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
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 journal › Article
}
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 -