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

TL;DR

This paper analytically investigates the non-equilibrium dynamics of two-dimensional Poissonian run-and-tumble particles with arbitrary velocity orientation distributions, revealing hyperdiffusive behavior and the influence of initial states on dispersion rates.

Contribution

It generalizes the Montroll-Weiss formula for RTPs with arbitrary angular distributions and uncovers hyperdiffusive scaling in their mean squared displacement.

Findings

Displacement moments depend on finite angular velocity reorientation moments.

Derived the angular distribution of velocity reorientation for one-state RTPs.

Identified hyperdiffusive scaling with an anomalous exponent between 2 and 3.

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.