Active processes in one dimension

We consider the thermal and athermal overdamped motion of particles in one-dimensional geometries where discrete internal degrees of freedom (spin) are coupled with the translational motion. Adding a driving velocity that depends on the time-dependent spin constitutes the simplest model of active particles (run-and-tumble processes) where the violation of the equipartition principle and of the Sutherland-Einstein relation can be studied in detail even when there is generalized reversibility. We give an example (with four spin values) where the irreversibility of the translational motion manifests itself only in higher-order (than two) time correlations. We derive a generalized telegraph equation as the Smoluchowski equation for the spatial density for an arbitrary number of spin values. We also investigate the Arrhenius exponential law for run-and-tumble particles; due to their activity the slope of the potential becomes important in contrast to the passive diffusion case and activity enhances the escape from a potential well (if that slope is high enough). Finally, in the absence of a driving velocity, the presence of internal currents such as in the chemistry of molecular motors may be transmitted to the translational motion and the internal activity is crucial for the direction of the emerging spatial current.

Figure 1

See formula (37). The plot gives ${\lambda }_2$ for $\mathcal{E}=T=1$ as function of the driving $c/\mathcal{E}$ for various amounts of persistence. The escape time (37) decreases with $c/\mathcal{E}$, and faster for larger persistence $\propto 1/a$. The ordering of the graphs (from top to bottom) agrees with the order of the labels in the legend.

Figure 2

${\lambda }_2$ as function of persistence $1/a$ for several driving amplitudes $c/\mathcal{E}$ while again $\mathcal{E}=T=1$. Larger driving decreases the escape time (37) for all values of persistence. The lower curve is $1/a\mapsto \frac{\mathcal{E}}{T+\frac{c^2}{2a}}$, in good agreement for small persistence, here compared with ${\lambda }_2$ for $c=1$. For large persistence the asymptotic values are $(\mathcal{E}-c)/T$. The ordering of the graphs (from top to bottom) agrees with the order of the labels in the legend.

Figure 3

Internal states making the pearls on a necklace. Here six internal states are associated to each discrete position. Trajectory A expends as much entropy as trajectory B, leaving it undecided by the second law whether the spatial motion will be toward the right or towards the left. Here six internal states are associated to each discrete position. (Courtesy Stichelbaut [28].)