Hollerbach, R and Kim, E-J (2019) Information Geometry of Spatially Periodic Stochastic Systems. Entropy, 21 (7). 681. ISSN 1099-4300
Abstract
We explore the effect of different spatially periodic, deterministic forces on the information geometry of stochastic processes. The three forces considered are f0=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 μ is qualitatively similar to the force, including the differences between f0=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.
Metadata
Item Type: | Article |
---|---|
Authors/Creators: |
|
Copyright, Publisher and Additional Information: | © 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: | stochastic processes; Fokker–Planck equation; information length |
Dates: |
|
Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Mathematics (Leeds) > Applied Mathematics (Leeds) |
Depositing User: | Symplectic Publications |
Date Deposited: | 11 Jul 2019 09:36 |
Last Modified: | 25 Jun 2023 21:54 |
Status: | Published |
Publisher: | MDPI |
Identification Number: | 10.3390/e21070681 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:148437 |