The Eyring - Kramers Law for Potentials with Nonquadratic Saddles
2010, v.16, Issue 3, 549-598
The Eyring - Kramers law describes the mean transition time of an overdamped Brownian particle between local minima in a potential landscape. In the weak-noise limit, the transition time is to leading order exponential in the potential difference to overcome. This exponential is corrected by a prefactor which depends on the principal curvatures of the potential at the starting minimum and at the highest saddle crossed by an optimal transition path. The Eyring - Kramers law, however, does not hold whenever one or more of these principal curvatures vanishes, since it would predict a vanishing or infinite transition time. We derive the correct prefactor up to multiplicative errors that tend to one in the zero-noise limit. As an illustration, we discuss the case of a symmetric pitchfork bifurcation, in which the prefactor can be expressed in terms of modified Bessel functions, as well as bifurcations with two vanishing eigenvalues. The corresponding transition times are studied in a full neighbourhood of the bifurcation point. These results extend work by Bovier, Eckhoff, Gayrard and Klein [A. Bovier, M. Eckhoff, V. Gayrard and M. Klein, Metastability in reversible diffusion processes. I. Sharp asymptotics for capacities and exit times. J. Euro. Math. Soc., 2004, v. 6, N 4, pp. 399-424], who rigorously analysed the case of quadratic saddles, using methods from potential theory.
Keywords: stochastic differential equations,exit problem,transition times,most probable transition path,large deviations,Wentzell - Freidlin theory,metastability,potential theory,capacities,subexponential asymptotics,pitchfork bifurcation