### Abstract

We consider the classical double-well model of stochastic resonance, in which a particle in a potential V(x,t)=[−x
^{2}
∕2+x
^{4}
∕4−Asin(ωt)x] is subject to an additional stochastic forcing that causes it to occasionally jump between the two wells at x≈±1. We present direct numerical solutions of the Fokker–Planck equation for the probability density function p(x,t), for ω=10
^{−2}
to 10
^{−6}
, and A∈[0,0.2]. Previous results that stochastic resonance arises if ω matches the average frequency at which the stochastic forcing alone would cause the particle to jump between the wells are quantified. The modulation amplitudes A necessary to achieve essentially 100% saturation of the resonance tend to zero as ω→0. From p(x,t) we next construct the information length L(t)=∫[∫(∂
_{t}
p)
^{2}
∕pdx]
^{1∕2}
dt, measuring changes in information associated with changes in p. L shows an equally clear signal of the resonance, which can be interpreted in terms of the underlying meaning of L. Finally, we present escape time calculations, where the Fokker–Planck equation is solved only for x≥0, and find that resonance shows up less clearly than in either the original p or L.

Original language | English |
---|---|

Pages (from-to) | 1313-1322 |

Number of pages | 10 |

Journal | Physica A: Statistical Mechanics and its Applications |

Volume | 525 |

Early online date | 6 Apr 2019 |

DOIs | |

Publication status | Published - 1 Jul 2019 |

Externally published | Yes |

### Bibliographical note

NOTICE: this is the author’s version of a work that was accepted for publication in Physica A: Statistical Mechanics and its Applications. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Physica A: Statistical Mechanics and its Applications, 525, (2019) DOI: 10.1016/j.physa.2019.04.074© 2019, Elsevier. Licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International http://creativecommons.org/licenses/by-nc-nd/4.0/

### Keywords

- Fokker–Planck equation
- Information geometry
- Probability density function
- Stochastic resonance

### ASJC Scopus subject areas

- Statistics and Probability
- Condensed Matter Physics

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## Cite this

*Physica A: Statistical Mechanics and its Applications*,

*525*, 1313-1322. https://doi.org/10.1016/j.physa.2019.04.074