# Entropy Production in Non-Markovian Collision Models: Information Backflow vs. System-Environment Correlations

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## Abstract

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## 1. Introduction

## 2. Collision Model

**Strategy 1**- Before moving on to the next collision, we erase all the correlations between each ingredient of the model. In this case, the system state after the ith step of the considered model is given by$${\rho}_{i+1}^{S}={\mathrm{tr}}_{{E}_{i},{E}_{i+1}}\left[{\mathcal{U}}_{{E}_{i},{E}_{i+1}}{\mathcal{U}}_{S,{E}_{i}}\left({\rho}_{i}^{S}\otimes {\tilde{\rho}}_{i}^{{E}_{i}}\otimes {\rho}_{i}^{{E}_{i+1}}\right){\mathcal{U}}_{S,{E}_{i}}^{\u2020}{\mathcal{U}}_{{E}_{i},{E}_{i+1}}^{\u2020}\right],$$
**Strategy 2**- We keep the state of ${\rho}_{i}^{S{E}_{i}}$ untouched and use it as it is in the next iteration. Now, the system state after the ith step of the model is given as$${\rho}_{i+1}^{S}={\mathrm{tr}}_{{E}_{i},{E}_{i+1}}\left[{\mathcal{U}}_{{E}_{i},{E}_{i+1}}{\mathcal{U}}_{S,{E}_{i}}\left({\rho}_{i}^{S{E}_{i}}\otimes {\rho}_{i}^{{E}_{i+1}}\right){\mathcal{U}}_{S,{E}_{i}}^{\u2020}{\mathcal{U}}_{{E}_{i},{E}_{i+1}}^{\u2020}\right].$$Clearly, in such a case, any correlations established between S and ${E}_{i}$ on the $(i-1)$th step, i.e., before they directly interact, are carried over to the ith step.

## 3. Quantifying Non-Markovianity

## 4. Entropy Production

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Trace distance as a function of the number of collisions. While the dashed black lines show the behavior of the Markovian limit of both strategies, solid, blue lines display the behavior of the trace distance in the non-Markovian regime. Insets in both of the plots show the non-Markovianity measure, which is the sum of the amount of increases in the trace distance throughout the dynamics. The model parameters are chosen as ${T}_{E}=1$, system-environment interaction strength $\nu =0.05\times \pi /2$, and all particles in the model are resonant ${\omega}_{S}={\omega}_{E}=1$. For non-Markovian dynamics, intra-environment interaction strength is chosen as $\epsilon =0.95\times \pi /2$. The initial state pair used in the calculation of the non-Markovianity measure is given by the eigenstates of the Pauli operator ${\sigma}_{x}$.

**Figure 2.**Total irreversible entropy production (

**a**,

**b**) and its rate (

**c**,

**d**) for Markovian (blacked dashed lines) and non-Markovian (solid blue lines) evolution. The model parameters are chosen as ${T}_{S}=0.1$, ${T}_{E}=1$, system-environment interaction strength $\nu =0.05\times \pi /2$, and all particles in the model are resonant ${\omega}_{S}={\omega}_{E}=1$. For non-Markovian dynamics, intra-environment interaction strength is chosen as $\epsilon =0.95\times \pi /2$.

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**MDPI and ACS Style**

Şenyaşa, H.T.; Kesgin, Ş.; Karpat, G.; Çakmak, B.
Entropy Production in Non-Markovian Collision Models: Information Backflow vs. System-Environment Correlations. *Entropy* **2022**, *24*, 824.
https://doi.org/10.3390/e24060824

**AMA Style**

Şenyaşa HT, Kesgin Ş, Karpat G, Çakmak B.
Entropy Production in Non-Markovian Collision Models: Information Backflow vs. System-Environment Correlations. *Entropy*. 2022; 24(6):824.
https://doi.org/10.3390/e24060824

**Chicago/Turabian Style**

Şenyaşa, Hüseyin T., Şahinde Kesgin, Göktuğ Karpat, and Barış Çakmak.
2022. "Entropy Production in Non-Markovian Collision Models: Information Backflow vs. System-Environment Correlations" *Entropy* 24, no. 6: 824.
https://doi.org/10.3390/e24060824