# Information Geometry of Nonlinear Stochastic Systems

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## Abstract

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## 1. Introduction

## 2. Model

## 3. Analytic Solutions

#### 3.1. Exact Solutions for $D=0$

#### 3.2. Approximate Solutions for $D\ne 0$

#### 3.3. Final Stationary Distribution

## 4. Numerical Solutions

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The top row shows the peak amplitudes as functions of time, for the initial condition in Equation (34) with $\mu =0.65$, and $D={10}^{-6}$ to ${10}^{-9}$ as indicated. The three panels show $n=3,5,7$, as labeled. The thick dashed (magenta) lines correspond to the analytic result (Equation (15)) that applies in the nondiffusive phase. The bottom row shows the equivalent widths at half-peak, which are inversely proportional to the peak amplitudes. The standard deviation ${\langle {(x-\langle x\rangle )}^{2}\rangle}^{1/2}$ follows exactly the same pattern as the half-peak widths.

**Figure 3.**The top row shows the skewness $\int {[(x-\langle x\rangle )/\sigma ]}^{3}\phantom{\rule{0.166667em}{0ex}}p\phantom{\rule{0.166667em}{0ex}}dx$, and the bottom row shows the kurtosis $\int {[(x-\langle x\rangle )/\sigma ]}^{4}\phantom{\rule{0.166667em}{0ex}}p\phantom{\rule{0.166667em}{0ex}}dx$, as functions of time. The labeling of D and n is as in Figure 1 and Figure 2. The peaks in both quantities occur at times in essentially perfect agreement with ${t}_{2}$ in (32).

**Figure 4.**The heavy (blue) lines show the equilibrium profiles in Equation (30), for $D={10}^{-6}$, and $n=3,5,7$ as indicated. The lighter (red) lines show $p\left(x\right)$ at the $t=629,\phantom{\rule{4pt}{0ex}}3930,\phantom{\rule{4pt}{0ex}}9794$, for $n=3,5,7$, respectively. As shown in Figure 3, these are the times when the skewness reaches its maximum negative values. Results for other values of D are identical, once x and $p\left(x\right)$ are rescaled as in Equation (31), and t is shifted as in Figure 3 to consistently have the correct skewness values.

**Figure 5.**The top row shows $\mathcal{E}\left(t\right)$ and the bottom row shows $\mathcal{L}\left(t\right)$. The labeling of D and n is as in Figure 1, Figure 2 and Figure 3. The peaks are initially located at $\mu =0.65$, consistent with the initial plateau in $\mathcal{E}$ being reduced by a factor of $0.{65}^{4}=0.1785$ when comparing $n=3$ and 5, and similarly $n=5$ and 7.

**Figure 6.**${\mathcal{L}}_{\infty}$ as a function of the initial position $\mu $, for $D={10}^{-6}$ to ${10}^{-9}$ and $n=3,5,7$ as labeled.

**Table 1.**The first two rows show the overall peak amplitudes ${p}_{\mathbf{Max}}$ and the times ${t}_{\mathbf{Max}}$ at which they occur, for the results in Figure 1. $D={10}^{-8}$ and ${10}^{-9}$, and $n=3,5,7$ as indicated, and all results at $\mu =0.65$. The row labeled “Exponent” uses the ratios of the two D values to extract scaling exponents of the form ${D}^{-\alpha}$. The final row compares these numerically deduced exponents with the analytic predictions from Equations (26) and (29). That is, the exponent $\alpha $ should equal $\frac{n-1}{3n-1}$ for ${t}_{\mathbf{Max}}$, and $\frac{n}{3n-1}$ for ${p}_{\mathbf{Max}}$.

