An Efficient Algorithm to Estimate the Potential Barrier Height from Noise-Induced Escape Time Data

Abstract

AbstractAn algorithm is developed for determining the potential barrier height experimentally, provided that we have control over the noise strength $$\sigma $$ σ . We are concerned with the situation when the laboratory or numerical experiment requires large resources of time or computational power, respectively, and wish to find a protocol that provides the best estimate in a given amount of time. The optimal noise strength $$\sigma ^*$$ σ ∗ to use is found to be very simply related to the potential barrier height $$\Delta \Phi $$ Δ Φ as: $$y^*=\Delta \Phi ^{-1}$$ y ∗ = Δ Φ - 1 , $$y=\sigma ^{-2}-\sigma ^{-2}_\mathrm{a}$$ y = σ - 2 - σ a - 2 , with some “anchor point” $$\sigma ^{-2}_\mathrm{a}$$ σ a - 2 ; and, as a second ingredient, an iterative method is proposed for the estimation. For a numerical verification of the optimality, we apply the algorithm to a simple system of an over-damped particle confined to a double-well potential, when it is feasible to evaluate statistics of the estimator. Subsequently, we also apply it to a high-dimensional case of a diffusive energy balance model, when the potential barrier height—concerning e.g. the warm-to-snowball-climate transition—cannot be determined analytically, but we would have to resort to more sophisticated numerical methods.