Kinematics of persistent random walkers with two distinct modes of motion

We study the stochastic motion of active particles that undergo spontaneous transitions between two distinct modes of motion. Each mode is characterized by a velocity distribution and an arbitrary (anti)persistence. We present an analytical formalism to provide a quantitative link between these two microscopic statistical properties of the trajectory and macroscopically observable transport quantities of interest. For exponentially distributed residence times in each state, we derive analytical expressions for the initial anomalous exponent, the characteristic crossover time to the asymptotic diffusive dynamics, and the long-term diffusion constant. We also obtain an exact expression for the time evolution of the mean square displacement over all timescales and provide a recipe to obtain higher displacement moments. Our approach enables us to disentangle the combined effects of velocity, persistence, and switching probabilities between the two states on the kinematics of particles in a wide range of stochastic active or passive processes and to optimize the transport quantities of interest with respect to any of the particle dynamics properties.

Figure 1

Sketch of a sample trajectory with two states of motion. The selected directional changes along the trajectory represent the four possible turning angles introduced in the model: the directional changes within the states [characterized by the turning-angle distributions $R_{{}_{\text{I}}}\left(\varphi \right)$ and $R_{{}_{\text{II}}}\left(\varphi \right)$] and the changes in the direction of motion at the switching events [characterized by the turning-angle distributions $R_{{}_{\text{I}\to \text{II}}}\left(\varphi \right)$ and $R_{{}_{\text{II}\to \text{I}}}\left(\varphi \right)$]. The inset magnifies the trajectory around the successive time steps $t-\mathrm{\Delta }t$ and $t$. After a change $\varphi =\gamma -\beta$ in the direction of motion at $t-\mathrm{\Delta }t$, the particle arrives at position $(x,y)$ along the direction $\gamma$ at time $t$.

Figure 2

Time evolution of the MSD in log-log scale. The symbols denote simulation results and the lines correspond to analytical predictions via Eq. (13). A rich variety of dynamical regimes of anomalous motion can be observed at short and intermediate timescales. The velocity of each state is constant except for the blue (upper) dashed curve, which is obtained for broad uniform velocity distributions (with $\langle v^2\rangle =20{\langle v\rangle }^2$ in both states). All the MSD profiles belong to steady-state initial conditions except for the red (lower) dashed curve, which is obtained for the initial condition $q_0^{\text{I}}=1$, i.e., starting the motion in state I. In all plots we have chosen $a_{{}_{\text{I}\to \text{II}}}=a_{{}_{\text{II}}}$ and $a_{{}_{\text{II}\to \text{I}}}=a_{{}_{\text{I}}}$ for the directional changes at the switching events. An ensemble of ${10}^5$ realizations has been considered for the simulations and the four turning-angle distributions are uniformly distributed around the zero change in the turning angle.

Figure 3

Phase diagrams of the initial anomalous exponent according to Eq. (16). The color intensity reflects the magnitude of $\nu$, with red (blue) meaning superdiffusion (subdiffusion). The dotted regions denote oscillatory subdomains. We consider $a_{{}_{\text{I}\to \text{II}}}=a_{{}_{\text{II}}}$ and $a_{{}_{\text{II}\to \text{I}}}=a_{{}_{\text{I}}}$ in all panels and choose a constant velocity ($\langle v\rangle =\langle v^2\rangle =1$) in both states, except for (c). The initial anomalous exponent $\nu$ is plotted in the $(a_{{}_{\text{I}}},a_{{}_{\text{II}}})$ plane for (a) $f_{{}_{\text{I}\to \text{II}}}=f_{{}_{\text{II}\to \text{I}}}=0.5$ and (b) $f_{{}_{\text{I}\to \text{II}}}=0.1$ and $f_{{}_{\text{II}\to \text{I}}}=0.9$. By the asymmetric choices of $f_{{}_{\text{I}\to \text{II}}}$ and $f_{{}_{\text{II}\to \text{I}}}, \nu$ is influenced more strongly by the persistence of the state with a longer mean residence time. (c) Similar parameter values as in (b) but with broad velocity distributions ($\langle v\rangle =1$ and $\langle v^2\rangle =3$ in both states). Broader velocity distributions push the initial slope of the MSD towards $\nu =1$ (diffusion). The initial anomalous exponent $\nu$ in the $(f_{{}_{\text{I}\to \text{II}}},f_{{}_{\text{II}\to \text{I}}})$ plane for a combination of persistent and antipersistent motions characterized by (d) $a_{{}_{\text{I}}}=0.6$ and $a_{{}_{\text{II}}}=-0.6$ and (e) $a_{{}_{\text{I}}}=0.9$ and $a_{{}_{\text{II}}}=-0.9$. Here $\nu$ is enhanced by the switching probabilities that lead to a longer stay in the persistent mode.