$\mathit{n}=3$ | $\mathit{n}=5$ | $\mathit{n}=7$ | ||||
---|---|---|---|---|---|---|

${\mathit{t}}_{\mathbf{Max}}\phantom{\rule{4pt}{0ex}}$ | ${\mathit{p}}_{\mathbf{Max}}$ | ${\mathit{t}}_{\mathbf{Max}}\phantom{\rule{4pt}{0ex}}$ | ${\mathit{p}}_{\mathbf{Max}}$ | ${\mathit{t}}_{\mathbf{Max}}\phantom{\rule{4pt}{0ex}}$ | ${\mathit{p}}_{\mathbf{Max}}$ | |

$D={10}^{-8}$ | $4.42\phantom{\rule{4pt}{0ex}}$ | 2063 | $5.82\phantom{\rule{4pt}{0ex}}$ | 1661 | $8.17\phantom{\rule{4pt}{0ex}}$ | 1337 |

$D={10}^{-9}$ | $8.80\phantom{\rule{4pt}{0ex}}$ | 4888 | $12.55\phantom{\rule{4pt}{0ex}}$ | 3776 | $18.58\phantom{\rule{4pt}{0ex}}$ | 2998 |

Exponent | $0.30\phantom{\rule{4pt}{0ex}}$ | $0.375$ | $0.33\phantom{\rule{4pt}{0ex}}$ | $0.357$ | $0.36\phantom{\rule{4pt}{0ex}}$ | $0.349$ |

(26), (29) | $0.25\phantom{\rule{4pt}{0ex}}$ | $0.375$ | $0.29\phantom{\rule{4pt}{0ex}}$ | $0.357$ | $0.30\phantom{\rule{4pt}{0ex}}$ | $0.350$ |

**Table 2.**The first two rows are as in Table 1, but now for fixed $D={10}^{-9}$, $\mu =0.55$ and $0.75$, and $n=3,5,7$ as indicated. The row labeled “Exponent” uses the ratios of the two $\mu $ values to extract scaling exponents of the form ${\mu}^{\delta}$. The final row compares these numerically deduced exponents with the analytic predictions from Equations (26) and (29). That is, the exponent $\delta $ should equal $-\frac{2n(n-1)}{3n-1}$ for ${t}_{\mathbf{Max}}$, and $\frac{n(n-1)}{3n-1}$ for ${p}_{\mathbf{Max}}$.

$\mathit{n}=3$ | $\mathit{n}=5$ | $\mathit{n}=7$ | ||||
---|---|---|---|---|---|---|

${\mathit{t}}_{\mathbf{Max}}\phantom{\rule{4pt}{0ex}}$ | ${\mathit{p}}_{\mathbf{Max}}$ | ${\mathit{t}}_{\mathbf{Max}}\phantom{\rule{4pt}{0ex}}$ | ${\mathit{p}}_{\mathbf{Max}}$ | ${\mathit{t}}_{\mathbf{Max}}\phantom{\rule{4pt}{0ex}}$ | ${\mathit{p}}_{\mathbf{Max}}$ | |

$\mu =0.55$ | $11.17\phantom{\rule{4pt}{0ex}}$ | 4314 | $19.74\phantom{\rule{4pt}{0ex}}$ | 2975 | $35.87\phantom{\rule{4pt}{0ex}}$ | 2105 |

$\mu =0.75$ | $7.18\phantom{\rule{4pt}{0ex}}$ | 5439 | $8.48\phantom{\rule{4pt}{0ex}}$ | 4631 | $10.46\phantom{\rule{4pt}{0ex}}$ | 4035 |

Exponent | $-1.4\phantom{\rule{4pt}{0ex}}$ | $0.75$ | $-2.7\phantom{\rule{4pt}{0ex}}$ | $1.43$ | $-4.0\phantom{\rule{4pt}{0ex}}$ | $2.10$ |

(26), (29) | $-1.5\phantom{\rule{4pt}{0ex}}$ | $0.75$ | $-2.9\phantom{\rule{4pt}{0ex}}$ | $1.43$ | $-4.2\phantom{\rule{4pt}{0ex}}$ | $2.10$ |

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**MDPI and ACS Style**

Hollerbach, R.; Dimanche, D.; Kim, E.-j. Information Geometry of Nonlinear Stochastic Systems. *Entropy* **2018**, *20*, 550.
https://doi.org/10.3390/e20080550

**AMA Style**

Hollerbach R, Dimanche D, Kim E-j. Information Geometry of Nonlinear Stochastic Systems. *Entropy*. 2018; 20(8):550.
https://doi.org/10.3390/e20080550

**Chicago/Turabian Style**

Hollerbach, Rainer, Donovan Dimanche, and Eun-jin Kim. 2018. "Information Geometry of Nonlinear Stochastic Systems" *Entropy* 20, no. 8: 550.
https://doi.org/10.3390/e20080550