# Energy Dissipation and Information Flow in Coupled Markovian Systems

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## Abstract

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## 1. Introduction

## 2. Theoretical Background

## 3. Results

#### 3.1. Alternating Energy Levels

#### 3.2. Arbitrary System Rates

#### 3.3. Arbitrary Environment Rates

## 4. Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Discrete-time system and environmental dynamics. The system ${Y}_{t}$ and environment ${X}_{t}$ alternate steps, with system evolution during relaxation steps, and environment evolution during work steps.

**Figure 2.**Model kinetics. States and transition rates for models with two system states and two environment states. (

**a**) System equilibration rate and energy gap magnitude and environment transition rate are independent of environment state, but the direction of the energy gap switches with environment state; (

**b**) System equilibration rate and energy gap vary with environment state. Environment transition rate is fixed; (

**c**) System equilibration rate and energy gap and environment transition rate vary with environment state.

**Figure 3.**Nostalgia increases with energy gap and system equilibration rate. Nostalgia ${I}_{\mathrm{nos}}^{\mathrm{ss}}$ as a function of the energy gap $\beta \Delta E$ and transition ratio ${k}_{\mathrm{sys}}/{\kappa}_{\mathrm{env}}$ (${\kappa}_{\mathrm{env}}\Delta t={10}^{-12}$).

**Figure 4.**Dissipation ratio increases with energy gap and system equilibration rate. Steady-state dissipation ratio ${\varphi}^{\mathrm{ss}}\equiv {I}_{\mathrm{nos}}^{\mathrm{ss}}/\beta \u2329{W}_{\mathrm{diss}}^{ss}\u232a$ as a function of the energy gap $\beta \Delta E$ and transition ratio ${k}_{\mathrm{sys}}/{\kappa}_{\mathrm{env}}$ (${\kappa}_{\mathrm{env}}\Delta t={10}^{-12}$).

**Figure 5.**Lower bound on dissipation ratio for fixed environment transition rate. The steady-state dissipation ratio ${\varphi}^{\mathrm{ss}}$ is lower-bounded by the black curve (9) for all values of the transition ratio ${k}_{\mathrm{sys}}/{\kappa}_{\mathrm{env}}$. Each point corresponds to a particular set of parameters ${k}^{A}$, ${k}^{B}$, $\beta \Delta {E}^{A}$, and $\beta \Delta {E}^{B}$. (

**a**) Models in which the energy gaps $\Delta {E}^{A}$ and $\Delta {E}^{B}$ are anti-aligned; (

**b**) Models in which the energy gaps $\Delta {E}^{A}$ and $\Delta {E}^{B}$ are aligned (${\kappa}_{\mathrm{env}}\Delta t={10}^{-12}$).

**Figure 6.**Lower bound on dissipation ratio for varying environment transition rate. The steady-state dissipation ratio ${\varphi}^{\mathrm{ss}}$ is lower-bounded by the black curve (9) for all values of the transition ratio ${k}_{\mathrm{sys}}/{\kappa}_{\mathrm{env}}$. Each point corresponds to a particular set of parameters ${k}^{A}$, ${k}^{B}$, $\beta \Delta {E}^{A}$, and $\beta \Delta {E}^{B}$. The environment transition rates are ${\kappa}^{AB}=1.8{\overline{\kappa}}_{\mathrm{env}}$ and ${\kappa}^{BA}=0.2{\overline{\kappa}}_{\mathrm{env}}$. (

**a**) Models in which the energy gaps $\Delta {E}^{A}$ and $\Delta {E}^{B}$ are anti-aligned; (

**b**) Models in which the energy gaps $\Delta {E}^{A}$ and $\Delta {E}^{B}$ are aligned (${\overline{\kappa}}_{\mathrm{env}}\Delta t={10}^{-12}$).

**Table 1.**Limiting behavior of dissipation ratio. Steady-state dissipation ratio ${\varphi}^{\mathrm{ss}}$ in the various limits of driving strength and speed. These limits are given by the bound in Equation (10), valid in the limit of continuous time.

Driving Strength | Weak | Strong | |
---|---|---|---|

Driving Speed | ($\mathit{\beta}\mathbf{\Delta}\mathit{E}\ll \mathbf{1}$) | ($\mathit{\beta}\mathbf{\Delta}\mathit{E}\gtrsim \mathbf{1}$) | |

Quasi-static | (${\kappa}_{\mathrm{env}}\ll {k}_{\mathrm{sys}}$) | ${\varphi}^{\mathrm{ss}}=1$ | ${\varphi}^{\mathrm{ss}}=1$ |

Intermediate | (${\kappa}_{\mathrm{env}}\sim {k}_{\mathrm{sys}}$) | ${\varphi}^{\mathrm{ss}}={(1+{\kappa}_{\mathrm{env}}/{k}_{\mathrm{sys}})}^{-1}$ | ${(1+{\kappa}_{\mathrm{env}}/{k}_{\mathrm{sys}})}^{-1}\le {\varphi}^{\mathrm{ss}}\le 1$ |

Fast | (${\kappa}_{\mathrm{env}}\gg {k}_{\mathrm{sys}}$) | ${\varphi}^{\mathrm{ss}}={k}_{\mathrm{sys}}/{\kappa}_{\mathrm{env}}$ | ${k}_{\mathrm{sys}}/{\kappa}_{\mathrm{env}}\le {\varphi}^{\mathrm{ss}}\le 1$ |

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Quenneville, M.E.; Sivak, D.A. Energy Dissipation and Information Flow in Coupled Markovian Systems. *Entropy* **2018**, *20*, 707.
https://doi.org/10.3390/e20090707

**AMA Style**

Quenneville ME, Sivak DA. Energy Dissipation and Information Flow in Coupled Markovian Systems. *Entropy*. 2018; 20(9):707.
https://doi.org/10.3390/e20090707

**Chicago/Turabian Style**

Quenneville, Matthew E., and David A. Sivak. 2018. "Energy Dissipation and Information Flow in Coupled Markovian Systems" *Entropy* 20, no. 9: 707.
https://doi.org/10.3390/e20090707