Kramers rate for Brownian particles in excitable media with deformable double-well substrates

TL;DR

This paper studies the Kramers escape problem for Brownian particles in excitable media with deformable double-well substrates (DDWPs). It introduces three deformability schemes (DDWP I–III) controlled by a single parameter $\mu$, which reduces to the standard $\phi^4$ potential as $\mu\to 0$. The authors derive analytic escape rates using overdamped Langevin dynamics and the Smoluchowski equation, expanding the potential around the metastable minimum and the barrier to quartic order to obtain Gaussian and non-Gaussian contributions, with the non-Gaussian part expressed through a parabolic cylinder function. A central result is that non-Gaussian corrections become significant only for $\mu>\mu_c=\sqrt{3/2}$, where they can dramatically enhance the escape rate and are connected to a first-order classical-to-quantum crossover in tunneling; the study also relates to spectral problems and instanton-phonon scattering analyses for deformable bistable systems.

Abstract

We address the Kramers escape problem for Brownian particles in bistable substrates with deformable double-well shapes. The shape deformability is considered of three distinct forms: in one, the positions of the two degenerate minima can be shifted continuously without affecting the barrier height. In the second the minima positions are kept fix while the barrier height is continuously shifted. In the third the minima positions and barrier height can be tuned simultaneously by changing a single real parameter that we refer to as deformability parameter. We obtain the analytical expression of the Kramers escape rate for the three different double-well models, and identify a critical value of the deformability parameter above which non-Gaussian corrections become relevant. Remarkable enough in the latter region, statistical mechanics predicts a first-order transition in quantum tunneling using the functional action associated with the family of deformable double-well potentials.

Figures (4)

• Figure 1: The three members of the family of deformable dougle-well potentials given by eq. (\ref{['eq2']}), plotted versus $x$ for $\mu=0.1$ (solid line) $\mu=0.5$ (dash line), $\mu=1.0$ (dash-dotted line), $\mu=2.0$ (dot line), $a_0=1.2$. Left graph is the DDWP I, middle graph is the DDWP II, right graph is the DDWP III.
• Figure 2: Positive minima positions (left graph) and barrier height (right graph) of the three members of the family of DDWPs in eq. (\ref{['eq2']}), plotted versus $\mu$ for $a_0=1.2$. Dotted curve: DDWP I, solid line: DDWP II, dashed curve: DDWP III.
• Figure 3: Non-corrected (i.e. Gaussian) escape rate versus $\beta$ for four different values of $\mu$ namely $\mu=10^{-6}$ (solid line), $\mu=0.25$ (dot-dashed line), $\mu=1$ (dashed line) and $\mu=2.5$ (dotted line). Left graph is for DDWP I, middle graph for DDWP II and right graph for DDWP III.
• Figure 4: The corrected escape rate versus $\beta$ for four different values of $\mu$ namely $\mu=\sqrt{1.505}$ (solid line), $\mu=\sqrt{1.75}$ (dot-dashed line), $\mu=2$ (dashed line) and $\mu=2.5$ (dotted line). Left graph is for DDWP I, middle graph for DDWP II and right graph for DDWP III.