Two-dimensional active motion

The diffusion in two dimensions of noninteracting active particles that follow an arbitrary motility pattern is considered for analysis. A Fokker-Planck-like equation is generalized to take into account an arbitrary distribution of scattered angles of the swimming direction, which encompasses the pattern of active motion of particles that move at constant speed. An exact analytical expression for the marginal probability density of finding a particle on a given position at a given instant, independently of its direction of motion, is provided, and a connection with a generalized diffusion equation is unveiled. Exact analytical expressions for the time dependence of the mean-square displacement and of the kurtosis of the distribution of the particle positions are presented. The analysis is focused in the intermediate-time regime, where the effects of the specific pattern of active motion are conspicuous. For this, it is shown that only the expectation value of the first two harmonics of the scattering angle of the direction of motion are needed. The effects of persistence and of circular motion are discussed for different families of distributions of the scattered direction of motion.

Figure 1

Dimensionless mean-square displacement ${\mathrm{\Gamma }}_1^2\langle {\mathit{x}}^2\left(t\right)\rangle /v_0^2$ as function of the dimensionless time ${\mathrm{\Gamma }}_{1}t$ for different values of the ratio ${\mathrm{\Omega }}_1/{\mathrm{\Gamma }}_1$, namely, 0.1, 1, 10, 100.

Figure 2

The first 10 values of ${\mathrm{\Gamma }}_n/\mathrm{\Lambda }$ are shown for different unimodal distributions of scattered angles centered at the forward direction of motion. For the Lévy $\alpha$-stable distributions wrapped to the circle (circle symbols), ${\tilde{Q}}_{\text{wr}}\left(\phi \right), \alpha =2$ (wrapped Gaussian), 3/2, 1 (wrapped Lorentz), and 1/2 were chosen, all with parameters $\sigma =1, \beta =0$. For the Jones and Pewsey distributions of scattered angles (square symbols), ${\tilde{Q}}_{\text{JP},\sigma ,\varphi ,\psi }$ the values $\sigma =1, \varphi =0$, and $\psi =-1$ (cardioid), $-1/2$, 0 (von Misses), 1/2, and 1 (wrapped Cauchy) were chosen. Notice the saturation value ${\mathrm{\Gamma }}_n/\mathrm{\Lambda }=1$, which is half the maximum value allowed (see text in Sec. 3).

Figure 3

Kurtosis as function of the dimensionless time ${\mathrm{\Gamma }}_{1}t$ for symmetric scattering-angle distribution ${\tilde{Q}}_S\left(\phi \right)$. The values of the ratio ${\mathrm{\Gamma }}_2/{\mathrm{\Gamma }}_1=0.1$ (dashed red line), 1 [solid black line given by Eq. (63)], 10 (dashed-dotted blue line), and 100 (dashed-double-dotted magenta line) have been considered. Horizontal thin-dotted lines mark the values $\kappa =8$ and 4 that correspond to the cases for which the probability density $p_0(\mathit{x},t)$ is Gaussian in the long-time regime ($\kappa =8$) and a sharp pulse that propagates with speed $v_0$ ($\kappa =4$), respectively. The solid-golden line corresponds to the time dependence of the kurtosis for active Brownian motion with rotational diffusion constant equal to ${\mathrm{\Gamma }}_1$.

Figure 4

The ratios ${\mathrm{\Gamma }}_2/{\mathrm{\Gamma }}_1$ (solid line), ${\mathrm{\Omega }}_1/{\mathrm{\Gamma }}_1$ (dashed line) and ${\mathrm{\Omega }}_2/{\mathrm{\Gamma }}_1$ (dashed-dotted line) are shown as functions of the mode $\varphi$, when $\tilde{Q}\left(\phi \right)$ is given by the wrapped-Gaussian distribution (wrapped stable distribution with index $\alpha =2$), with values of the scale parameter $\sigma =0.25$ (thick dark lines) and 0.1 (thin fuzzy lines).

Figure 5

The time dependence of the kurtosis is shown for $\tilde{Q}\left(\phi \right)$ given by the wrapped-Gaussian distribution (wrapped stable distribution with index $\alpha =2$), with values of the scale parameter $\sigma =0.1$ (thick blue lines) and 0.25 (thin red lines). The values of the ratios: ${\mathrm{\Gamma }}_2/{\mathrm{\Gamma }}_1, {\mathrm{\Omega }}_1/{\mathrm{\Gamma }}_1$, and ${\mathrm{\Omega }}_2/{\mathrm{\Gamma }}_1$ correspond to those values of $\varphi \in [0,\pi ]$, for which ${\mathrm{\Omega }}_2/{\mathrm{\Gamma }}_1$ is maximum (solid blue and dashed red lines) and when is minimum (thick dotted-dashed and thin dotted-dashed lines), as shown in Fig. 4.

Figure 6

Level curves of constant ratio $\chi ={\mathrm{\Omega }}_1/{\mathrm{\Gamma }}_1$ in the plane $\nu -{\phi }_0$. The solid (red) line corresponds to the case ${\mathrm{\Omega }}_1/{\mathrm{\Gamma }}_1=1$ for which the effective diffusion coefficient is maximum. Dashed (blue) lines correspond to the cases for which $|{\mathrm{\Omega }}_1/{\mathrm{\Gamma }}_1|<1$ (shown $\chi =10$ and 2), while dashed-dotted (cyan) lines correspond to the cases for which $|{\mathrm{\Omega }}_1/{\mathrm{\Gamma }}_1|>1$ (shown $\chi =0.5$ and 0.1).

Figure 7

Time dependence of the kurtosis $\kappa \left(t\right)$ for the bimodal distribution of scattered angles (65). The values of ratios ${\mathrm{\Gamma }}_2/{\mathrm{\Gamma }}_1, {\mathrm{\Omega }}_2/{\mathrm{\Gamma }}_1$ correspond to the values of ${\phi }_0^{*}$ that make $\nu =1$ on the contour lines given in Fig. 6 for the values of $\chi =10$, 2, 1, 0.5, and 0.1.

Figure 8

The time dependence of the kurtosis is shown for the bimodal distribution with equally weighed modes at 0 and $\pi$, which corresponds to a particular case of the patterns of motion called run-and-reverse.