorientation for one-state RTPs and (ii) $\left \langle \mathbf{r}^{2}(t) \right \rangle$ over all timescales for two-state RTPs. In particular, we find the intricate time evolution of $\left \langle \mathbf{r}^{2}(t) \right \rangle$ that depends on initial dynamic states and, remarkably, detect hyperdiffusive scaling $\left \langle \mathbf{r}^{2}(t) \right \rangle \propto t^{\beta(t)}$ with an anomalous exponent $2<\beta(t)\leq 3$ in the short- and intermediate-time regimes. Our work suggests that the localization emerging within complex systems can increase the dispersion rate of active transport even beyond the ballistic limit.