Escape rate of active particles in the effective equilibrium approach

The escape rate of a Brownian particle over a potential barrier is accurately described by the Kramers theory. A quantitative theory explicitly taking the activity of Brownian particles into account has been lacking due to the inherently out-of-equilibrium nature of these particles. Using an effective equilibrium approach [Farage et al., Phys. Rev. E 91, 042310 (2015)] we study the escape rate of active particles over a potential barrier and compare our analytical results with data from direct numerical simulation of the colored noise Langevin equation. The effective equilibrium approach generates an effective potential that, when used as input to Kramers rate theory, provides results in excellent agreement with the simulation data.

Figure 1

Bare potential, Eq. (8), and analytic effective potential $V^{\mathrm{eff}}\left(x\right)$, Eq. (9), for different values of ${\mathcal{D}}_{\text{a}}$ (see legend). For the given parameters ${\omega }_0=10$, $\alpha =1$, $\tau =0.02$, and $D_{\text{t}}=1$, these results are indistinguishable from the numeric solution of Eq. (6). We denote by $x_{\text{a}}=0$ the local minimum of the potential from where particles escape over the barrier at $x_{\text{b}}$ to the sink located at arbitrary $x_{\text{c}}$. For numerical treatment $x_{\text{c}}$ will be obtained as the solution of $\beta V^{\mathrm{eff}}\left(x\right)=-20$. The orange vertical arrow indicates the decreasing potential barrier with increasing activity. The change in the curvature of the energy landscape is clearly evident. We use the curvature at $x_{\text{b,0}}={\omega }_0/\left(3\alpha \right)$ to approximate the curvature at the (effective) maximum, which shifts slightly toward larger values of $x$.

Figure 2

Escape rate of active particles as a function of the active diffusivity ${\mathcal{D}}_{\text{a}}$. Escape rate of active particles is expressed in units of the escape rate of passive particles over the barrier of the bare potential. The rate of escape is calculated for three different values of $\tau$ (see legend). The curvature of the bare potential at $x_{\text{a}}$ is fixed to ${\omega }_{\text{0}}=10$ and the nonlinear parameter is $\alpha =1$. The escape rate $r_{\text{act}}$ increases by several orders of magnitude with increasing ${\mathcal{D}}_{\text{a}}$. The circles represent the rate calculated within the effective equilibrium approach and is obtained by numerically solving Eq. (17). The squares correspond to the rate obtained by simulating the colored-noise Langevin Eq. (2). The solid black lines correspond to the analytic predictions of Eq. (20) using the effective potential in the Kramers approach. The excellent agreement between the predictions of Eq. (20) with the numerically obtained rate from Eq. (17) indicates the high-accuracy of the Kramers analytical approach used in calculation of Eq. (20). The escape rate calculated using the colored-noise Langevin equation [Eq. (2)] starts deviating from the prediction of Eq. (20) for large ${\mathcal{D}}_{\text{a}}$.

Figure 3

Escape rate in units of $\beta D_{\text{t}}$ for different values of the curvature ${\omega }_0$ of the bare potential Eq. (8) with (a) $\alpha ={\omega }_0/10$ chosen thus that the barrier is always located at a fixed $x_{\text{b,0}}$ and its height $\beta E_{\text{0}}$ increases with ${\omega }_0$, and (b) $\alpha =\sqrt{{{\omega }_0^3}^{\phantom{0}}}/35$ such that $x_{\text{b,0}}$ moves toward the origin with increasing ${\omega }_0$, while maintaining a constant barrier height. The data in (a) are plotted on a logarithmic scale and in (b) on linear scale to highlight the exponential and almost linear behavior of the escape rate as a function of ${\omega }_{\text{0}}$, respectively. The parameters are set to $\tau =0.02$ and ${\mathcal{D}}_{\text{a}}=2/3$. The dashed black line indicates a linearly increasing $r_{\text{act}}$ with a slope $m=\beta D_{\text{t}}exp\left[-\beta E_{\text{0}}/(1+{\mathcal{D}}_{\text{a}})\right]/\left(2\pi \right)$ [Eq. (21)]. For such small values of $r_{\text{act}}$, statistical fluctuations in the numerically measured escape rate (squares) make it difficult to ascertain the functional dependence of $r_{\text{act}}$ on ${\omega }_{\text{0}}$. It appears that $r_{\text{act}}$ becomes slightly nonlinear with increasing ${\omega }_{\text{0}}$ as it gets closer to the solid black line, which corresponds to Eq. (20). Nevertheless, the approximately linear dependence of $r_{\text{act}}$ on ${\omega }_{\text{0}}$ is clearly evident. The inset of (b) is a plot of the escape rate in Eq. (20) for different values of ${\mathcal{D}}_{\text{a}}$ for fixed $\beta E_{\text{0}}$. In the direction of the arrow ${\mathcal{D}}_{\text{a}}$ is 0.5, 1, 2, 4, and 6. $r_{\text{act}}$ is normalized with respect to the slope $m$ to highlight the nonlinearity with increasing ${\omega }_{\text{0}}$. The dash-dotted line of unit slope corresponds to the exactly linear variation of $r_{\text{act}}$ with ${\omega }_{\text{0}}$ in the limit of ${\mathcal{D}}_{\text{a}}=0$.