# Eigentimes and Very Slow Processes

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## Abstract

**:**

## 1. Introduction

## 2. No Wind

## 3. No Temperature

## 4. Linking Phenomenology to Mathematics: Closure

## 5. Resolution of Chemical Reactions

## 6. Eigentimes

## 7. Evolutionary Modeling

## 8. Discussion

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Two images of the same Niagara Falls downstream flow. The image (

**a**) is an exposure of 0.4 s, while the image (

**b**) is exposed for 50 s. Note the flow features visible in (b) (stream lines, bow waves, standing waves, vortices, etc.) that are not clearly visible or simply invisible in (a).

**Figure 2.**The velocity distribution $p(v;w,\psi )$ of Equation (4) for $w=2.5$ and center value $\psi =1$ (red). A pure Gaussian thermal distribution is shown in green for comparison. The frame (

**a**) is a normal linear plot and the frame (

**b**) is a semilog plot where the agreement between the slow time distribution (red) and a thermal distribution (green) for small velocities but large discrepancy at large velocities is even more evident.

**Figure 3.**The free energy landscape of the four species $A,B,C,E$ and the transition states between them for collision rates Z in the range ${10}^{12}$ to ${10}^{-3}$ ${s}^{-1}$. This shows that for timescales larger than $\tau =1/Z=1$ s (red curve) states B and C are equilibrated and should rather be thought of as a single state. The portions representing negative $\Delta {G}^{\u2021}$ have been dotted; on these timescales the two species should be combined to give one effective species.

**Figure 4.**Decay curves for 3 parallel processes of distinctly different eigentimes, in plot (

**a**) on a linear scale, in plot (

**b**) on a semilog scale. The 3 exponential decays are most clearly separatred on a semilog plot (b), where the green lines indicate each of the exponential decays separately.

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Andresen, B.; Essex, C.
Eigentimes and Very Slow Processes. *Entropy* **2017**, *19*, 492.
https://doi.org/10.3390/e19090492

**AMA Style**

Andresen B, Essex C.
Eigentimes and Very Slow Processes. *Entropy*. 2017; 19(9):492.
https://doi.org/10.3390/e19090492

**Chicago/Turabian Style**

Andresen, Bjarne, and Christopher Essex.
2017. "Eigentimes and Very Slow Processes" *Entropy* 19, no. 9: 492.
https://doi.org/10.3390/e19090492