A study of the Fokker–Planck equation of bistable systems by the method of state-dependent diagonalization

We study the Fokker–Planck equation of one-dimensional bistable systems with symmetric as well as asymmetric potentials. In this work, the method of state-dependent diagonalization (SDD) is proposed for solving the eigenvalue problem of the associated Schrödinger equation. The example of a quartic bistable potential is used to illustrate its validity. Unlike the conventional exact diagonalization, the SDD method is found to be quite efficient for the high eigenstates. We use the eigenfunctions and eigenvalues obtained by this method to calculate the propagator, the spectral density of the autocorrelation function, and the response of the system to a weak periodic driving force for various noise intensities. The asymmetry of the potential is found to play an important role, and it strongly influences the signal-to-noise ratio.