Diffusion of bistable oscillators driven by colored noise: Short-correlation-time approximation versus Hänggi’s ansatz

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Abstract
The problem of the activation rate in overdamped (Smoluchowski) bistable oscillators subject to colored noise with correlation time τ is discussed in detail. We show that a nontrivial prescription (the so-called Hänggi’s ansatz) for truncating the τ expansion of the Fokker-Planck equation describing this class of processes can reproduce the results of both analog and digital simulations in the regime of large activation energies. Such a prescription predicts a τ dependence of the Arrhenius factor appearing in the determination of the relevant mean first-passage time but cannot be applied to reproduce the correct τ dependence of the prefactor which may be dominant in the regime of rather small activation energies. An improved expression for the mean first-passage time in a double-well potential coupled to a time-correlated thermal bath is obtained on generalizing Kramer’s theory of the activation rates accordingly. The new determination of the mean first-passage time is meant to bridge previous approximations only valid in the regime of either large or small activation energies.