Speeding up Brownian escape via intermediate finite potential barriers

TL;DR

The paper addresses accelerating thermally activated barrier crossing by reshaping a single energy barrier into a sequence of intermediate barriers while preserving the total height $\Delta U$. Using the overdamped Langevin framework and the backward Fokker–Planck formalism, it derives MFPT expressions for linear and harmonic base potentials and demonstrates that introducing intermediate barriers can significantly reduce the MFPT, with further reductions as the number of barriers increases (notably for even numbers of barriers). The work provides both analytical results (for linear and harmonic segments) and numerical demonstrations, and highlights experimental feasibility via optical trapping. It also discusses reductions in FPT variance and outlines open questions, including rigorous proofs of MFPT reduction and extensions to more complex dynamics. Overall, the findings establish intermediate barriers as a practical control strategy for speeding thermally activated transitions in diffusive systems.

Abstract

The mean first-passage time (MFPT) for a Brownian particle to surmount a potential barrier of height $ΔU$ is a fundamental quantity governing a wide array of physical and chemical processes. According to the Arrhenius Law, the MFPT typically grows exponentially with increasing barrier height, reflecting the rarity of thermally activated escape events. In this work, we demonstrate that the MFPT can be significantly reduced by reshaping the original single-barrier potential into a structured energy landscape comprising multiple intermediate barriers of lower heights, while keeping the total barrier height $ΔU$ unchanged. Furthermore, this counterintuitive result holds across both linear and nonlinear potential profiles. Our findings suggest that tailoring the energy landscape -- by introducing well-placed intermediate barriers -- can serve as an effective control strategy to accelerate thermally activated transitions. These predictions are amenable to experimental validation using optical trapping techniques.

Figures (8)

• Figure 1: Panel (a): Schematic of a Brownian particle (large sphere) that starts at $x=0$, which is a reflecting boundary, and is eventually absorbed at $x=1$, which is the absorbing boundary, after overcoming a potential barrier of height $\Delta U$. By altering the potential structure by introducing intermediate potential barriers as in panel (b), we would like to examine whether the mean first-passage time to reach the absorbing boundary can be reduced than that from the unaltered potential configuration as in panel (a). The Brownian particle is assumed to be in contact with a thermal bath surrounded by the bath particles (indicated by the smaller spheres).
• Figure 2: Panels (a) $\&$ (c): Piecewise linear and harmonic potential with a single kink at $x=a=0.3$. Panel (b): The inverse speed up factor, $\mathcal{T}_1^\text{L}/\mathcal{T}_0^\text{L}$, for linear case is plotted as a function of $k_1$ with $\Delta U = 3$ and $a=0.3$. For a range of $k_1$ values, $\mathcal{T}_1^\text{L}/\mathcal{T}_0^\text{L}$ is less than $1$, implying that the modification has accelerated the process completion in these regimes. Panel (d): The inverse speed-up factor, $\mathcal{T}_1^\text{H}/\mathcal{T}_0^\text{H}$, for harmonic case is plotted as a function of $k_1$ for $\Delta U = 3$ and $a=0.3$. Similar to the linear case, $\mathcal{T}_1^\text{H}/\mathcal{T}_0^\text{H}<1$ for several $k_1$ values. Simulation data are shown as points, and the analytical solution is plotted as continuous curve and $k_BT$ is taken to be $1$.
• Figure 3: Panels (a) $\&$ (c) show the optimal potential profiles (for various values of $a$) for the case of one kink in linear and harmonic potentials respectively. Panel (b) shows the variation of the inverse maximal speed-up factor and the corresponding optimal strengths with $a$, for the linear case, with $\Delta U=3$. The left vertical axis represents this dimensionless factor, while the right vertical axis indicates the corresponding optimal strengths, $k_1^*$ and $k_2^*$. For $a=0.5$, we observe $k_1^*=k_2^*$, implying that the optimal potential is the original linear potential itself in this case from panel (a). Panel (d) depicts the variation of these quantities for harmonic case, for $\Delta U=3$. It shows that the inverse maximal speed-up factor never reaches unity. Similar to the linear case, there exists a range of $a$ values for which $k_1^* > k_2^*$, while for other values, $k_2^* > k_1^*$ as shown in panel (d). The MFPTs are determined analytically and the corresponding optimal values have been found numerically, while fixing $k_BT$ to unity.
• Figure 4: The figure presents the linear and harmonic potentials (in blue) along with their corresponding modified potentials after introducing $N = 2n = 2$ kinks (in black), as shown in panels (a) and (b), respectively. The modified potential consists of segments with alternating strengths $k_1$ and $k_2$. In both cases, the kinks are equally spaced and positioned at $x = a$ and $x = 2a$, where $a = 1/3$.
• Figure 5: Variation of the inverse maximal speed-up factor, $\mathcal{T}_N^{\text{L}^*}/\mathcal{T}_0^\text{L}$, and the optimal strengths, with the number of kinks $N$. The left vertical axis represents the inverse maximal speed-up factor, while the right vertical axis corresponds to optimal strengths, $k_1^*$ and $k_2^*$. The barrier height is fixed at $\Delta U=5$ and $k_BT=1$. To obtain these results, the MFPT for a given $N$ is first computed analytically using Mathematica and the corresponding optimal MFPT is evaluated by numerical minimization. Panel (a) shows that for any odd $N$, the inverse maximal speed-up factor is unity, indicating no reduction in the MFPT. In these cases, the optimal strengths satisfy $k_1^* = k_2^*$. Panel (b) illustrates the behavior for even values of $N = 2n$. The inverse maximal speed-up factor is less than $1$ and decreases monotonically, indicating successive reduction of the MFPT with increasing $N$. The case $N = 0$ corresponds to the unmodified linear potential.