Figure 20.3 — Stochastic resonance Figure 20.3 — Stochastic resonance. (a) Stochastic resonance in a symmetric double well potential. Sketch of the double-well potential U(x) = 0.25 bx4 – 0.5 ax2 the minima are located at x± = ± √a/b and are separated by a potential ￼￼￼barrier ΔU± = a2/(4b), which is located at xs = 0. The cartoon shows how, in the presence of periodic driving, the double-well potential is tilted back and forth, thereby successively raising and lowering the potential barriers of the right and the left well, respectively, in an antisymmetric manner: a suitable dose of noise (i.e., when the period of the driving equals approximately twice the noise-induced escape time) will make the ‘sad face’ happy by allowing synchronised hopping to the globally stable state. The graph on the right shows one of the principal signatures of stochastic resonance, the fact that the amplitude of the periodic component of the response of a bistable system simultaneously to noise and to a weak periodic forcing reaches a sharp maximum as a function of the noise intensity. (b, c) Experimental escape-time distributions of a particle in the double-well potential generated by a two-beam optical trap. The time is measured in units of the mean escape time τescape (from one potential minimum to the other). The period of the forcing was chosen as (b) 3.08 and (c) 0.76 times the escape time τescape. Whereas in (c) the peaks are clearly located at odd multiples of half the forcing period, the second peak in (b) is clearly shifted to the left.Figure (a) is reprinted from Gammaitoni et al., Rev. Mod. Phys. 70, 223–87. Copyright (1988) by the American Physical Society.Figures (b) and (c) are reprinted from Simon and Libchaber, Phys. Rev. Lett. 68, 3375–8. Copyright (1992) by the American Physical Society.