### 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 language | English |
---|---|

Article number | 681 |

Number of pages | 12 |

Journal | Entropy |

Volume | 21 |

Issue number | 7 |

DOIs | |

Publication status | Published - 12 Jul 2019 |

Externally published | Yes |

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### 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

*Entropy*,

*21*(7), [681]. https://doi.org/10.3390/e21070681

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

Research output: Contribution to journal › Article

*Entropy*, vol. 21, no. 7, 681. https://doi.org/10.3390/e21070681

}

TY - JOUR

T1 - Information Geometry of Spatially Periodic Stochastic Systems

AU - Hollerbach, Rainer

AU - Kim, Eun-jin

N1 - 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/).

PY - 2019/7/12

Y1 - 2019/7/12

N2 - 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.

AB - 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.

KW - Fokker-Planck equation

KW - Information length

KW - Stochastic processes

UR - http://www.scopus.com/inward/record.url?scp=85068929707&partnerID=8YFLogxK

U2 - 10.3390/e21070681

DO - 10.3390/e21070681

M3 - Article

VL - 21

JO - Entropy

JF - Entropy

SN - 1099-4300

IS - 7

M1 - 681

ER -