| Summary: | 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.
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