Information Geometry of Spatially Periodic Stochastic Systems
We explore the effect of different spatially periodic, deterministic forces on the information geometry of stochastic processes. The three forces considered are <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi mathvariant="bold&qu...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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MDPI AG
2019-07-01
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| Series: | Entropy |
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| Online Access: | https://www.mdpi.com/1099-4300/21/7/681 |
| _version_ | 1818014795848220672 |
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| author | Rainer Hollerbach Eun-jin Kim |
| author_facet | Rainer Hollerbach Eun-jin Kim |
| author_sort | Rainer Hollerbach |
| collection | DOAJ |
| description | We explore the effect of different spatially periodic, deterministic forces on the information geometry of stochastic processes. The three forces considered are <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi mathvariant="bold">f</mi> <mn mathvariant="bold">0</mn> </msub> <mo>=</mo> <mo form="prefix">sin</mo> <mrow> <mo>(</mo> <mi>π</mi> <mi>x</mi> <mo>)</mo> </mrow> <mo>/</mo> <mi>π</mi> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi mathvariant="bold">f</mi> <mo>±</mo> </msub> <mspace width="3.33333pt"></mspace> <mo>=</mo> <mspace width="3.33333pt"></mspace> <mo form="prefix">sin</mo> <mrow> <mo>(</mo> <mi>π</mi> <mi>x</mi> <mo>)</mo> </mrow> <mo>/</mo> <mi>π</mi> <mo>±</mo> <mo form="prefix">sin</mo> <mrow> <mo>(</mo> <mn>2</mn> <mi>π</mi> <mi>x</mi> <mo>)</mo> </mrow> <mo>/</mo> <mn>2</mn> <mi>π</mi> </mrow> </semantics> </math> </inline-formula>, with <inline-formula> <math display="inline"> <semantics> <msub> <mi mathvariant="bold">f</mi> <mo>-</mo> </msub> </semantics> </math> </inline-formula> chosen to be particularly flat (locally cubic) at the equilibrium point <inline-formula> <math display="inline"> <semantics> <mrow> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics> </math> </inline-formula>, and <inline-formula> <math display="inline"> <semantics> <msub> <mi mathvariant="bold">f</mi> <mo>+</mo> </msub> </semantics> </math> </inline-formula> particularly flat at the unstable fixed point <inline-formula> <math display="inline"> <semantics> <mrow> <mi>x</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula>. We numerically solve the Fokker−Planck equation with an initial condition consisting of a periodically repeated Gaussian peak centred at <inline-formula> <math display="inline"> <semantics> <mrow> <mi>x</mi> <mo>=</mo> <mi>μ</mi> </mrow> </semantics> </math> </inline-formula>, with <inline-formula> <math display="inline"> <semantics> <mi>μ</mi> </semantics> </math> </inline-formula> in the range <inline-formula> <math display="inline"> <semantics> <mrow> <mo>[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>]</mo> </mrow> </semantics> </math> </inline-formula>. The strength <i>D</i> of the stochastic noise is in the range <inline-formula> <math display="inline"> <semantics> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>4</mn> </mrow> </msup> </semantics> </math> </inline-formula>−<inline-formula> <math display="inline"> <semantics> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>6</mn> </mrow> </msup> </semantics> </math> </inline-formula>. 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 <inline-formula> <math display="inline"> <semantics> <mrow> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics> </math> </inline-formula>, the peaks largely maintain their shape while moving. In contrast, for initial positions sufficiently close to the unstable point <inline-formula> <math display="inline"> <semantics> <mrow> <mi>x</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula>, 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 <inline-formula> <math display="inline"> <semantics> <msub> <mi mathvariant="script">L</mi> <mo>∞</mo> </msub> </semantics> </math> </inline-formula>, 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 <inline-formula> <math display="inline"> <semantics> <msub> <mi mathvariant="script">L</mi> <mo>∞</mo> </msub> </semantics> </math> </inline-formula> as a function of initial position <inline-formula> <math display="inline"> <semantics> <mi>μ</mi> </semantics> </math> </inline-formula> is qualitatively similar to the force, including the differences between <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi mathvariant="bold">f</mi> <mn mathvariant="bold">0</mn> </msub> <mo>=</mo> <mo form="prefix">sin</mo> <mrow> <mo>(</mo> <mi>π</mi> <mi>x</mi> <mo>)</mo> </mrow> <mo>/</mo> <mi>π</mi> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi mathvariant="bold">f</mi> <mo>±</mo> </msub> <mo>=</mo> <mo form="prefix">sin</mo> <mrow> <mo>(</mo> <mi>π</mi> <mi>x</mi> <mo>)</mo> </mrow> <mo>/</mo> <mi>π</mi> <mo>±</mo> <mo form="prefix">sin</mo> <mrow> <mo>(</mo> <mn>2</mn> <mi>π</mi> <mi>x</mi> <mo>)</mo> </mrow> <mo>/</mo> <mn>2</mn> <mi>π</mi> </mrow> </semantics> </math> </inline-formula>, illustrating the value of information length as a useful diagnostic of the underlying force in the system. |
| first_indexed | 2024-04-14T06:48:54Z |
| format | Article |
| id | doaj.art-cbf3a829383346dab3b12ff863b7eb10 |
| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-04-14T06:48:54Z |
| publishDate | 2019-07-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Entropy |
