Thermal decay of a metastable state: Influence of rescattering on the quasistationary dynamical rate

We study the effect of backscattering of the Brownian particles as they escape out of a metastable state overcoming the potential barrier. For this aim, we model this process numerically using the Langevin equations. This modeling is performed for the wide range of the friction constant covering both the energy and spatial diffusion regimes. It is shown how the influence of the descent stage on the quasistationary decay rate gradually disappears as the friction constant decreases. It is found that, in the energy diffusion regime, the rescattering absents and the descent stage does not influence the decay rate. As the value of friction increases, the descent alters the value of the rate by more than 50% for different values of thermal energy and different shapes of the potential. To study the influence of the backscattering on the decay rate, four potentials have been considered which coincide near the potential well and the barrier but differ beyond the barrier. It is shown that the potential for which the well and the barrier are described by two smoothly joined parabolas (“the parabolic potential”) plays a role of a dividing range for the mutual layout of the quasistationary dynamical rate and the widely used in the literature Kramers rate. Namely, for the potentials with steeper tails, the Kramers rate $R_{\mathrm{KM}}$ underestimates the true quasistationary dynamical rate $R_D$, whereas for the less steep tails the opposite holds (inversion of $R_D/R_{\mathrm{KM}}$). It is demonstrated that the mutual layout of the values of $R_D$ for different potentials is explained by the rescattering of the particles from the potential tail.

Figure 1

The potential energy $U_P\left(q\right)$ with the metastable state ($q_c$). $q_b$ indicates the position of the barrier; $q_a$ reflects the possible location of the absorption point; $q_l$ shows the left turning point [according to Eq. (5) $U_P\left(q_l\right)=U_b$].

Figure 2

The time dependence of the dynamical rates for four values of the dimensionless damping coefficient $\beta$ (and separation parameter $\gamma$). The horizontal lines indicate the quasistationary dynamical rates. These calculations are performed for $\varepsilon =3.74$.

Figure 3

The ratio of the dynamical quasistationary rates found at the fixed barrier point $q_b$ and at the varied absorption point $q_a$, $R_{Db}/R_{Da}$, versus $q_a$. The horizontal lines correspond to $r_s=R_{Db}/R_{Das}$ (see text).

Figure 4

The decay rates (a) and the saturated value of the rate ratio $r_s$ (b) versus the dimensionless damping coefficient $\beta$. In panel (a) the solid circles, triangles up, and lines correspond to $R_{Das}$, $R_{Db}$, $R_{\mathrm{KM}}$ (the right part), and $R_{\mathrm{KL}}$ (the left part), respectively. $\varepsilon =3.00$.

Figure 5

The same as in Fig. 4 but for two values of $\varepsilon$ indicated in panel (a). The lines in that panel correspond to $R_{\mathrm{KL}}$ (left side) and $R_{\mathrm{KM}}$ (right side). Symbols in panel (a) stand for $R_{Das}$.

Figure 6

The same as in Fig. 5 but for two types of the potential energy: parabolic (circles) and cubic (diamonds). The analytical rates $R_{\mathrm{KL}}$ (left side) and $R_{\mathrm{KM}}$ (right side) are shown by the solid lines in upper panel only for the parabolic potential. $\varepsilon =3.00$.

Figure 7

Four dimenmsionless potentials used in the present work: “flat,” “linear,” “parabolic,” and “steep” [see Eqs. (16), (19, 20, 21)]. $q_c=1.0$, $q_b=1.6$, $q_j=1.7$.

Figure 8

Typical behavior of the decay rate $R_a\left(t\right)$ for the potentials presented in Fig. 7. The horizontal lines indicate the quasistationary dynamical rates. $\varepsilon =3.75$, $q_j=1.7$, $q_a=2.6$, $\beta =5.07$.

Figure 9

(a) The quasistationary dynamical rates (symbols) and integral Kramers rates (lines) versus $\varepsilon$ for the potentials presented in Fig. 7. Zero-order Kramers rate is shown as well but it is indistinguishable from $R_{\mathrm{KI}}$ for the parabolic potential. (b) The fractional difference ${\xi }_{\mathrm{MD}}$ defined by Eq. (22), (c) the fractional difference ${\xi }_{\mathrm{ID}}$ defined by Eq. (25). $q_j=1.7$, $q_a=2.6$, $\beta =5.07$.

Figure 10

The number of rescattered particles evaluated numerically over the absorbed particles number (the diamonds) and over the total number of particles involved in the modeling (other symbols) as functions of the modeling time for the potentials presented in Fig. 7. The horizontal lines represent the saturated values of $N_{\mathrm{rs}\mathrm{num}}/N_{\mathrm{tot}}$ (see text for details). $\varepsilon =3.74$.

Figure 11

The quasistationary dynamical rates (symbols) and the rates calculated according to the integral Kramers formula (lines) versus the absorption point coordinate for four potentials presented in Fig. 7. The notations are the same as in Fig. 9. The thick solid line in the upper part corresponds to the doubled zero-order Kramers rate. $\varepsilon =3.17$, $q_j=1.7$.

Figure 12

The same as in Fig. 11 but versus the difference $q_j-q_b$. Results for the parabolic potential are not shown because $q_j$ is not applicable in this case. $\varepsilon =3.17$, $q_a=2.6$.