# Far-From-Equilibrium Time Evolution between Two Gamma Distributions

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## Abstract

**:**

## 1. Introduction

## 2. Stochastic Logistic Model

## 3. Diagnostics

## 4. Results

#### 4.1. $\gamma >D$

#### 4.2. $D>\gamma $

#### 4.3. $Q\ne 0$

## 5. Conclusions

- If $D<\gamma $, so that stationary solutions exist, but D is also sufficiently close to $\gamma $ that a gamma distribution differs significantly from a Gaussian, then the time-dependent PDFs will also differ significantly from gamma distributions.
- If $D>\gamma $, stationary gamma distributions do not exist at all. Instead, peaks move ever closer to the origin and in the process increasingly differ from gamma distributions.
- If the initial condition is a peak right on the origin—either as a result of adding additive noise to produce stationary solutions even for $D>\gamma $, or simply as an arbitrary initial condition—then any evolution away from the origin will differ significantly from gamma distributions. Unlike the previous two items, which become more pronounced for larger D, this effect is most clearly visible for smaller D, where the mismatch between the naturally narrower peaks and the extreme broadening seen in Figure 11 becomes increasingly significant.

## Author Contributions

## Conflicts of Interest

## Appendix A. Derivation of the Fokker–Planck Equations

## Appendix B. Time-Dependent Analytical Solutions of Equation (3)

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**Figure 1.**(

**a**) shows the result of switching $\gamma =0.5\to 0.05$; (

**b**) $\gamma =0.05\to 0.5$, both at fixed $\u03f5=1$ and $D=0.02$. The initial (red) and final (blue) gamma distributions are shown as heavy lines. The four intermediate lines are when the time-dependent solutions have $\langle x\rangle =0.1,\phantom{\rule{4pt}{0ex}}0.2,\phantom{\rule{4pt}{0ex}}0.3,\phantom{\rule{4pt}{0ex}}0.4$. The arrows are a reminder of the direction of motion, inward on the left and outward on the right.

**Figure 2.**(

**a**) shows $\langle x\rangle $ as a function of time; (

**b**) shows $\mathcal{E}\xb7D$ (to indicate the $\mathcal{E}\sim {D}^{-1}$ scaling); (

**c**) shows $\mathcal{L}\xb7{D}^{1/2}$ (to indicate the $\mathcal{L}\sim {D}^{-1/2}$ scaling). Solid lines denote $\gamma =0.5\to 0.05$, dashed lines the reverse. Each solid or dashed “line” is in fact three, occasionally barely distinguishable, lines with $D=0.01,\phantom{\rule{4pt}{0ex}}0.02,\phantom{\rule{4pt}{0ex}}0.04$. The dots on the $\langle x\rangle $ curves correspond to the PDFs shown in Figure 1.

**Figure 3.**(

**a**) shows $\mathcal{L}\xb7{D}^{1/2}$ and (

**b**) entropy, both as functions of $\langle x\rangle $. Solid lines denote $\gamma =0.5\to 0.05$, dashed lines the reverse. Numbers besides curves indicate $D=0.01,\phantom{\rule{4pt}{0ex}}0.02,\phantom{\rule{4pt}{0ex}}0.04$. The arrows on the entropy plot are a reminder of the direction of inward/outward motion.

**Figure 4.**(

**a**) $\sigma /{D}^{1/2}$, (

**b**) (skewness $/{D}^{1/2}$) and (

**c**) (kurtosis $/D$), as functions of $\langle x\rangle $. Solid lines denote $\gamma =0.5\to 0.05$, dashed lines the reverse. Numbers besides curves indicate $D=0.01,\phantom{\rule{4pt}{0ex}}0.02,\phantom{\rule{4pt}{0ex}}0.04$. The heavy green curves are $\sqrt{\langle x\rangle}$, $2/\sqrt{\langle x\rangle}$ and $6/\langle x\rangle $, respectively, and indicate the behaviour expected for exact gamma distributions.