| spelling | doaj.art-cbf3a829383346dab3b12ff863b7eb102022-12-22T02:07:05ZengMDPI AGEntropy1099-43002019-07-0121768110.3390/e21070681e21070681Information Geometry of Spatially Periodic Stochastic SystemsRainer Hollerbach0Eun-jin Kim1Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, UKSchool of Mathematics and Statistics, University of Sheffield, Sheffield S3 7RH, UKWe explore the effect of different spatially periodic, deterministic forces on the information geometry of stochastic processes. The three forces considered are <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi mathvariant="bold">f</mi> <mn mathvariant="bold">0</mn> </msub> <mo>=</mo> <mo form="prefix">sin</mo> <mrow> <mo>(</mo> <mi>π</mi> <mi>x</mi> <mo>)</mo> </mrow> <mo>/</mo> <mi>π</mi> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi mathvariant="bold">f</mi> <mo>±</mo> </msub> <mspace width="3.33333pt"></mspace> <mo>=</mo> <mspace width="3.33333pt"></mspace> <mo form="prefix">sin</mo> <mrow> <mo>(</mo> <mi>π</mi> <mi>x</mi> <mo>)</mo> </mrow> <mo>/</mo> <mi>π</mi> <mo>±</mo> <mo form="prefix">sin</mo> <mrow> <mo>(</mo> <mn>2</mn> <mi>π</mi> <mi>x</mi> <mo>)</mo> </mrow> <mo>/</mo> <mn>2</mn> <mi>π</mi> </mrow> </semantics> </math> </inline-formula>, with <inline-formula> <math display="inline"> <semantics> <msub> <mi mathvariant="bold">f</mi> <mo>-</mo> </msub> </semantics> </math> </inline-formula> chosen to be particularly flat (locally cubic) at the equilibrium point <inline-formula> <math display="inline"> <semantics> <mrow> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics> </math> </inline-formula>, and <inline-formula> <math display="inline"> <semantics> <msub> <mi mathvariant="bold">f</mi> <mo>+</mo> </msub> </semantics> </math> </inline-formula> particularly flat at the unstable fixed point <inline-formula> <math display="inline"> <semantics> <mrow> <mi>x</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula>. We numerically solve the Fokker−Planck equation with an initial condition consisting of a periodically repeated Gaussian peak centred at <inline-formula> <math display="inline"> <semantics> <mrow> <mi>x</mi> <mo>=</mo> <mi>μ</mi> </mrow> </semantics> </math> </inline-formula>, with <inline-formula> <math display="inline"> <semantics> <mi>μ</mi> </semantics> </math> </inline-formula> in the range <inline-formula> <math display="inline"> <semantics> <mrow> <mo>[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>]</mo> </mrow> </semantics> </math> </inline-formula>. The strength <i>D</i> of the stochastic noise is in the range <inline-formula> <math display="inline"> <semantics> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>4</mn> </mrow> </msup> </semantics> </math> </inline-formula>−<inline-formula> <math display="inline"> <semantics> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>6</mn> </mrow> </msup> </semantics> </math> </inline-formula>. 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 <inline-formula> <math display="inline"> <semantics> <mrow> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics> </math> </inline-formula>, the peaks largely maintain their shape while moving. In contrast, for initial positions sufficiently close to the unstable point <inline-formula> <math display="inline"> <semantics> <mrow> <mi>x</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula>, 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 <inline-formula> <math display="inline"> <semantics> <msub> <mi mathvariant="script">L</mi> <mo>∞</mo> </msub> </semantics> </math> </inline-formula>, 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 <inline-formula> <math display="inline"> <semantics> <msub> <mi mathvariant="script">L</mi> <mo>∞</mo> </msub> </semantics> </math> </inline-formula> as a function of initial position <inline-formula> <math display="inline"> <semantics> <mi>μ</mi> </semantics> </math> </inline-formula> is qualitatively similar to the force, including the differences between <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi mathvariant="bold">f</mi> <mn mathvariant="bold">0</mn> </msub> <mo>=</mo> <mo form="prefix">sin</mo> <mrow> <mo>(</mo> <mi>π</mi> <mi>x</mi> <mo>)</mo> </mrow> <mo>/</mo> <mi>π</mi> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi mathvariant="bold">f</mi> <mo>±</mo> </msub> <mo>=</mo> <mo form="prefix">sin</mo> <mrow> <mo>(</mo> <mi>π</mi> <mi>x</mi> <mo>)</mo> </mrow> <mo>/</mo> <mi>π</mi> <mo>±</mo> <mo form="prefix">sin</mo> <mrow> <mo>(</mo> <mn>2</mn> <mi>π</mi> <mi>x</mi> <mo>)</mo> </mrow> <mo>/</mo> <mn>2</mn> <mi>π</mi> </mrow> </semantics> </math> </inline-formula>, illustrating the value of information length as a useful diagnostic of the underlying force in the system.https://www.mdpi.com/1099-4300/21/7/681stochastic processesFokker–Planck equationinformation length |
| spellingShingle | Rainer Hollerbach Eun-jin Kim Information Geometry of Spatially Periodic Stochastic Systems Entropy stochastic processes Fokker–Planck equation information length |
| title | Information Geometry of Spatially Periodic Stochastic Systems |
| title_full | Information Geometry of Spatially Periodic Stochastic Systems |
| title_fullStr | Information Geometry of Spatially Periodic Stochastic Systems |
| title_full_unstemmed | Information Geometry of Spatially Periodic Stochastic Systems |
| title_short | Information Geometry of Spatially Periodic Stochastic Systems |
| title_sort | information geometry of spatially periodic stochastic systems |
| topic | stochastic processes Fokker–Planck equation information length |
| url | https://www.mdpi.com/1099-4300/21/7/681 |
| work_keys_str_mv | AT rainerhollerbach informationgeometryofspatiallyperiodicstochasticsystems AT eunjinkim informationgeometryofspatiallyperiodicstochasticsystems |