# Broken versus Non-Broken Time Reversal Symmetry: Irreversibility and Response
## 4 The Dissipation Function
### t-Mixing and Irreversibility  
Let us rewrite Equation (58) in a more convenient way as follows:  
\[\frac{1}{\tau}\ln\frac{\mu_{t}(\overline{\Omega}_{0,\tau}^{(0)}|_{(A)_{\delta}}) }{\mu_{t}(\overline{\Omega}_{0,\tau}^{(0)}|_{(-A)_{\delta}})}=A+\epsilon( \delta,t,A,\tau)-\frac{1}{\tau}\ln\mathcal{C}_{0}(A,\delta,t,\tau)\, \tag{59}\]  
where we have introduced a new symbol \(\mathcal{C}_{0}\) to stress its role in the following:  
\[\mathcal{C}_{0}(A,\delta,t,\tau):=\left\langle e^{-\Omega_{0t}^{(0)}-\Omega_{ t+\tau 2+\tau}^{(0)}}\right\rangle_{\overline{\Omega}_{t+\tau}^{(0)}\in(A)_{\delta}}^{(0 )}=\left\langle\mathcal{Q}_{t}\left(\mathcal{Q}_{t}\circ S^{t+\tau}\right) \right\rangle_{\overline{\Omega}_{t+\tau}^{(0)}\in(A)_{\delta}}^{(0)} \tag{60}\]  
as a kind of auto-correlation function for the following observable:  
\[\mathcal{Q}_{t}(\Gamma):=\exp\left[-\int_{0}^{t}\Omega^{(0)}(S^{u}\Gamma) \mathrm{d}u\right]. \tag{61}\]  
Note that, in Equation (60), the average is computed with respect to the initial probability distribution \(\mu_{0}\). Suppose now to take the \(t\to\infty\) limit at fixed \(A\), \(\delta\), \(\tau\) of Equation (59). If the logarithm of \(\mathcal{C}_{0}\) diverges, one of the probabilities of the left-hand side of (59) goes to zero, which means that either \(A\) or \(-A\) are not observable at the steady state, i.e., when the probability distribution is \(\mu_{\infty}\). Clearly, if no pair \((A,-A)\) is observable, the steady-state FR is of no interest. On the contrary, if observable pairs \((A,-A)\) do exist, it means that, for such values, a steady-state FR in fact holds, i.e., the ratio of the probabilities to observe such values with respect to \(\mu_{\infty}\) must satisfy the following inequalities, for any \(\delta>0\):  
\[A-\delta\leq\frac{1}{\tau}\ln\frac{\mu_{\infty}(\overline{\Omega}_{0,\tau}^{(0 )}|_{(A)_{\delta}})}{\mu_{\infty}(\overline{\Omega}_{0,\tau}^{(0)}|_{(-A)_{ \delta}})}\leq A+\delta. \tag{62}\]  
The relation (62) evidently implies that the \(t\to\infty\) limit of the term containing the auto-correlation function \(\mathcal{C}_{0}\) is negligible. Then, relation (62) follows in the \(\tau\to\infty\) limit, if \(\mathcal{C}_{0}\) does not grow exponentially fast with \(\tau\). This points out a necessary condition for the steady-state FR to hold for a pair \((A,-A)\), i.e., the decay in the auto-correlation with respect to the initial (non-invariant) distribution of the variable defined by Equation (61). Thanks to that, Equation (60) can be factorized as follows:  
\[\left\langle e^{-\Omega_{0,t}^{(0)}-\Omega_{t+\tau 2+\tau}^{(0)}}\right\rangle ^{(0)}=\left\langle e^{-\Omega_{0,t}^{(0)}}\right\rangle^{(0)}\left\langle e ^{-\Omega_{t+\tau 2+\tau}^{(0)}}\right\rangle^{(0)}. \tag{63}\]  
Equation (63) is one instance of the following property, which we call _weak t-mixing_:  
\[\lim_{t\to\infty}\left[\left\langle\mathcal{Q}\left(\mathcal{O}\circ S^{t} \right)\right\rangle^{(0)}-\left\langle\mathcal{Q}\right\rangle^{(0)}\left\langle \mathcal{O}\circ S^{t}\right\rangle^{(0)}\right]=0. \tag{64}\]  
The name derives from the fact that this property is a form of mixing, but it concerns the probability measure associated with the transient initial state. Taking \(\mathcal{Q}=\Omega^{(0)}\) in Equation (64), and recalling that \(\langle\Omega^{(0)}\rangle^{(0)}=0\) because \(\Omega^{(0)}\) is odd for even \(f_{0}\), Equation (64) amounts to what we call _weak \(\Omega t\)-mixing_ condition:  
\[\lim_{t\to\infty}\left\langle\Omega^{(0)}\left(\mathcal{O}\circ S^{t}\right) \right\rangle^{(0)}=0. \tag{65}\]  
Then, the _response formula_ in Equation (24) can be written in asymptotic form as:  
\[\langle\mathcal{O}\rangle^{\infty}=\langle\mathcal{O}\rangle^{(0)}+\int_{0}^{ \infty}\mathrm{d}s\ \left\langle\Omega^{(0)}\left(\mathcal{O}\circ S^{s}\right)\right\rangle^{(0)}, \tag{66}\]which is the response of the initial ensemble to some perturbation codified within the evolution operator \(S^{t}\). We call \(\Omega t\)_-mixing_ the condition under which the integral in (66) exists.