Information Geometry of Spatially Periodic Stochastic Systems

Rainer Hollerbach, Eun-jin Kim

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Abstract

We explore the effect of different spatially periodic, deterministic forces on the information geometry of stochastic processes. The three forces considered are f 0 = sin(πx)/π and f ± = sin(πx)/π ± sin(2πx)/2π, with f_ chosen to be particularly flat (locally cubic) at the equilibrium point x = 0, and f + particularly flat at the unstable fixed point x = 1. We numerically solve the Fokker-Planck equation with an initial condition consisting of a periodically repeated Gaussian peak centred at x = μ, with μ in the range [0, 1]. The strength D of the stochastic noise is in the range 10 -4-10 -6. We study the details of how these initial conditions evolve toward the final equilibrium solutions and elucidate the important consequences of the interplay between an initial PDF and a force. For initial positions close to the equilibrium point x = 0, the peaks largely maintain their shape while moving. In contrast, for initial positions sufficiently close to the unstable point x = 1, there is a tendency for the peak to slump in place and broaden considerably before reconstituting itself at the equilibrium point. A consequence of this is that the information length L , the total number of statistically distinguishable states that the system evolves through, is smaller for initial positions closer to the unstable point than for more intermediate values. We find that L as a function of initial position m is qualitatively similar to the force, including the differences between f 0 = sin(πx)/π and f ± = sin(πx)/π ± sin(2πx)/2π, illustrating the value of information length as a useful diagnostic of the underlying force in the system.

Original languageEnglish
Article number681
Number of pages12
JournalEntropy
Volume21
Issue number7
DOIs
Publication statusPublished - 12 Jul 2019
Externally publishedYes

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geometry
stochastic processes
Fokker-Planck equation
tendencies

Bibliographical note

c 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

Keywords

  • Fokker-Planck equation
  • Information length
  • Stochastic processes

ASJC Scopus subject areas

  • Physics and Astronomy(all)

Cite this

Information Geometry of Spatially Periodic Stochastic Systems. / Hollerbach, Rainer; Kim, Eun-jin.

In: Entropy, Vol. 21, No. 7, 681, 12.07.2019.

Research output: Contribution to journalArticle

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