Persistent-random-walk approach to anomalous transport of self-propelled particles

The motion of self-propelled particles is modeled as a persistent random walk. An analytical framework is developed that allows the derivation of exact expressions for the time evolution of arbitrary moments of the persistent walk's displacement. It is shown that the interplay of step length and turning angle distributions and self-propulsion produces various signs of anomalous diffusion at short time scales and asymptotically a normal diffusion behavior with a broad range of diffusion coefficients. The crossover from the anomalous short-time behavior to the asymptotic diffusion regime is studied and the parameter dependencies of the crossover time are discussed. Higher moments of the displacement distribution are calculated and analytical expressions for the time evolution of the skewness and the kurtosis of the distribution are presented.

Figure 1

Trajectory of the walker during two successive steps.

Figure 2

Illustration of symmetric rotations with respect to the incoming direction, in two (left) and three (right) dimensions.

Figure 3

Time evolution of the mean-square displacement $\langle r^2\rangle$ scaled by $n$ for different values of $\lambda ,p$, and $\mathcal{R}$. The solid lines correspond to analytical predictions via Eq. (21), and the symbols denote simulation results.

Figure 4

(a) The crossover time $n_{{}_c}$ to the asymptotic diffusive regime versus self-propulsion $p$. (b) The asymptotic diffusion coefficient $D$ in terms of self-propulsion $p$ (light gray curves) or anisotropy $\mathcal{R}$ (dark gray curves). The solid, dashed, and dotted lines correspond to $\lambda =1,2,5$, respectively.

Figure 5

Time evolution of $\langle r^2\rangle$ for two different distributions $\mathcal{F}\left(\ell \right)$ with the same moments $\langle \ell \rangle =2$ and $\langle {\ell }^2\rangle =5 \left(\lambda =1.25\right)$. Inset: Comparison between the exponential distribution $\mathcal{F}\left(\ell \right)=e^{1-\ell } \left(\ell \in [1,\infty ]\right)$ and the uniform distribution $\mathcal{F}\left(\ell \right)=\frac{H(\ell -{\ell }_{\text{min}})+H({\ell }_{\text{max}}-\ell )-1}{{\ell }_{\text{max}}-{\ell }_{\text{min}}} ({\ell }_{\text{min}}=0.268,{\ell }_{\text{max}}=3.732)$.

Figure 6

(a) Mean-square displacement $\langle r^2\rangle$ vs $n$ for three different distributions $R\left(\varphi \right)$ with $\mathcal{R}=0$ given in the text. The step-length distribution is chosen to be either $\mathcal{F}\left(\ell \right)=\delta (\ell -1)$ (with $p=0$) or $\mathcal{F}\left(\ell \right)=e^{1-\ell }$ (with $p=0.9$). Insets: The possible directions of motion at the next step for each $R\left(\varphi \right)$. (b) A few sample distributions $R\left(\varphi \right)$ with $\mathcal{R}\simeq 0.4$ (top) and $\mathcal{R}\simeq -0.4$ (bottom). The dotted lines show the arrival direction and the arrows indicate the possible directions in the next step, with length being proportional to the probability.

Figure 7

Typical trajectories at $p=0$ and $\lambda =1$ for isotropic (left), forward symmetric (middle), and forward asymmetric (right) turning-angle distributions, after the same number of steps. The arrows show possible directions of motion in the next step.

Figure 8

Time evolution of the fourth moment of displacement for different values of $\lambda ,p$, and $\mathcal{R}$. The solid lines correspond to analytical predictions and the symbols denote simulation results.

Figure 9

The fourth cumulant of $x$ (in units of $\langle {\ell }^4\rangle$) in terms of $n$ for a random walk with $p=A_{{}_1}=A_{{}_2}=0$. Inset: The corresponding fourth cumulant of the net displacement $r$ vs $n$.

Figure 10

The kurtosis measure ${\beta }_2$ in terms of the step number $n$ for different turning-angle distributions. The lines correspond to analytical predictions via Eq. (36) and the symbols denote simulation results.

Figure 11

(a) Comparison between $\langle x^2{y}^2\rangle$ (solid lines) and $\langle x^2\rangle \langle y^2\rangle$ (dashed lines) (both presented in units of $\langle {\ell }^4\rangle$), obtained from the simulations where the short-time motion is sub- $\left(\mathcal{R}=-0.9\right)$, normal $\left(\mathcal{R}=0\right)$, or superdiffusion $\left(\mathcal{R}=0.9\right)$. (b) $\langle r^4\rangle$ vs $n$ from simulations (symbols) or via Eq. (37) (solid lines).

Figure 12

(a) Probability distribution of the distance $r$ (in units of $\langle \ell \rangle$) from the origin at $n=60,190,1400,$ and 4400 (solid, dashed, dash-dotted, and dotted lines, respectively), separately shown for persistent walks with short-time sub- (left), normal (middle), and superdiffusive motion (right). Insets: Collapse of $P\left(r\right)\sqrt{n}$ vs $r/\sqrt{n}$. (b) Comparison between persistent walks with short-time sub- (dashed lines), normal (dash-dotted lines), and superdiffusive motion (dotted lines) at the early stages of the walk (left) and after a long time (right). Insets: $P\left(r\right)\sqrt{Dn}$ in terms of $r/\sqrt{Dn}$, where $D$ is the asymptotic diffusion coefficient. The solid lines are obtained from Eq. (39).

Figure 13

(a) Evolution of the angular distribution $f_{{}_n}\left(\alpha \right)$ of the direction of motion $\alpha$ in the lab frame. The walker initially arrives along the $+x$ direction. A comparison is made between motions with short-time sub- $\left(\mathcal{R}=-0.9\right)$, normal $\left(\mathcal{R}=0\right)$, and superdiffusion $\left(\mathcal{R}=0.9\right)$. The lines are obtained from Eq. (41) and symbols denote simulation results. The gray (dotted) lines are guides to eye. (b) Probability $f_{{}_n}(\pm \frac{\pi }{2})$ of turning to a perpendicular direction after $n$ steps from simulations (symbols) or via Eq. (41) (dashed lines).