Critical Exponent Crossovers in Escape near a Bifurcation Point

In periodically driven systems, near a bifurcation (critical) point the period-averaged escape rate $\overline{W}$ scales with the field amplitude $A$ as $|\ln \overline{W}|\propto (A_c-A{)}^{\xi }$, where $A_c$ is a critical amplitude. We find three scaling regions. With increasing field frequency or decreasing $|A_c-A|$, the critical exponent $\xi$ changes from $\xi =3/2$ for a stationary system to a dynamical value $\xi =2$ and then again to $\xi =3/2$. Monte Carlo simulations agree with the scaling theory.

Figure 1

(a) An oscillating potential barrier. In the limit of slow modulation, the stable and unstable periodic states $q_a$ and $q_b$ are the instantaneous positions of the potential minimum and barrier top, respectively. (b) For slow modulation, when the driving amplitude $A$ is close to $A_c^{\mathrm{a}\mathrm{d}}$, the states $q_{a,b}(t)$ come close to each other once per period. (c) As $A$ further approaches $A_c$, the states $q_{a,b}(t)$ become skewed compared to the adiabatic picture, to avoid crossing.

Figure 2

Scaling crossover of the escape activation energy $\tilde{R}$ vs the distance to the bifurcation point $\eta \propto A_c^{\mathrm{s}\mathrm{l}}-A$ (9) for slow modulation. The escape rate $\overline{W}\propto \exp (-\tilde{R}/\tilde{D})$. The thick solid lines show the numerical solution of Eqs. (10, 11). They display the occurrence of two scaling regions $\tilde{R}\propto {\eta }^{\xi }$. The dash-dot lines show the nonadiabatic scaling (14), with $\tilde{R}\propto {\eta }^2$. The dashed lines show the adiabatic result $\tilde{R}=(4/3)(\eta -1{)}^{3/2}$; its asymptote $\tilde{R}\propto {\eta }^{3/2}$ is shown by the thin solid line. The analytical and numerical results agree in the regions $\eta \ll 1$ and $\eta \gg 1$.

Figure 3

The results showing the nonadiabatic $\xi =2$ scaling of the activation energy, $R\propto (A_c-A{)}^{\xi }$, over a wide range of modulation amplitude $A$ and the crossover to the $\xi =3/2$ scaling at larger modulation frequency. The data refer to a driven Brownian particle in a time-dependent potential $U(q,t)=-q^3/3+q^2/2-Aq\cos {\omega }_{F}t$, with ${\omega }_F=0.25$ and ${\omega }_F=0.5$; the relaxation time in the absence of modulation is $t_{\mathrm{r}}=1$; the escape rate $\overline{W}\propto \exp (-R/D)$. The solid line shows the results of a numerical minimization of the functional $\mathcal{R}$ (2). The dots show the results of Monte Carlo simulations. The dash-dot line shows the $\xi =2$ scaling (14) with appropriate parameters. The dashed line shows the high-frequency $\xi =3/2$ scaling.