Niveau: Supérieur, Doctorat, Bac+8
The Eyring–Kramers law for potentials with nonquadratic saddles Nils Berglund and Barbara Gentz Abstract The Eyring–Kramers law describes the mean transition time of an overdamped Brown- ian 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 opti- mal 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 [BEGK04], who rigorously analysed the case of quadratic saddles, using methods from potential theory. Date. April 7, 2009. Revised. October 29, 2009. 2000 Mathematical Subject Classification. 60J45, 31C15 (primary), 60J60, 37H20 (secondary) Keywords and phrases.
- bifurcation
- double-zero eigenvalue
- breaking bifurcations
- elapsed between
- been given
- hessian has
- transition times
- weak-noise limit
- limit