# Anomalous Diffusion with an Apparently Negative Diffusion Coefficient in a One-Dimensional Quantum Molecular Chain Model

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## Abstract

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## 1. Introduction

## 2. Model and Quantum Transport Equation

## 3. H-Theorem for the Quantum Transport Equation

## 4. Example of the Exciton Propagation with Apparent Negative Diffusion Coefficient

## 5. Summary and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Time evolution of ${D}^{\left(x\right)}\left(t\right)$ under two different initial conditions in units where $m=1$, $c=1$, $\hslash =1$, ${k}_{\mathrm{B}}=1$, $g{{\Delta}}_{0}=1$, and ${\rho}_{M}=1$. The transport coefficients $\sigma \left(P\right)$ and $D\left(P\right)$ are calculated at the temperature $T=1$. We chose the initial peak positions of the Gaussian wave packets as $\{({\overline{X}}_{1},{\overline{P}}_{1}),({\overline{X}}_{2},{\overline{P}}_{2})\}=\{(0,0.7),(40,-0.7)\}$ for the solid-line, and $\{({\overline{X}}_{1},{\overline{P}}_{1}),({\overline{X}}_{2},{\overline{P}}_{2})\}=\{(0,-0.2),(40,0.5)\}$ for the dashed-line. The width of each Gaussian wave packet is given as ${\Delta}{X}_{\alpha =1,2}=3$. Note that the origin of time t is ${\tau}_{\mathrm{rel}}$.

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**MDPI and ACS Style**

Nakade, S.; Kanki, K.; Tanaka, S.; Petrosky, T.
Anomalous Diffusion with an Apparently Negative Diffusion Coefficient in a One-Dimensional Quantum Molecular Chain Model. *Symmetry* **2021**, *13*, 506.
https://doi.org/10.3390/sym13030506

**AMA Style**

Nakade S, Kanki K, Tanaka S, Petrosky T.
Anomalous Diffusion with an Apparently Negative Diffusion Coefficient in a One-Dimensional Quantum Molecular Chain Model. *Symmetry*. 2021; 13(3):506.
https://doi.org/10.3390/sym13030506

**Chicago/Turabian Style**

Nakade, Sho, Kazuki Kanki, Satoshi Tanaka, and Tomio Petrosky.
2021. "Anomalous Diffusion with an Apparently Negative Diffusion Coefficient in a One-Dimensional Quantum Molecular Chain Model" *Symmetry* 13, no. 3: 506.
https://doi.org/10.3390/sym13030506