Soft trapping lasts longer: Dwell time of a Brownian particle varied by potential shape

It is often regarded that the dwell time (or residence time, escape time, trapping duration) of trapped Brownian particles is described by the multiplication of two separate factors, i.e., the diffusive traveling time of the trapping domain size without taking into account the trapping force, and the stochastic event of overcoming the trapping energy by thermal one instantaneously. However, we show that the ratio of dwell time to the typical traveling time for the trapping domain size depends on the shape of the force field. The shape of the trapping potential affects this ratio even if the trapping energy gap is the same and the smooth potential has a single minimum. Our finding suggests the possible application of the potential shape to realize the desired trapping characteristics.

Figure 1

Schematic diagram of the model system. A Brownian particle with a diameter $d_{\mathrm{p}}$ is suspended in a fluid. The particle is subject to a trapping force field of harmonic potential that depends only on the distance $r$ with its origin defined at the minimum potential energy. The trapping force field acts in the finite domain with a width of $d_{\mathrm{s}}$, and the trapping energy is denoted as $\mathrm{\Delta }E\left(r\right)$. The escape event is judged with the position of the central position of the particle. $d_{\mathrm{p}}$ affects only the diffusion coefficient $D$ through the Stokes-Einstein relation [cf. Eq. (4)].

Figure 2

Dependence of dwell time on (a) the trapping energy $\alpha \equiv \mathrm{\Delta }E(d_{\mathrm{s}}/2)/\left(k_{\mathrm{B}}T\right)$ and (b) the potential shape represented by the exponent $n$ of Eq. (8). ${\tau }_{\mathrm{TST}}$ and ${\tau }_{\mathrm{A}}$ correspond to Eqs. (2) and (13), respectively.

Figure 3

Deviations of numerical results from analytical solution as a function of the time resolution when the exponent $n$ of Eq. (8) is (a) 2 and (b) 20.

Figure 4

Dependence of (a) potential shape and (b) generalized spring constant on the exponent $n$ in Eqs. (8) and (9), respectively. The spring constant $k_{\mathrm{s}}$ in (b) is evaluated for the case where $d_{\mathrm{s}}={10}^{-6}$ m.

Figure 5

Distribution of ${\tau }_{\mathrm{d}}$ for different $\alpha \equiv \mathrm{\Delta }E(d_{\mathrm{s}}/2)/\left(k_{\mathrm{B}}T\right)$ when the exponent $n$ of the potential shape is (a) 2, (b) 4, (c) 10, and (d) 20, respectively. $\langle {\tau }_{\mathrm{d}}\rangle$ indicates the average of ${\tau }_{\mathrm{d}}$ for each combination of $\alpha$ and $n$.

Figure 6

Dependence of the standard deviation of the dwell time on (a) the trapping energy $\alpha \equiv \mathrm{\Delta }E(d_{\mathrm{s}}/2)/\left(k_{\mathrm{B}}T\right)$ and (b) the potential shape represented by the exponent $n$ of Eq. (8).