# Information Geometry of Spatially Periodic Stochastic Systems

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## Abstract

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## 1. Introduction

## 2. Model

## 3. Results

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

## References

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**Figure 1.**The three profiles ${\mathbf{f}}_{\mathbf{0}}$, ${\mathbf{f}}_{-}$ and ${\mathbf{f}}_{+}$. Note how ${\mathbf{f}}_{-}$ is chosen to be flat at the attractor $x=0$, and ${\mathbf{f}}_{+}$ at the unstable fixed point $x=1$. All three choices are anti-symmetric in x, and periodic with period 2.

**Figure 2.**Peak amplitudes as functions of time, for the three choices ${\mathbf{f}}_{*}$ as labelled. The initial positions are at $\mu =0.5$ in the top row, and $\mu =0.95$ in the bottom row, with initial widths ${D}_{0}={10}^{-3}$ in all cases. $D={10}^{-4}$ to ${10}^{-6}$ as labelled, also colour-coded as blue, red, black. Note also that some panels have t on a linear scale, indicating a very rapid adjustment process, whereas others have t on a logarithmic scale, corresponding to much slower dynamics.

**Figure 3.**The solutions $p(x,t)$ for ${\mathbf{f}}_{\mathbf{0}}$, $D={10}^{-5}$, and initial condition $\mu =0.95$ and ${D}_{0}={10}^{-3}$. The numbers beside individual curves indicate the times, from $t=0$ to 9. The different colours are for clarity only, but do not indicate a specific colour ↔ time relationship.

**Figure 4.**The solutions $p(x,t)$ for ${\mathbf{f}}_{-}$, $D={10}^{-5}$, and initial condition $\mu =0.95$ and ${D}_{0}={10}^{-3}$. The numbers beside individual curves indicate the times, from $t=0$ to 300. Different colours are again only to help distinguish between the different lines.

**Figure 5.**The solutions $p(x,t)$ for ${\mathbf{f}}_{+}$, $D={10}^{-5}$, and initial condition $\mu =0.95$ and ${D}_{0}={10}^{-3}$. The numbers beside individual curves indicate the times, from $t=0$ to 300.

**Figure 6.**$\mathcal{E}$ as a function of time, for the six configurations as in Figure 2. Note again how the scale for t is sometimes linear and sometimes logarithmic.

**Figure 7.**The total information length ${\mathcal{L}}_{\infty}$ as a function of initial position $\mu $. The labels ${\mathbf{f}}_{*},\mathbf{n}$ indicate the particular profile ${\mathbf{f}}_{*}$, and the initial width ${D}_{0}={10}^{-n}$. Within each panel the three curves are $D={10}^{-4}$ (lowest, blue), $D={10}^{-5}$ (middle, red), and $D={10}^{-6}$ (top, black).

**Figure 8.**As in Figure 7, but now showing the time ${T}_{\infty}$ that it takes to achieve $\mathcal{E}={10}^{-8}$, that is, a measure of the time it takes to reach ${\mathcal{L}}_{\infty}$. Note how ${T}_{\infty}$ has a linear scale for ${\mathbf{f}}_{\mathbf{0}}$, but logarithmic scales for ${\mathbf{f}}_{-}$ and ${\mathbf{f}}_{+}$. The colour-coding is again $D={10}^{-4}$ blue, $D={10}^{-5}$ red, $D={10}^{-6}$ black.

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**MDPI and ACS Style**

Hollerbach, R.; Kim, E.-j.
Information Geometry of Spatially Periodic Stochastic Systems. *Entropy* **2019**, *21*, 681.
https://doi.org/10.3390/e21070681

**AMA Style**

Hollerbach R, Kim E-j.
Information Geometry of Spatially Periodic Stochastic Systems. *Entropy*. 2019; 21(7):681.
https://doi.org/10.3390/e21070681

**Chicago/Turabian Style**

Hollerbach, Rainer, and Eun-jin Kim.
2019. "Information Geometry of Spatially Periodic Stochastic Systems" *Entropy* 21, no. 7: 681.
https://doi.org/10.3390/e21070681