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Open AccessArticle

Information Geometry of Nonlinear Stochastic Systems

1
Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, UK
2
School of Mathematics and Statistics, University of Sheffield, Sheffield S3 7RH, UK
3
Institut National des Sciences Appliquées de Rouen, 76801 Saint-Étienne-du-Rouvray CEDEX, France
*
Author to whom correspondence should be addressed.
Entropy 2018, 20(8), 550; https://doi.org/10.3390/e20080550
Received: 28 June 2018 / Revised: 20 July 2018 / Accepted: 23 July 2018 / Published: 25 July 2018
(This article belongs to the Special Issue Entropy: From Physics to Information Sciences and Geometry)
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 δ-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)μm, where μ 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. View Full-Text
Keywords: stochastic processes; Fokker-Planck equation; information length stochastic processes; Fokker-Planck equation; information length
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Hollerbach, R.; Dimanche, D.; Kim, E.-J. Information Geometry of Nonlinear Stochastic Systems. Entropy 2018, 20, 550.

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