Run-and-tumble particles in two dimensions: Marginal position distributions

We study a set of run-and-tumble particle (RTP) dynamics in two spatial dimensions. In the first case of the orientation $\theta$ of the particle can assume a set of $n$ possible discrete values, while in the second case $\theta$ is a continuous variable. We calculate exactly the marginal position distributions for $n=3,4$ and the continuous case and show that in all cases the RTP shows a crossover from a ballistic to diffusive regime. The ballistic regime is a typical signature of the active nature of the systems and is characterized by nontrivial position distributions which depend on the specific model. We also show that the signature of activity at long times can be found in the atypical fluctuations, which we also characterize by computing the large deviation functions explicitly.

Figure 1

Schematic representation of the three different RTP dynamics considered (upper panel) and the corresponding trajectories (lower panel). Panels (a) and (d) correspond to the three-state model, panels (b) and (e) correspond to the four-state model, and panels (c) and (f) correspond to the continuous model.

Figure 2

Plot of the 2D position distribution $P(x,y;t)$, obtained from numerical simulations, of $n=3$ [(a)–(c)], $n=4$ [(d)–(f)], and continuous [(g)–(i)] models for $\gamma =0.1$ The left, middle, and right panels correspond to $t\ll {\gamma }^{-1}$ [(a), (d), (g)], $t\sim {\gamma }^{-1}$ [(b), (e), (h)], and $t\gg {\gamma }^{-1}$ [(c), (f), (i)], respectively. The lighter gray shade indicates lower values of $P(x,y;t)$, and darker shades indicate progressively higher values. Here we have used $v_0=1.$

Figure 3

The effective two-state jump process characterizing the time evolution of ${\sigma }_x$ for the three-state model.

Figure 4

Three-state model: (a) Plot of $P(x,t)$ for $\gamma =1$ and for different values of $t.$ The solid black lines correspond to the analytical prediction Eq. (19), and the symbols correspond to the data obtained from the numerical simulations. For better visibility we have excluded the $\delta$ functions at the two boundaries. (b) Plot of $P(x,t)$ obtained from numerical simulations, as a function of the scaled variable $w=x/\sqrt{t}$ for different (large) values of $t$ and $\gamma =1.$ The red dashed line shows the Gaussian distribution [see Eq. (26)]. The inset shows the same data as a function of $x/t.$ The solid black lines there correspond to the analytical prediction (19). Here we have used $v_0=1.$

Figure 5

Time evolution of the skewness of the $x$-marginal distribution for the three-state process for different values of $\gamma .$ The solid black lines indicate the analytical prediction and the symbols show the data from numerical simulations.

Figure 6

Three-state model: Schematic representation of the jump-process governing the time evolution of ${\sigma }_y.$

Figure 7

Three-state model: (a) Plot of $P(y,t)$ for $\gamma =1$ and for different values of $t.$ The solid black lines correspond to the analytical predictions, Eq. (43) for $t=0.5,1$ and Eq. (40) for the other cases, and the symbols correspond to the data obtained from the numerical simulations. For better visibility we have excluded the $\delta$ functions at the origin and the boundaries. (b) Plot of $P(y,t)$ obtained from numerical simulations, as a function of the scaled variable $w=x/\sqrt{t}$ for different (large) values of $t$ and $\gamma =1.$ The red dashed line shows the Gaussian distribution [see Eq. (46)]. The inset shows the same data as a function of $x/t.$ The solid black lines there correspond to the analytical prediction (40). Here we have used $v_0=1.$

Figure 8

The effective three-state jump process for the four-state model

Figure 9

Four-state model: (a) Plot of $x$ marginal for different values of t and $\gamma =1.0$ The solid black lines correspond to the analytical prediction (59), and the symbols correspond to the data from numerical simulations. For better visibility we have excluded the $\delta$ functions at the origin and the boundaries. (b) Plot of $P(x,t)$ obtained from numerical simulations, as a function of the scaled variable $w=x/\sqrt{t}$ for different (large) values of $t$ and $\gamma =1.$ The red dashed lines shows the corresponding Gaussian distribution. The inset shows the same data as a function of $x/t.$ The solid black lines there correspond to the analytical prediction (61). Here we have used $v_0=1.$

Figure 10

Continuous $\theta$ model: (a) Plot of $x$ marginal for different values of t and $\gamma =1.0$ The solid black lines correspond to the analytical prediction (79), and the symbols correspond to the data from numerical simulations. For better visibility we have excluded the $\delta$ functions at the origin and the boundaries. (b) Plot of $P(x,t)$ obtained from numerical simulations, as a function of the scaled variable $w=x/\sqrt{t}$ for different (large) values of $t$ and $\gamma =1.$ The red dashed line shows the corresponding Gaussian distribution. The inset shows the same data as a function of $x/t.$ The solid black lines there correspond to the analytical prediction (81). Here we have used $v_0=1.$

Figure 11

Illustration of the contour used to evaluate the integral (B7).