Most probable path of active Ornstein-Uhlenbeck particles

Using the path integral representation of the nonequilibrium dynamics, we compute the most probable path between arbitrary starting and final points that is followed by an active particle driven by persistent noise. We focus our attention on the case of active particles immersed in harmonic potentials, where the trajectory can be computed analytically. Once we consider the extended Markovian dynamics where the self-propulsive drive evolves according to an Ornstein-Uhlenbeck process, we can compute the trajectory analytically with arbitrary conditions on position and self-propulsion velocity. We test the analytical predictions against numerical simulations and we compare the analytical results with those obtained within approximated equilibriumlike dynamics.

Figure 1

(a) We consider a particle located at the initial time $t_0$ in $(t_0,{\varphi }_0)$ that reaches the final point $(t_1,{\varphi }_1)$ at the time $t_1$. Once we introduce the probability $\mathcal{P}({\varphi }_1,t_1,{\varphi }_0,t_0)$ as a sum over the paths connecting the two points, the most probable path corresponds to the saddle-point solution of the path integral. (b) The most probable trajectories return the value $S_{SP}$ of the dynamical action at the saddle point. For downhill trajectories, i.e., starting from ${\varphi }_0\ne 0$ and ending in ${\varphi }_1=0$, the dynamical action vanishes so that $S_{SP}=0$. In the case of the uphill trajectories connecting ${\varphi }_0=0$ with ${\varphi }_1\ne 0, S_{SP}$ returns a finite positive value.

Figure 2

Active Ornstein-Uhlenbeck particle in a harmonic potential. (a) The most probable path connecting $(t_0=0,{\varphi }_0=0)$ with $(t_1=0.95,{\varphi }_1=0.1)$ ($\tau =2$) for different values of ${\phi }_1\in [-0.6,0.8]$ (see legend). Colors from violet (dark in greyscale) to yellow (bright in greyscale) refer to numerical simulations, and dashed red curves are the analytical solution. (b) The most probable path from $(0,0)$ to $(10,1)$ for different values of $\tau$ (see legend). Red dashed curves are the theoretical prediction Eq. (53). The dotted black curve is the Brownian limit Eq. (16). (c) The color map indicates the number of trajectories connecting ${\varphi }_0$ with ${\varphi }_1$ (from the numerical data, $\tau =2$). The number of trajectories increases from red (dark in greyscale) to blue (bright in greyscale). The dashed green curve is the average path, and the dotted red curve is the theoretical prediction.

Figure 3

Most probable path within the effective equilibrium action. (a) Solid curves are the analytical solution $\varphi \left(t\right)$, dashed curves the most probable trajectories ${\varphi }_{UCN}\left(t\right)$ within an effective equilibrium approach (increasing values of $\tau$ from violet (dark in greyscale) to yellow (bright in greyscale), see legend). (b) The difference between $\varphi$ and ${\varphi }_{UCN}$ reveals the discrepancy between the two dynamics almost everywhere in a wide range of $\tau$ values.

Figure 4

Fraction of trajectories around the final point ${\varphi }_1=\varphi \left(t_1\right)$, with ${\varphi }_1=1, t_1=10$, and ${\varphi }_0=\varphi \left(0\right)=0$, as a function of $\tau$.