Scaling and crossovers in activated escape near a bifurcation point

Near a bifurcation point a system experiences a critical slowdown. This leads to scaling behavior of fluctuations. We find that a periodically driven system may display three scaling regimes and scaling crossovers near a saddle-node bifurcation where a metastable state disappears. The rate of activated escape $W$ scales with the driving field amplitude $A$ as $\ln W\propto {(A_c-A)}^{\xi }$, where $A_c$ is the bifurcational value of $A$. With increasing field frequency the critical exponent $\xi$ changes from $\xi =3∕2$ for stationary systems to a dynamical value $\xi =2$ and then again to $\xi =3∕2$. The analytical results are in agreement with the results of asymptotic calculations in the scaling region. Numerical calculations and simulations for a model system support the 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 modulation amplitude $A$ is close to its adiabatic bifurcational value $A_c^{\mathrm{ad}}$, the states $q_{a,b}\left(t\right)$ come close to each other once per period. (c) As $A$ further increases beyond $A_c^{\mathrm{ad}}$, the barrier of $U$ disappears for a portion of the modulation period, but the system may still have coexisting periodic states $q_{a,b}\left(t\right)$. As seen in (d), they become skewed compared to the adiabatic picture, to avoid crossing. In the critical range, the form of $q_{a,b}\left(t\right)$ is model independent.

Figure 2

Nonadiabatic stable and unstable states $Q_a\left(\tau \right)$ and $Q_b\left(\tau \right)=-Q_a(-\tau )$ for slow modulation as given by the equation $dQ∕d\tau =G(Q,\eta ,\tau )$ for $\eta =0.2$. The functions $Q_{a,b}\left(\tau \right)$ are strongly asymmetric, in contrast to the adiabatic states (19) which are even functions of $\tau$.

Figure 3

The stable and unstable states, ${\mathbf{q}}_a\left(t\right)$ and ${\mathbf{q}}_b\left(t\right)$, close to the bifurcation point. For ${\omega }_F{t}_{\mathrm{r}}⪢1$ the states are close to each other throughout the modulation period. The figure refers to a one-dimensional overdamped particle in a potential $U(q,t)=\frac{1}{4}q-\frac{1}{3}{q}^3-Aq\cos \left({\omega }_{F}t\right)$ for $(A_c-A)∕A_c\approx 0.01$. The modulation is comparatively slow, ${\omega }_F{t}_{\mathrm{r}}^{\left(0\right)}=1$, but for chosen $A$ the relaxation time becomes long, ${\omega }_F{t}_{\mathrm{r}}\approx 9.8$.

Figure 4

The activation energy $\tilde{R}$ vs $\eta \propto A_c-A$ for slow driving, ${\omega }_F{t}_{\mathrm{r}}⪡1$. The thick solid line shows the numerical solution of Eq. (28). The dashed line is the adiabatic activation energy (33), ${\tilde{R}}^{\mathrm{ad}}\propto {(\eta -1)}^{3∕2}$. The thin solid line shows the corrected adiabatic activation energy ${\tilde{R}}^{\mathrm{ad}}+\delta \tilde{R}$. It is close to the numerical result for $\eta \gtrsim 3$. The correction $\delta \tilde{R}$ diverges at the adiabatic bifurcation point $\eta =1$.

Figure 5

The activation energy $\tilde{R}=-\tilde{D}\ln \overline{W}$ on a logarithmic and linear scale (inset) vs $\eta \propto A_c-A$ for slow modulation, ${\omega }_F{t}_{\mathrm{r}}⪡1$. Thick solid lines show the numerical solution of the variational problem (28). It describes the crossover between different scaling regions. The thin solid line shows the adiabatic scaling for large $\eta$, $\tilde{R}\propto {\eta }^{\xi }$ with $\xi =3∕2$. The full result of the adiabatic approximation is shown by the dashed line. The dash-dot line shows the nonadiabatic result (38) that applies for $\eta ⪡1$; here, $\tilde{R}\propto {\eta }^{\xi }$ with $\xi =2$.

Figure 6

The critical amplitude $A_c$ as a function of the modulation frequency ${\omega }_F$ for the system (40). Numerical results are shown by thick solid lines. The dashed line shows the linear in ${\omega }_F$ nonadiabatic correction to $A_c$ described by Eq. (16). The thin solid line in the inset describes a correction obtained from the self-consistent local analysis, Eq. (42). The dash-dot line describes the high-frequency asymptotic that follows from Eq. (43).

Figure 7

The activation energy of escape $R$ vs. modulation amplitude $A$ on the logarithmic and linear (inset) scales for a Brownian particle in a modulated potential (40). The values of ${\omega }_F$ are indicated on each panel. The thick solid lines show the results of the numerical solution of the variational problem for $R$. The dashed lines for ${\omega }_F=0.1,0.25$ show the adiabatic approximation, whereas for ${\omega }_F=0.5,1.0$ they show the approximation of effectively fast oscillations: in both cases the scaling exponent is $\xi =3∕2$ (for ${\omega }_F=0.1$ this asymptotic scaling is shown by the thin solid line). The dash-dot lines show the $\xi =2$ scaling (38). The dots show the results of numerical simulations of Eq. (40).

Figure 8

The scaled probability density of the dwell time ${\tilde{p}}_{\mathrm{dw}}\left(t\right)=p_{\mathrm{dw}}\left(t\right){\tau }_F$ obtained by numerical simulations of a Brownian particle in a modulated potential, Eq. (40). The parameters are $A=0.1$, $D=0.05$, ${\omega }_F=0.25$ (solid line). The dashed line shows the exponential fit of the envelope with decrement $\overline{W}{\tau }_F=0.008$.