Journal
ANNALES HENRI POINCARE
Volume 17, Issue 12, Pages 3499-3532Publisher
SPRINGER INTERNATIONAL PUBLISHING AG
DOI: 10.1007/s00023-016-0507-4
Keywords
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Funding
- European Research Council under the European Union's seventh Framework Programme [616811]
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In the small noise regime, the average transition time between metastable states of a reversible diffusion process is described at the logarithmic scale by Arrhenius' law. The Eyring-Kramers formula classically provides a subexponential prefactor to this large deviation estimate. For irreversible diffusion processes, the equivalent of Arrhenius' law is given by the Freidlin-Wentzell theory. In this paper, we compute the associated prefactor and thereby generalise the Eyring-Kramers formula to irreversible diffusion processes. In our formula, the role of the potential is played by Freidlin-Wentzell's quasipotential, and a correction depending on the non-Gibbsianness of the system along the minimum action paths is highlighted. Our study assumes some properties for the vector field: (1) attractors are isolated points, (2) the dynamics restricted to basin of attraction boundaries are attracted to single points (which are saddle-points of the vector field). We moreover assume that the minimum action paths that connect attractors to adjacent saddle-points (the instantons) have generic properties that are summarised in the conclusion. At a technical level, our derivation combines an exact computation for the first-order WKB expansion around the instanton and an exact computation of the first-order match asymptotics expansion close to the saddle-point. While the results are exact once a formal expansion is assumed, the validity of these asymptotic expansions remains to be proven.
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