Reactive flux theory for finite potential barriers

Motivated by developing a simple, accurate, and widely applicable approach to incorporate the finite barrier correction in analytical calculation of the escape rate, the reactive flux theory for finite barriers is proposed. For higher temperatures, instead of at the top of the barrier in the original reactive flux theory, the starting point of the trajectories of Brownian particles is removed into a position inside the potential well where the probability distribution can be regarded as an equilibrium one, and the potential barrier is replaced with an equivalent parabolic potential barrier. The equivalent potential barrier frequency can be obtained by two schemes. The population is also calculated more realistically for finite barriers. The theoretical method is tested by a Brownian particle moving in a cubic metastable potential and subjected to Gaussian white noise. The numerical simulation results confirm the approach satisfactorily until lower reduced barrier heights.

Figure 1

The original potential (solid line) and the equivalent paraboloic potential (dashed line). $x_0$ is the starting point of the trajectories in reactive flux theory for finite barrier.

Figure 2

The escape rate as a function of damping, where $V_b=2$, (a) $T=0.4,{\varpi }_b^2=1.286$, (b) $T=0.6,{\varpi }_b^2=1.260$, (c) $T=0.8, {\varpi }_b^2=1.231$, (d) $T=1,{\varpi }_b^2=1.194$. The solid lines are theoretical results, the dashed lines are numerical simulation results, and the dotted lines are the original Kramers theoretical results.

Figure 3

The logarithm of escape rate as a function of inverse temperature. The circles are theoretical results, the triangles are numerical simulation results, where $V_b=2,\gamma =3$. The straight line is used to guide the eyes.