Thermally induced passage and current of particles in a highly unstable optical potential

We discuss the statistics of first-passage times of a Brownian particle moving in a highly unstable nonlinear potential proportional to an odd power of position. We observe temperature-induced shortening of the mean first-passage time and its dependence on the power of nonlinearity. We propose a passage-time fraction as both a simple and experimentally detectable witness of the nonlinearity. It is advantageously independent of all other parameters of the experiment and observable for a small number of trajectories. To better characterize the stochastic passage in the unstable potential, we introduce an analogy of the signal-to-noise ratio for the statistical distribution of the first-passage times. Interestingly, the upper bound for the signal-to-noise ratio is temperature independent in the unstable potential. Finally, we describe the nonequilibrium steady state of the particle cyclically passing through unstable odd nonlinearity. The maximum of the steady-state probability distribution shifts against the directions of the current and this counterintuitive effect increases with temperature. All these thermally induced effects are very promising targets for experimental tests of highly nonlinear stochastic dynamics of particles placed into optical potential landscapes of shaped optical tweezers.

Figure 1

The bottom panel shows three trajectories of Brownian motion in the nonlinear cubic potential $U\left(x\right)=\mu x^n$, $n=3$, schematically illustrated in the top panel. Within the interval $(-l,l)$, $l={(k_{B}T/\mu )}^{1/n}$ [see Eq. (17)], thermal fluctuations have a pronounced effect on the motion of the particle. The dynamics out of $(-l,l)$ is rather fast and for $\left|x\right|\gg l$ it is close to the deterministic one since the thermal noise is negligible compared to the rapidly increasing (absolute) value of $U\left(x\right)$. The top panel indicates also the scaling of the mean first-passage time with the temperature $T$ as discussed in Sec. 3b.

Figure 2

Simulated PDF $f_a\left(t\right|y)$ of the first-passage time $\mathbf{T}(y\to a)$ for the cubic potential and three temperatures. The parameters used are $y=4$, $a=-4$, and $\gamma =k_B=\mu =1$. The trajectories of the Brownian motion were obtained by a numerical integration of the Langevin equation (2) using the Euler-Maruyama method [29]. Each histogram of the FPTs was computed from ${10}^6$ simulated trajectories ($N_{\mathrm{traj}}={10}^6$).

Figure 3

Shown on top is the simulated PDF $f_0\left(t\right|y)$ of the FPT $\mathbf{T}(y\to 0)$ and on bottom the simulated PDF $f_a\left(t\right|0)$ of the FPT $\mathbf{T}(0\to a)$ for the cubic potential and three temperatures. The parameters used are $y=4$, $a=-4$, $\gamma =k_B=\mu =1$, and $N_{\mathrm{traj}}={10}^6$.

Figure 4

The MFPT $〈\mathbf{T}(y\to a)〉$ as a function of the temperature $T$ computed from 1000 simulated trajectories. The parameters used are $a=-y$ and $\gamma =k_B=\mu =1$.

Figure 5

Even for a small number of trajectories, the fractions (27) are obtained in the simulation with sufficient precision. We have used $n=3$, $N_{\mathrm{traj}}=1000$, $y=5$, $a=-5$, and $dt={10}^{-4}$.

Figure 6

The SNR as a function of the temperature $T$ computed from 1000 simulated trajectories for $n=3$ and the same parameters as in Fig. 4. The upper bound on the SNR is given by the temperature-independent result (32). The theoretical curves (red solid lines) are computed as $〈\mathbf{T}(y\to a)〉/\sqrt{〈{\left[\mathrm{\Delta }\mathbf{T}(\infty \to -\infty )\right]}^2〉}$, where the first moment of the FPT $〈\mathbf{T}(y\to a)〉$ is given by Eq. (25) and its asymptotic standard deviation by Eq. (30).

Figure 7

Stationary PDF (36) for three temperatures. The parameters used are $k_B=\gamma =\mu =1$.