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Kramers theory in the relaxation dynamics of a tilted asymmetric periodic potential

Takaaki Monnai1,*, Ayumu Sugita2,†, and Katsuhiro Nakamura2,‡

  • 1Department of Applied Physics, Waseda University, 3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555, Japan
  • 2Department of Applied Physics, Osaka City University, 3-3-138 Sugimoto, Sumiyoshi-ku, Osaka 558-8585, Japan

  • *Electronic address: monnai@suou.waseda.jp
  • †Electronic address: sugita@a-phys.eng.osaka-cu.ac.jp
  • ‡Electronic address: nakamura@a-phy.eng.osaka-cu.ac.jp
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Phys. Rev. E 76, 031140 – Published 28 September, 2007

DOI: https://doi.org/10.1103/PhysRevE.76.031140

Abstract

We investigate the low-temperature relaxation dynamics toward a nonequilibrium steady state in a tilted asymmetric periodic potential based on the WKB analysis and the numerical diagonalization of the Fokker-Planck operator. Due to the tilting, the Fokker-Planck operator, and thus the Schrödinger operator associated with it, are non-Hermitian. Therefore, we evaluate the decay rate based on the WKB analysis both for real- and complex-valued eigenvalues. In the tilting range where the double-humped barrier exists, the decay rate is shown to obey a law which is a subtle nonequilibrium extension of the so-called Kramers escape rate. The decay rate for the single-humped barrier case is analyzed as well. The large tilting regime where the barriers no longer exist is also investigated.

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Phys. Rev. E 76, 031140– Published 28 September, 2007
  • Received 10 April 2007

© 2007 American Physical Society

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