We derive an approximate but fully explicit formula for the mean first-passage time (MFPT) to a small absorbing target of arbitrary shape in a general elongated domain in the plane. Our approximation combines conformal mapping, boundary homogenisation, and Fick-Jacobs equation to express the MFPT in terms of diffusivity and geometric parameters. A systematic comparison with a numerical solution of the original problem validates its accuracy when the starting point is not too close to the target. This is a practical tool for a rapid estimation of the MFPT for various applications in chemical physics and biology.

中文翻译：

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到细长平面域中小吸收目标的平均首次通过时间

我们推导出一个近似但完全明确的公式，用于平均首次通过时间（MFPT）到平面中一般细长域中任意形状的小吸收目标。我们的近似结合了保角映射、边界均匀化和 Fick-Jacobs 方程，以根据扩散率和几何参数来表达 MFPT。当起点不太接近目标时，与原始问题的数值解的系统比较验证了其准确性。这是在化学物理和生物学的各种应用中快速估计 MFPT 的实用工具。