### Abstract

We investigate the phenomenon of asymptotic decay towards the nonequilibrium steady state of the thermal diffusion in the presence of a tilted periodic potential. The parameter dependence of the decay rate is revealed by investigating the Fokker-Planck (FP) equation in the low temperature case under the spatially periodic boundary condition (PBC). We apply the WKB method to the associated Schrödinger equation. While eigenvalues of the non-Hermitian FP operator are complex in general, in a small tilting case the imaginary parts of the eigenvalues are almost vanishing. Then the Schrödinger equation is solved with PBC. The decay rate is analyzed in the context of quantum tunneling through a triple-well effective periodic potential. In a large tilting case, the imaginary parts of the eigenvalues of the FP operator are crucial. We apply the complex-valued WKB method to the Schrödinger equation with the absorbing boundary condition, finding that the decay rate saturates and depends only on the temperature, the potential periodicity and the viscous constant. The intermediate tilting case is also explored. The analytic results agree well with the numerical data for a wide range of tilting.

Original language | English |
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Pages (from-to) | 177-183 |

Number of pages | 7 |

Journal | Physica D: Nonlinear Phenomena |

Volume | 219 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2006 Jul 15 |

### Keywords

- Decay rate
- Fokker-Planck equation
- Thermal diffusion
- Tilted periodic potential
- WKB analysis

### ASJC Scopus subject areas

- Applied Mathematics
- Statistical and Nonlinear Physics

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## Cite this

*Physica D: Nonlinear Phenomena*,

*219*(2), 177-183. https://doi.org/10.1016/j.physd.2006.06.004