Information geometry of nonlinear stochastic systems

Rainer Hollerbach, Donovan Dimanche, Eun Jin Kim

Research output: Contribution to journalArticlepeer-review

10 Citations (Scopus)
7 Downloads (Pure)


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 -xn (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 mm, 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 languageEnglish
Article number550
Issue number8
Publication statusPublished - 25 Jul 2018
Externally publishedYes


  • Fokker-Planck equation
  • Information length
  • Stochastic processes

ASJC Scopus subject areas

  • Physics and Astronomy(all)


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