# Broken versus Non-Broken Time Reversal Symmetry: Irreversibility and Response
## 4 The Dissipation Function
### Fluctuation Relations: Symmetry in Dissipative Dynamics
In this sense, transient FRs close the circle with the fluctuation-dissipation relations which, on the contrary, get information on nonequilibrium properties from equilibrium experiments. For the behaviour in a steady-state, one may consider the following expression:  
\[\frac{\mu_{0}(\overline{\mathcal{O}}_{t,t+\tau}|_{(-A)_{\delta}})}{\mu_{0}( \overline{\mathcal{O}}_{t,t+\tau}|_{(A)_{\delta}})}=\frac{\int_{\overline{ \mathcal{O}}_{t,t+\tau}|_{(-A)_{\delta}}}f_{0}(\Gamma)d\Gamma}{\int_{\overline {\mathcal{O}}_{t,t+\tau}|_{(A)_{\delta}}}f_{0}(\Gamma)d\Gamma}. \tag{51}\]  
in the \(t\to\infty\) limit. For any finite \(t\), one has:  
\[\overline{\mathcal{O}}_{t,t+\tau}|_{(-A)_{\delta}}=\mathcal{I}S^{2t+\tau} \overline{\mathcal{O}}_{t,t+\tau}|_{(A)_{\delta}}, \tag{52}\]  
which reduces to the previous expression (47) for \(t=0\). Therefore, by introducing \(X\in\mathcal{M}\) so that \(\Gamma=iS^{2t+\tau}X\) and by using the parity of \(f_{0}\) with respect to \(TRI\), we obtain:  
\[\int_{\overline{\mathcal{O}}_{t,t+\tau}|_{(-A)_{\delta}}}f_{0}( \Gamma)\mathrm{d}\Gamma = \int_{\overline{\mathcal{O}}_{t,t+\tau}|_{(A)_{\delta}}}f_{0}(X )e^{-\Omega_{02t+\tau}(X)}\mathrm{d}X\,. \tag{53}\]  
Substituting Equation (53) in Equation (51), the result is:  
\[\frac{\mu_{0}(\overline{\mathcal{O}}_{t,t+\tau}|_{(-A)_{\delta}})}{\mu_{0}( \overline{\mathcal{O}}_{t,t+\tau}|_{(A)_{\delta}})}=\Big{\langle}\exp\left(- \Omega_{0,2t+\tau}^{(0)}\right)\Big{\rangle}_{\overline{\mathcal{O}}_{t,t+ \tau}\in(A)_{\delta}}^{(0)}, \tag{54}\]  
where the right-hand side is an average over the phase space trajectory segments for which \(\overline{\mathcal{O}}_{t,t+\tau}(\Gamma)\) is close to \(A\). This is still a transient identity, fundamentally based only TRI for \(f_{0}\). We may now use the generalized Liouville equation for the conservation of probability to replace the initial distribution of density \(f_{0}\) in (54) with the distribution \(\mu_{t}\) of density \(f_{t}\), which is reached by evolving \(f_{0}\) for a time \(t\):  
\[\mu_{t}(S^{t}E)=\mu_{0}(E)\quad\text{i.e.,}\quad\int_{S^{t}E}f_{t}(X)dX=\int_{ E}f_{0}(X)dX\]  
\[\overline{\mathcal{O}}_{0,\tau}\mid_{(-A)_{\delta}}=iS^{\tau}\,\overline{ \mathcal{O}}_{0,\tau}\mid_{(A)_{\delta}}. \tag{47}\]0  
By combining Equations (54) and (55), we get the following relation:  
\[\overline{\mathcal{O}}_{0,\tau}\mid_{(-A)_{\delta}}=iS^{\tau}\,\overline{ \mathcal{O}}_{0,\tau}\mid_{(A)_{\delta}}. \tag{47}\]1  
which, in the case \(\mathcal{O}=\Omega^{(0)}\), eventually leads to:  
\[\overline{\mathcal{O}}_{0,\tau}\mid_{(-A)_{\delta}}=iS^{\tau}\,\overline{ \mathcal{O}}_{0,\tau}\mid_{(A)_{\delta}}. \tag{47}\]2 \[\overline{\mathcal{O}}_{0,\tau}\mid_{(-A)_{\delta}}=iS^{\tau}\,\overline{ \mathcal{O}}_{0,\tau}\mid_{(A)_{\delta}}. \tag{47}\]3  
where only part of the integral on the right-hand side is constrained. This relation is exact and holds for all values of \(t\), \(\tau\) and all pairs of \(A\) and \(-A\). Moreover, the probability measures are no longer computed with respect to the initial density \(f_{0}\), like in the previous transient FR, but they refer to the density \(f_{t}\) that may eventually approach in time the steady-state distribution. The existence of a steady-state FR implies that the expression (58) tends to a given value as \(t\to\infty\), while the ratio of probabilities in expression (57) does. Assuming that such a steady-state FR holds, we may now investigate which conditions are necessarily verified.