The random walk of intermittently self-propelled particles

TL;DR

This work develops a general two-state renewal model for intermittently self-propelled particles that switch between active runs and turns according to arbitrary waiting-time distributions. By deriving the velocity autocorrelation and mean-square displacement within renewal theory, it provides exact expressions for the diffusion coefficient and identifies conditions for sub-, normal-, and superdiffusive long-time transport, including special limits corresponding to active Brownian motion, run-and-tumble, and Lévy-walk-like behaviors. The results reveal how the lifetimes and reorientation statistics of the motility modes control ergodicity, short-time ballistic scaling, and intermediate-to-long-time transport, with tempered tails yielding intermediate anomalous diffusion. The framework unifies multiple transport processes and offers explicit predictions for experimental tracking data, with extensions to higher dimensions and potential applications to biased motion in fields or gradients.

Abstract

Motivated by various recent experimental findings, we propose a dynamical model of intermittently self-propelled particles: active particles that recurrently switch between two modes of motion, namely an active run-state and a turn state, in which self-propulsion is absent. The durations of these motility modes are derived from arbitrary waiting-time distributions. We derive the expressions for exact forms of transport characteristics like mean-square displacements and diffusion coefficients to describe such processes. Furthermore, the conditions for the emergence of sub- and superdiffusion in the long-time limit are presented. We give examples of some important processes that occur as limiting cases of our system, including run-and-tumble motion of bacteria, Lévy walks, hop-and-trap dynamics, intermittent diffusion and continuous time random walks.

Figures (5)

• Figure 1: Illustration of the model dynamics. In panel (a), a schematic trajectory is drawn; run and turn episodes are highlighted in blue and red, respectively. The process is observed after the aging time $t_a$. Panel (b) shows different changes of the orientation vector $\mathbf{ e }$ [cf. inset in panel (a)]: the rotational diffusion coefficient depends on the motility state (first two panels); the last panel illustrates instantaneous reorientations by an angle $\chi$ (flip). The random switching of speed and rotational diffusion is depicted in panels (c) and (d), respectively. The times $t_r$ and $t_t$ are drawn from the waiting-time distributions $\psi_r(t)$ and $\psi_t(t)$. Panel (e) illustrates a time series of the direction of motion $\varphi(t)$, where the effect of different rotational diffusion coefficients and sudden flips are visible by the variance of temporal fluctuations and instantaneous jumps.
• Figure 2: The cartoons show three examples of different random walk patterns which are contained in the model [Eq. \ref{['eq:runeq']}]. (a): run-and-tumble motion, characterized by highly persistent runs, interrupted by abrupt directional changes; (b): self-propelled motion at constant speed with a finite persistence time, interrupted by turns; (c): mobile-immobile dynamics (see main text for a detailed description). In all cases, turn episodes are highlighted by red dots; the blue dot indicates the begin of a trajectory.
• Figure 3: The cartoon shows the temporal sequence of run and turn episodes for one particular realization of the diffusion process, as discussed in the context of the velocity auto-correlation function [Eq. \ref{['eqn:Cvv_time']}]. The process starts at $t=0$ and the observation begins at the aging time $t_a$. Within the lag time $\Delta$, there are $N=2$ turn episodes. The total time spent in the two motility states are $\tau_r = \sum_{i=1}^{N+1} \tau_r^{(i)}$ and $\tau_t = \sum_{i=1}^{N} \tau_t^{(i)}$, respectively. The first and the last observed run is truncated (censored).
• Figure 4: Anomalous diffusion exponent $\gamma$ of the ensemble-averaged MSD, $m_2(t_a=0,\Delta) \sim \Delta^{\!\!\:\gamma}$, shown in color as predicted by Eq. \ref{['eqn:m2_nonequilibrium']}, for run- and turn-time distributions with power-law decay [cf. Eq. \ref{['eqn:psi_power_laws']}]. Panel (a) shows the scaling for runs with a finite persistence time ($D_{\varphi}^{(r)} > 0$); if runs are perfectly straight ($D_{\varphi}^{(r)} = 0$), the phase behavior is richer as shown in (b). The background pattern indicates the type of diffusion process: subdiffusion ($\gamma < 1$), normal diffusion ($\gamma = 1$), superdiffusion ($1 <\gamma < 2$) and ballistic motion ($\gamma = 2$). The diffusion process equilibrates, as discussed in the context of Eq. \ref{['eqn:m2_eq']}, if $\alpha > 1$ and $\beta > 1$---the waiting time distributions possess a finite mean in this case. Note that Brownian diffusion was neglected ($D=0$) to produce the plots above; if isotropic Brownian motion is present during the turn state, subdiffusion is not observed but the corresponding regions are replaced by normal diffusion ($\gamma = 1$).
• Figure 5: MSD of intermittently self-propelled particles with exponentially distributed run-times [Eq. \ref{['eqn:exponential_wtpdf']}] and turn-times that follow a tempered power-law distribution [Eq. \ref{['eqn:temp_power_law']}], obtained by numerical inverse Laplace transform noteSI of Eqs. (\ref{['eqn:msd']}, \ref{['eqn:Cvv_su']}). Panel (a): ensemble-averaged MSD $m_2(t_a,\Delta)$ as a function of the lag time $\Delta$ for various aging times $t_a$. For small aging times, the MSD shows three regimes: ballistic motion---the processes starts with a run---followed by subdiffusion as predicted by Fig. \ref{['fig:long_time_exponents']}; in the long-time limit, there is a crossover to normal diffusion due to the exponential cutoff of waiting times in the distribution $\psi_t(t)$. Panel (b): For the same data, the MSD divided by the lag time $\Delta$ is shown, underlining the sub-linear scaling at intermediate timescales and linear scaling of the MSD for large lag times. The MSD $m_2$ for large aging time ($t_a \rightarrow \infty$) are identical to the ensemble-average of the time-averaged MSD $\langle \delta^2_T(t_a,\Delta) \rangle$, cf. Eq. \ref{['eqn:ergo']}. Parameters: mean run-time $1/\lambda = 1$, timescale $\tau_{\alpha} = 1$, cutoff timescale $\tau_c = 10^6$, exponent $\alpha = 1/2$, rotational diffusion coefficients $D_{\varphi}^{(r)} = 10^{-5}$, $D_{\varphi}^{(t)} = 0$, speed $v_0 = 1$, $\langle \cos \chi \rangle = 0$, isotropic diffusion coefficient $D = 0$.