# Fluctuation Theorem of Information Exchange between Subsystems that Co-Evolve in Time

## Abstract

**:**

## 1. Introduction

## 2. Results

#### 2.1. Theoretical Framework

#### 2.2. Proof of Fluctuation Theorem of Information Exchange

#### 2.3. Corollary

## 3. Examples

#### 3.1. Measurement

#### 3.2. Feedback Control

## 4. Conclusions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Paradox in thermodynamics of information (

**a**) Maxwell’s demon (orange cat) uses information on the speed of the particles in the box: He opens/closes the small hole (orange line) without expenditure of energy such that fast particles (red filled circles) are gathered in the upper-half of the box and slow particles (blue filled circles) are gathered in the lower-half of the box. Since temperature is the average velocity of the particles, the demon’s action results in spontaneous flow of heat from colder places to hotter places, which violates the second-law of thermodynamics. (

**b**) A cycle of Szilard’s engine is represented. A lever (green curved arrow) is controlled such that a weight can be lifted during the wall moves quasi-statically in the direction that the particle pushes. This engine harnesses heat from the heat reservoir (yellow region around each boxes) and convert it into mechanical work, cyclically, and thus corresponds to a perpetual-motion engine of the second kind, which is prohibited by the second-law of thermodynamics.

**Figure 2.**Measurement and feedback control: system X is, for example, a measuring device and system Y is a measured system. X and Y co-evolve, as they interact weakly, along trajectories $\left\{{x}_{t}\right\}$ and $\left\{{y}_{t}\right\}$, respectively. (

**a**) Coupling is being established during the measurement process so that ${I}_{t}({x}_{t},{y}_{t})$ for $0\le t\le \tau $ may be increased (not necessarily monotonically). (

**b**) Established correlation is being used as a source of work through external parameter ${\lambda}_{t}$ so that ${I}_{t}({x}_{t},{y}_{t})$ for $\tau \le t\le {\tau}^{\prime}$ may be decreased (not necessarily monotonically).

**Table 1.**The joint probability distribution of x and y at an intermediate time t: Here we assume for simplicity that both systems X and Y have two states, 0 (left) and 1 (right).

$\mathit{X}\setminus \mathit{Y}$ | 0 (Left) | 1 (Right) |
---|---|---|

0 (Left) | 1/3 | 1/6 |

1 (Right) | 1/6 | 1/3 |

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Jinwoo, L.
Fluctuation Theorem of Information Exchange between Subsystems that Co-Evolve in Time. *Symmetry* **2019**, *11*, 433.
https://doi.org/10.3390/sym11030433

**AMA Style**

Jinwoo L.
Fluctuation Theorem of Information Exchange between Subsystems that Co-Evolve in Time. *Symmetry*. 2019; 11(3):433.
https://doi.org/10.3390/sym11030433

**Chicago/Turabian Style**

Jinwoo, Lee.
2019. "Fluctuation Theorem of Information Exchange between Subsystems that Co-Evolve in Time" *Symmetry* 11, no. 3: 433.
https://doi.org/10.3390/sym11030433