Theory of diffusion of active particles that move at constant speed in two dimensions

Starting from a Langevin description of active particles that move with constant speed in infinite two-dimensional space and its corresponding Fokker-Planck equation, we develop a systematic method that allows us to obtain the coarse-grained probability density of finding a particle at a given location and at a given time in arbitrary short-time regimes. By going beyond the diffusive limit, we derive a generalization of the telegrapher equation. Such generalization preserves the hyperbolic structure of the equation and incorporates memory effects in the diffusive term. While no difference is observed for the mean-square displacement computed from the two-dimensional telegrapher equation and from our generalization, the kurtosis results in a sensible parameter that discriminates between both approximations. We carry out a comparative analysis in Fourier space that sheds light on why the standard telegrapher equation is not an appropriate model to describe the propagation of particles with constant speed in dispersive media.

Figure 1

Mean-square displacement in units of ${(v_0/\gamma )}^2$ vs $\gamma t.$ The solid (red) line is the analytical approximation given by expression (16) and the data shown by symbols were obtained by averaging the numerical solution of Eqs. (2) considering ${10}^5$ trajectories and integrating over $2\times {10}^6$ time steps.

Figure 2

Kurtosis $\kappa$ for the circularly symmetric solutions of the TE (9) (dashed magenta line), Eq. (11) (solid red line), and the exact solution obtained from numerical simulations of Eqs. (2) (crosses) vs $t\gamma$. Dotted lines mark the values 8, 4, and $\frac{8}{3}\simeq 2.6667$ that correspond to the values of $\kappa$ for the two-dimensional Gaussian distribution, the two-dimensional distribution of particles that move with constant speed, and the circularly symmetric solutions of the two-dimensional wave equation, respectively. The insets show snapshots obtained from numerical simulations of the particles distribution $P_0(\mathbf{x},t)$ when they start to move from the origin at four different values of $t\gamma ,$ whose values of $\kappa$ are characteristic (closed red triangles): (a) $t\gamma =0.1$ and $\kappa =4.003$, (b) $t\gamma =1.25$ and $\kappa =4.33$, (c) $t\gamma =7.5$ and $\kappa =6.352$, and (d) $t\gamma =225$ and $\kappa =7.938$ (the radius of the distribution has been scaled to fit the viewing area). Numerical simulations were performed by averaging ${10}^5$ trajectories computed from Eqs. (2) and integrating over $2\times {10}^6$ time steps. The dash-dotted green line shows the time dependence of the kurtosis of the symmetric solution of the nonhomogeneous TE obtained in Ref. [30].

Figure 3

Plots showing the probability density ${\tilde{P}}_0(k,t)$ as a function of $k=\left|\mathbf{k}\right|$ at four different times, namely, (a) $\gamma t=0.1$, (b) $\gamma t=1$, (c) $\gamma t=10$, and (d) $\gamma t=100$. The solid red line corresponds to the solution of Eq. (11) and the dashed magenta line is the solution of the TE (9). The numerical solution, considering up to seven Fourier modes in Eq. (7), is shown by the dash-dotted line. In (a) the dashed gray line corresponds to the normalized solution to the two-dimensional wave equation with propagation speed $v_0\sqrt{2}$, namely, $cos{v}_{0}kt/\sqrt{2}$, and the solid gray line corresponds to the short-time limit of the Laplace inversion of expression (13) given by Eq. (C6).