The kinetics of escaping of Brownian particles from a potential well for different space dimensionality. The effect of external force

TL;DR

This paper develops a generalized two-state model to analyze how external forces influence the escape kinetics of Brownian particles from potential wells in 2D and 3D, providing explicit expressions and studying force effects in different regimes.

Contribution

A generalized two-state model incorporating external force effects is proposed, accurately describing escape kinetics in 2D and 3D potential wells with simple analytical expressions.

Findings

Escape rate depends on the parameter φ = Fa/(2k_b T)

Weak and strong force limits are characterized in both 2D and 3D

Model aligns well with experimental data for certain well shapes

Abstract

The kinetics of two (2D) and three (3D) dimensional diffusion-assisted escaping of Brownian particles from a potential well in the presence of an external force is analyzed in detail. The kinetics is studied within the two-state model (TSM) proposed for processes in the absence of external force. The generalized variant of this model, taking into account the force effect, is proposed which is shown to be quite accurate for some shapes of the well both for 2D and 3D processes. Within the generalized TSM simple expressions for the well depopulation kinetics and, in particular, for the escape rate are obtained. The effect of the force ($F$) is shown to manifest itself in the escape rate dependence on the only parameter $\varphi = Fa/(2k_b T)$, where $a$ is the Onsager radius of the attractive part of the well $U(r)$, defined by the relation $|U(a)| \approx k_b T$. The limiting behavior of this dependence in the cases of weak and strong force is studied in detail both in 2D and 3D processes. Some applications of obtained results to the analysis of experiments are briefly discussed.