Abstract
The eigenvalues and eigenfunctions of the Fokker-Planck equation describing the extremely underdamped Brownian motion in a symmetric double-well potential are investigated. By transforming the Fokker-Planck equation to energy and position coordinates and by performing a suitable averaging over the position coordinate, a differential equation depending only on energy is derived. For finite temperatures this equation is solved by numerical integration, whereas in the weak-noise limit an analytic result for the lowest nonzero eigenvalue is obtained. Furthermore, by using a boundary-layer theory near the critical trajectory, the correction term to the zero-friction-limit result is found.
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Risken, H., Voigtlaender, K. Eigenvalues and eigenfunctions of the Fokker-Planck equation for the extremely underdamped Brownian motion in a double-well potential. J Stat Phys 41, 825–863 (1985). https://doi.org/10.1007/BF01010006
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DOI: https://doi.org/10.1007/BF01010006