Abstract
We consider two-dimensional overdamped double-well systems perturbed by white noise. In the weak-noise limit the most probable fluctuational path leading from either point attractor to the separatrix (the most probable escape path, or MPEP) must terminate on the saddle between the two wells. However, as the parameters of a symmetric double-well system are varied, a unique MPEP may bifurcate into two equally likely MPEPs. At the bifurcation point in parameter space, the activation kinetics of the system become non-Arrhenius. We quantify the non-Arrhenius behavior of a system at the bifurcation point, by using the Maslov-WKB method to construct an approximation to the quasistationary probability distribution of the system that is valid in a boundary layer near the separatrix. The approximation is a formal asymptotic solution of the Smoluchowski equation. Our construction relies on a new scaling theory, which yields “critical exponents” describing weak-noise behavior at the bifurcation point, near the saddle.
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Maier, R.S., Stein, D.L. A scaling theory of bifurcations in the symmetric weak-noise escape problem. J Stat Phys 83, 291–357 (1996). https://doi.org/10.1007/BF02183736
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DOI: https://doi.org/10.1007/BF02183736
Key Words
- Boundary layer
- caustics
- double well
- Fokker-Planck equation
- Lagrangian manifold
- large deviation theory
- large fluctuations
- Maslov-WKB method
- non-Arrhenius behavior
- nongeneric caustics
- Pearcey function
- singular perturbation theory
- Smoluchowski equation
- stochastic escape problem
- stochastic exit problem
- stochastically perturbed dynamical systems
- weak noise
- Wentzell-Freidlin theory
- WKB approximation