### Abstract

We elucidate the effect of different deterministic nonlinear forces on geometric structure of stochastic processes by investigating the transient relaxation of initial PDFs of a stochastic variable x under forces proportional to -x^{n} (n = 3, 5, 7) and different strength D of d-correlated stochastic noise. We identify the three main stages consisting of nondiffusive evolution, quasi-linear Gaussian evolution and settling into stationary PDFs. The strength of stochastic noise is shown to play a crucial role in determining these timescales as well as the peak amplitude and width of PDFs. From time-evolution of PDFs, we compute the rate of information change for a given initial PDF and uniquely determine the information length L(t) as a function of time that represents the number of different statistical states that a system evolves through in time. We identify a robust geodesic (where the information changes at a constant rate) in the initial stage, and map out geometric structure of an attractor as L(t → ∞)α μ ^{m}m, where m is the position of an initial Gaussian PDF. The scaling exponent m increases with n, and also varies with D (although to a lesser extent). Our results highlight ubiquitous power-laws and multi-scalings of information geometry due to nonlinear interaction.

Original language | English |
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Article number | 550 |

Journal | Entropy |

Volume | 20 |

Issue number | 8 |

DOIs | |

Publication status | Published - 25 Jul 2018 |

Externally published | Yes |

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### Keywords

- Fokker-Planck equation
- Information length
- Stochastic processes

### ASJC Scopus subject areas

- Physics and Astronomy(all)

### Cite this

*Entropy*,

*20*(8), [550]. https://doi.org/10.3390/e20080550