We investigate asymptotic decay phenomena towards the nonequilibrium steady state of the thermal diffusion in a periodic potential in the presence of a constant external force. 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 accompanied with local minima, 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 period of the potential and the damping coefficient. The intermediate tilting case is also explored. The analytic results well agree with the numerical data for a wide range of tilting. Finally, in the case that the potential includes a higher Fourier component, we report the slow relaxation, which is taken as the resonance tunneling. In this case, we analytically obtain the Kramers type decay rate. To cite this article: T. Monnai et al., C. R. Physique 8 (2007).
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