**Figure 5.**(

**a**) shows the difference (13) between the actual PDF and the equivalent gamma distribution, as functions of $\langle x\rangle $. Solid lines denote $\gamma =0.5\to 0.05$, dashed lines the reverse, with arrows also indicating the direction of motion. The dots at $\langle x\rangle =0.3$ for $\gamma =0.05\to 0.5$ and $\langle x\rangle =0.1$ for $\gamma =0.5\to 0.05$, correspond to the other two panels: (

**b**) compares the $\gamma =0.05\to 0.5$ PDF with its equivalent gamma distribution; (

**c**) compares the $\gamma =0.5\to 0.05$ PDF with its equivalent gamma distribution. The actual PDFs in each case are solid (red), and the equivalent gamma distributions are dashed (blue). $D=0.04$ for both sets.

**Figure 6.**The initial condition is a gamma distribution with $\gamma =0.5$, $\u03f5=1$ and $D={10}^{-3}$; $\gamma $ is then switched to zero, and the solution is evolved according to Equation (3). Numbers besides curves indicate time, from the initial condition at $t=0$ to the final time 1000. The dashed curves indicate the equivalent gamma distributions having the same $\langle x\rangle $ and $\sigma $.

**Figure 7.**As in Figure 2, (

**a**) shows $\langle x\rangle $; (

**b**) shows $\mathcal{E}\xb7D$; and (

**c**) $\mathcal{L}\xb7{D}^{1/2}$. Solid lines denote $\gamma =0.5\to 0$ for $D={10}^{-3}$, dashed lines the previous $\gamma =0.5\to 0.05$ for $D=0.01$. Note how the scalings of $\mathcal{E}$ and $\mathcal{L}$ with D are still preserved even when D is changed by a factor of 10.

**Figure 8.**(

**a**) Entropy, (

**b**) $\sigma /{D}^{1/2}$, (

**c**) (skewness $/{D}^{1/2}$) and (

**d**) (kurtosis $/D$), as functions of $\langle x\rangle $, for the $\gamma =0.5\to 0$ calculation from Figure 6. The heavy green curves in the last three panels are $\sqrt{\langle x\rangle}$, $2/\sqrt{\langle x\rangle}$ and $6/\langle x\rangle $, respectively, and indicate the behaviour expected for exact gamma distributions.

**Figure 9.**(

**a**) shows the result of switching $\gamma =0.5\to 0.1$, (

**b**) $\gamma =0.1\to 0.5$, both at fixed $\u03f5=1$, $D={10}^{-3}$ and $Q={10}^{-5}$. The initial (red) and final (blue) gamma distributions are shown as heavy lines. The three intermediate lines are when the time-dependent solutions have $\langle x\rangle =0.2,\phantom{\rule{4pt}{0ex}}0.3,\phantom{\rule{4pt}{0ex}}0.4$. ${\mathcal{L}}_{\infty}=25$ on the left and 16 on the right.

**Figure 10.**(

**a**) the result of switching $\gamma =0.5\to 0$, (

**b**) $\gamma =0\to 0.5$, both at fixed $\u03f5=1$, $D={10}^{-3}$ and $Q={10}^{-5}$. The initial (red) and final (blue) gamma distributions are shown as heavy lines. The four intermediate lines are when the time-dependent solutions have $\langle x\rangle =0.1,\phantom{\rule{4pt}{0ex}}0.2,\phantom{\rule{4pt}{0ex}}0.3,\phantom{\rule{4pt}{0ex}}0.4$. ${\mathcal{L}}_{\infty}=35$ on the left and 9.5 on the right.

**Figure 11.**The $\gamma =0\to 0.5$ process as in Figure 10, but now shown in more detail. The dashed (magenta) curves are the gamma distributions that best fit the two thicker curves at intermediate times. Note how even a “best-fit” is a rather poor approximation to the actual PDFs.

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**MDPI and ACS Style**

Kim, E.-j.; Tenkès, L.-M.; Hollerbach, R.; Radulescu, O. Far-From-Equilibrium Time Evolution between Two Gamma Distributions. *Entropy* **2017**, *19*, 511.
https://doi.org/10.3390/e19100511

**AMA Style**

Kim E-j, Tenkès L-M, Hollerbach R, Radulescu O. Far-From-Equilibrium Time Evolution between Two Gamma Distributions. *Entropy*. 2017; 19(10):511.
https://doi.org/10.3390/e19100511

**Chicago/Turabian Style**

Kim, Eun-jin, Lucille-Marie Tenkès, Rainer Hollerbach, and Ovidiu Radulescu. 2017. "Far-From-Equilibrium Time Evolution between Two Gamma Distributions" *Entropy* 19, no. 10: 511.
https://doi.org/10.3390/e19100511