# Fluctuation, Dissipation and the Arrow of Time

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## Abstract

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**PACS**1.70.+w 05.40.-a 05.70.Ln

## 1. Introduction

**Equilibrium Principle**- An isolated, macroscopic system which is placed in an arbitrary initial state within a finite fixed volume will attain a unique state of equilibrium.
**Second Law**(**Clausius**)- For a non-quasi-static process occurring in a thermally isolated system, the entropy change between two equilibrium states is non-negative.

**Figure 1.**Autonomous vs. nonautonomous dynamics. Top: Autonomous evolution of a gas from a non-equilibrium state to an equilibrium state (Minus-First Law). Bottom: Nonautonomous evolution of a thermally isolated gas between two equilibrium states. The piston moves according to a pre-determined protocol specifying its position ${\lambda}_{t}$ in time. The entropy change is non-negative (Second Law).

**Second Law**(**Kelvin**)- No work can be extracted from a closed equilibrium system during a cyclic variation of a parameter by an external source.

## 2. The Fluctuation Theorem

#### 2.1. Autonomous Dynamics

#### 2.2. Nonautonomous Dynamics

**Figure 2.**Microreversibility for nonautonomous classical (Hamiltonian) systems. The initial condition ${\Gamma}_{0}$ evolves to ${\Gamma}_{\tau}$ under the protocol λ, following the path Γ. The time-reversed final condition $\epsilon {\Gamma}_{\tau}$ evolves to the time-reversed initial condition $\epsilon {\Gamma}_{0}$ under the protocol $\tilde{\lambda}$, following the path $\tilde{\Gamma}$.

**Second Law**(**Fluctuation Theorem**)- Injecting some amount of energy ${W}_{0}$ into a thermally insulated system at equilibrium at temperature T by the cyclic variation of a parameter, is exponentially (i.e., by a factor ${e}^{{W}_{0}/\left({k}_{B}T\right)}$) more probable than withdrawing the same amount of energy from it by the reversed parameter variation.

## 3. Dissipation: Kubo’s Formula

## 4. Implications for the Arrow of Time Question

**Figure 3.**Degree of belief $P(+|{W}_{0})$ that a movie showing the nonautonomous evolution of a system is shown in the same temporal order as it was filmed, given that the work ${W}_{0}$ was observed and that the direction of the movie was decided by the tossing of an unbiased coin.

## 5. Remarks

## Acknowledgments

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**MDPI and ACS Style**

Campisi, M.; Hänggi, P.
Fluctuation, Dissipation and the Arrow of Time. *Entropy* **2011**, *13*, 2024-2035.
https://doi.org/10.3390/e13122024

**AMA Style**

Campisi M, Hänggi P.
Fluctuation, Dissipation and the Arrow of Time. *Entropy*. 2011; 13(12):2024-2035.
https://doi.org/10.3390/e13122024

**Chicago/Turabian Style**

Campisi, Michele, and Peter Hänggi.
2011. "Fluctuation, Dissipation and the Arrow of Time" *Entropy* 13, no. 12: 2024-2035.
https://doi.org/10.3390/e13122024