Figure 4

Characteristic time $t_{{\text{c}}_{+}}$ via Eq. (14) in terms of (a) $a_{{}_{\text{I}}}$ and (b) $f_{{}_{\text{II}\to \text{I}}}$. The switching persistencies are chosen as $a_{{}_{\text{I}\to \text{II}}}=a_{{}_{\text{II}\to \text{I}}}=1$ in both panels. The crossover time grows by several orders of magnitude by increasing the persistence of the states or increasing the residence time in the more persistent mode. (a) Results for several values of $a_{{}_{\text{II}}}$ for a given set of $f_{{}_{\text{I}\to \text{II}}}$ and $f_{{}_{\text{II}\to \text{I}}}$ parameters. (b) The $a_{{}_{\text{I}}}$ and $a_{{}_{\text{II}}}$ are fixed and $t_{{\text{c}}_{+}}$ is shown vs $f_{{}_{\text{II}\to \text{I}}}$ for several values of $f_{{}_{\text{I}\to \text{II}}}$.

Figure 5

Plot of $D_{\infty }$ in the (a) $(a_{{}_{\text{I}}},a_{{}_{\text{II}}})$ and (b) $(f_{{}_{\text{I}\to \text{II}}},f_{{}_{\text{II}\to \text{I}}})$ planes, scaled by $v^2\mathrm{\Delta }t$, for a constant velocity $v_{\text{I}}=v_{\text{II}}=v$ and the parameter values (unless varied) $f_{{}_{\text{I}\to \text{II}}}=0.1, f_{{}_{\text{II}\to \text{I}}}=0.9, a_{{}_{\text{I}}}=0.9, a_{{}_{\text{II}}}=-0.9, a_{{}_{\text{II}\to \text{I}}}=a_{{}_{\text{I}}}$, and $a_{{}_{\text{I}\to \text{II}}}=a_{{}_{\text{II}}}$. By the chosen asymmetric switching probabilities in (a), the walker spends longer times in state I and $D_{\infty }$ is more sensitive to $a_{{}_{\text{I}}}$. (b) Tuning the residence time in each state via $f_{{}_{\text{I}\to \text{II}}}$ and $f_{{}_{\text{II}\to \text{I}}}$ in a combination of persistent and antipersistent motions dramatically influences $D_{\infty }$. (c) Variations of $D_{\infty }$ and $t_{{\text{c}}_{+}}$ in a run-and-tumble process in terms of the mean tumble-to-run turning angle ${\langle \varphi \rangle }_{\text{tumble}\to \text{run}}$ (correspondingly $a_{\text{tumble}\to \text{run}}={\langle cos\varphi \rangle }_{\text{tumble}\to \text{run}}$). The other parameter values are $f_{\text{run}\to \text{tumble}}=f_{\text{tumble}\to \text{run}}=0.1, a_{\text{run}}=0.9, a_{\text{tumble}}=0, a_{\text{run}\to \text{tumble}}=1$, and constant velocities $v_{\text{run}}=2v_{\text{tumble}}$.

Figure 6

(a) Typical trajectory of a single-state persistent walker with an asymmetric turning-angle distribution uniformly distributed over the range $\varphi \in [-\frac{\pi }{6},\frac{2\pi }{6}]$. The asymmetry of the turning angles leads to a trajectory with frequent clockwise spirals. The resulting persistence parameter has real and imaginary parts $a_{\text{Re}}\simeq 0.9$ and $a_{\text{Im}}\simeq 0.2$, respectively. (b) Subset of optimal switching probabilities $f_{{}_{\text{I}\to \text{II}}}^{\text{opt}}$ and $f_{{}_{\text{II}\to \text{I}}}^{\text{opt}}$ that maximize the asymptotic diffusion constant according to Eqs. (20) and (21) in the $(f_{{}_{\text{I}\to \text{II}}},f_{{}_{\text{II}\to \text{I}}})$ plane for various choices of the velocity moments in a combination of ballistic and diffusive processes.