# Broken versus Non-Broken Time Reversal Symmetry: Irreversibility and Response
## 4 The Dissipation Function
### t-Mixing and Irreversibility
This is the necessary and sufficient condition for an ensemble of systems to reach steady-state after the driving. Equation (66) is exact, unlike the Green-Kubo linear response formula, which is first order in the perturbation. Note that ensemble results are the standard results in response theory because the behaviour of single macroscopic systems has been hard to treat so far. Still, in experiments as well as in every day life, we observe the evolution of single systems, rather than that of large collections of identical objects. Ensemble results are accepted under the assumption that the evolution of a single system would not differ significantly from the average one, which is then called _typical_ behaviour. Typical is the behaviour shared by the vast majority of systems. For Hamiltonian macroscopic systems, this is then the behaviour of probability 1, with respect to the invariant, equilibrium distribution, i.e., the counterpart of \"counting\" in continuous spaces. In the case of dissipative dynamics, this is not possible because the corresponding invariant measures are singular with respect to the Lebesgue measure, and they attribute probability 1 to sets of zero phase space volume, i.e. to sets that are negligible from the _counting_ standpoint. Nevertheless, the weak t-mixing condition allows us to make some progress in this direction. For any observable \(\mathcal{O}\) and any integer \(n\), let  
\[E_{n}^{\mathcal{O},\delta}=\left\{\Gamma:\overline{\mathcal{O}}(\Gamma)=\lim _{T\to\infty}\frac{1}{T}\int_{0}^{T}\mathcal{O}(S^{t}\Gamma)\;\mathrm{d}t\in \left[n\delta,(n+1)\delta\right)\right\} \tag{67}\]  
be the set of initial conditions \(\Gamma\in\mathcal{M}\) such that a measurement that lasts very many microscopic times yields a given value within a tolerance \(\delta\). By construction, these sets are disjoint and invariant, i.e., a trajectory starting in one set remains inside it forever:  
\[S^{-t}E_{n}^{\mathcal{O},\delta}=E_{n}^{\mathcal{O},\delta}.\]  
Now suppose that the time-average \(\overline{\mathcal{O}}(\Gamma)\) exists for almost all \(\Gamma\), in the sense of the Lebesgue measure. This is a delicate assumption because, apart from some idealized situation, one can prove existence of time averages only for \(\mu_{\infty}\)-almost all phases, where \(\mu_{\infty}\) is the steady-state probability distribution. This is not the desired case, since \"almost all\" could mean a set of vanishing volume in \(\mathcal{M}\). At the same time, it is rather common for systems of physical interest that \(\overline{\mathcal{O}}(\Gamma)\) exists almost everywhere in the sense of phase space volumes. Therefore, we assume this to be the case. We may then write:  
\[\mathcal{C}_{0}(A,\delta,t,\tau):=\left\langle e^{-\Omega_{0t}^{(0)}-\Omega_{ t+\tau 2+\tau}^{(0)}}\right\rangle_{\overline{\Omega}_{t+\tau}^{(0)}\in(A)_{\delta}}^{(0 )}=\left\langle\mathcal{Q}_{t}\left(\mathcal{Q}_{t}\circ S^{t+\tau}\right) \right\rangle_{\overline{\Omega}_{t+\tau}^{(0)}\in(A)_{\delta}}^{(0)} \tag{60}\]0  
Let us now introduce the characteristic function \(\chi_{n}^{\mathcal{O},\delta}\) of the sets \(E_{n}^{\mathcal{O},\delta}\) defined as:  
\[\chi_{E}(\Gamma)=\left\{ \begin{array}{ll}1&\text{if }\Gamma\in E,\\ 0&\text{if }\Gamma\
otin E.\end{array} \right.\]  
If we assume that the characteristic functions verify weak t-mixing, we have:  
\[\mu_{0}\left(E_{n}^{\mathcal{O},\delta}\cap S^{-t}E_{n}^{\mathcal{O},\delta} \right)-\mu_{0}\left(S^{-t}E_{n}^{\mathcal{O},\delta}\right)\mu_{0}\left(E_{ n}^{\mathcal{O},\delta}\right)\longrightarrow 0 \tag{69}\]  
in the \(t\to\infty\) limit.