# Broken versus Non-Broken Time Reversal Symmetry: Irreversibility and Response

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## Abstract

**:**

## 1. Introduction

## 2. Standard Picture for Irreversible Relaxation to Equilibrium

## 3. Time Reversible Dissipative Dynamics

## 4. The Dissipation Function

#### 4.1. Time-Dependent Perturbations

#### 4.2. Fluctuation Relations: Symmetry in Dissipative Dynamics

**Transient**$\mathcal{O}$

**-FR:**

**Transient Ω-FR:**

#### 4.3. t-Mixing and Irreversibility

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix

- For completeness, we report some mathematical derivations, as given in Ref. [24]. For the phase-space average of any observable $\mathcal{O}$, the following holds:$$\begin{array}{cc}\hfill {\langle \mathcal{O}\rangle}_{t+s}=\int \mathcal{O}(\Gamma ){f}_{t+s}(\Gamma )\Gamma & =\int \mathcal{O}\left({S}^{s}({S}^{-s}\Gamma )\right){f}_{t+s}\left({S}^{s}({S}^{-s}\Gamma )\right)\left|\frac{\partial \Gamma}{\partial {S}^{-s}\Gamma}\right|d({S}^{-s}\Gamma )\hfill \\ & =\int \mathcal{O}\left({S}^{s}({S}^{-s}\Gamma )\right){f}_{t+s}\left({S}^{s}({S}^{-s}\Gamma )\right)exp\left\{{\Lambda}_{-s,0}(\Gamma )\right\}\phantom{\rule{4pt}{0ex}}d({S}^{-s}\Gamma )\hfill \\ & =\int \mathcal{O}\left({S}^{s}({S}^{-s}\Gamma )\right){f}_{t+s}\left({S}^{s}({S}^{-s}\Gamma )\right)exp\left\{{\Lambda}_{0,s}({S}^{-s}\Gamma )\right\}\phantom{\rule{4pt}{0ex}}d({S}^{-s}\Gamma )\hfill \\ & =\int \mathcal{O}({S}^{s}\Gamma ){f}_{t+s}({S}^{s}\Gamma )exp\left\{{\Lambda}_{0,s}(\Gamma )\right\}\phantom{\rule{4pt}{0ex}}d\Gamma \hfill \\ & =\int \mathcal{O}({S}^{s}\Gamma ){f}_{t}(\Gamma )d\Gamma ={\langle \mathcal{O}\circ {S}^{s}\rangle}_{t},\hfill \end{array}$$
- We give here some guidelines to obtain Equation (23):$$\begin{array}{cc}\hfill \frac{d}{ds}{\u2329\mathcal{O}\u232a}_{{f}_{s}}& =\underset{h\to 0}{lim}\frac{1}{h}\left[{\u2329\mathcal{O}\u232a}_{{f}_{s+h}}-{\u2329\mathcal{O}\u232a}_{{f}_{s}}\right]\hfill \\ & =\underset{h\to 0}{lim}\frac{1}{h}\int \left[\mathcal{O}(\Gamma ){f}_{s+h}(\Gamma )-\mathcal{O}(\Gamma ){f}_{s}(\Gamma )\right]d\Gamma \hfill \\ & =\underset{h\to 0}{lim}\frac{1}{h}\int \mathcal{O}(\Gamma )\left[{f}_{r}(\Gamma )exp\{{\Omega}_{r-s-h,0}^{{f}_{r}}(\Gamma )\}-{f}_{r}(\Gamma )exp\{{\Omega}_{r-s,0}^{{f}_{r}}(\Gamma )\}\right]d\Gamma \hfill \\ & =\underset{h\to 0}{lim}\frac{1}{h}\int \mathcal{O}(\Gamma ){f}_{r}(\Gamma )exp\left\{{\Omega}_{r-s,0}^{{f}_{r}}(\Gamma )\right\}\left[exp\{{\Omega}_{r-s-h,r-s}^{{f}_{r}}(\Gamma )\}-1\right]d\Gamma .\hfill \end{array}$$$$\underset{h\to 0}{lim}\frac{1}{h}\left[exp\{{\Omega}_{r-s-h,r-s}^{{f}_{r}}(\Gamma )\}-1\right]={\Omega}^{{f}_{r}}({S}^{r-s}\Gamma ),$$$$\begin{array}{}(74)& \hfill \frac{d}{ds}{\u2329\mathcal{O}\u232a}_{{f}_{s}}& =\int \mathcal{O}(\Gamma ){f}_{r}(\Gamma )exp\left\{{\Omega}_{r-s,0}^{{f}_{r}}(\Gamma )\right\}\left[{\Omega}^{{f}_{r}}({S}^{r-s}\Gamma )\right]d\Gamma ,\hfill (75)& & =\int \mathcal{O}(\Gamma ){f}_{s}(\Gamma ){\Omega}^{{f}_{r}}({S}^{r-s}\Gamma )d\Gamma ={\u2329\mathcal{O}\xb7({\Omega}^{{f}_{r}}\circ {S}^{r-s})\u232a}_{{f}_{s}}.\hfill \end{array}$$

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**Figure 1.**The phase space $\mathcal{M}$ is imaginatively represented by an oblong figure. Let F be a phase variable on $\mathcal{M}$. For one many particles system, the region corresponding to the equilibrium value a of F, measured with accuracy j, is by far the largest region of $\mathcal{M}$: most of $\mathcal{M}$ is occupied by the phases Γ such that $F(\Gamma )\in (a-j,a+j)$ (white region). The regions corresponding to values $b,c,d,\ne a$ are represented by thin strips, that are much smaller than the equilibrium region. A trajectory γ in $\mathcal{M}$ is represented by a blue line, with an arrow at its upper end (B), indicating the direction of time. Let the (purely conventional) initial condition Γ of γ lie in the purple strip within the black ring, where γ is oriented upward. Evolving in the direction from A to B, the Γ in the nonequilibrium region (low entropy state), rapidly reaches the equilibrium much larger region (highest entropy state), and remains there, apart from fleeting deviations, when it enters the slim colored strips. Tracing γ backward in time means going in the direction opposite to the arrows (direction from B to A); even in this backward direction one rapidly moves out of whatever low entropy colored region, and falls in a state of higher entropy. For the Hamiltonian description, Ref. [3] states that both forward and backward evolutions out of any colored strip go from low to high entropy states (see also Reference [2] Section 5.5 and Reference [11] Section 27.5).

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Dal Cengio, S.; Rondoni, L.
Broken versus Non-Broken Time Reversal Symmetry: Irreversibility and Response. *Symmetry* **2016**, *8*, 73.
https://doi.org/10.3390/sym8080073

**AMA Style**

Dal Cengio S, Rondoni L.
Broken versus Non-Broken Time Reversal Symmetry: Irreversibility and Response. *Symmetry*. 2016; 8(8):73.
https://doi.org/10.3390/sym8080073

**Chicago/Turabian Style**

Dal Cengio, Sara, and Lamberto Rondoni.
2016. "Broken versus Non-Broken Time Reversal Symmetry: Irreversibility and Response" *Symmetry* 8, no. 8: 73.
https://doi.org/10.3390/sym8080073