# Broken versus Non-Broken Time Reversal Symmetry: Irreversibility and Response
## 4 The Dissipation Function
### Fluctuation Relations: Symmetry in Dissipative Dynamics  
While the main purpose of the dissipation function is to describe the response to perturbations, and the relaxation of dynamical systems, it has been introduced to express one symmetry of TRI dissipative dynamics: the FR. Let \(\mathcal{O}\) be a physical observable that is odd with respect to the time reversal operator \(i\), \(\mathcal{O}(ix)=-\mathcal{O}(x)\). For \(\delta>0\), introduce the interval \((A)_{\delta}=(A-\delta,A+\delta)\), and for a system with initial probability density \(f_{0}\), consider the probability that the average of \(\mathcal{O}\) in the time interval \((0,\tau)\), \(\overline{\mathcal{O}}_{0,\tau}\) say, belongs to \((A)_{\delta}\). By definition, this probability is given by \(\int_{\overline{\mathcal{O}}_{0,\tau}\mid_{(A)_{\delta}}}f_{0}(\Gamma)d\Gamma\), where \(\overline{\mathcal{O}}_{0,\tau}\mid_{(A)_{\delta}}\) is the set of phase points for which \(\overline{\mathcal{O}}_{0,\tau}\) lies in \((A)_{\delta}\). Then, we have:  
\[\frac{\mu_{0}(\overline{\mathcal{O}}_{0,\tau}\mid_{(-A)_{\delta}})}{\mu_{0}( \overline{\mathcal{O}}_{0,\tau}\mid_{(A)_{\delta}})}=\frac{\int_{\overline{ \mathcal{O}}_{0,\tau}\mid_{(-A)_{\delta}}}f_{0}(\Gamma)d\Gamma}{\int_{ \overline{\mathcal{O}}_{0,\tau}\mid_{(A)_{\delta}}}f_{0}(\Gamma)d\Gamma}\, \tag{46}\]  
which can be computed observing that  
\[\overline{\mathcal{O}}_{0,\tau}\mid_{(-A)_{\delta}}=iS^{\tau}\,\overline{ \mathcal{O}}_{0,\tau}\mid_{(A)_{\delta}}. \tag{47}\]  
Hence, introducing the coordinate transformation \(\Gamma=iS^{\tau}x\), whose Jacobian is \(\left|\frac{\partial\Gamma}{\partial x}\right|=\exp\left\{\Lambda_{0,\tau}(x)\right\}\), cf. Ref. [23] Equation (46) becomes:  
**Transient \(\mathcal{O}\)-FR:**  
\[\frac{\mu_{0}(\overline{\mathcal{O}}_{0,\tau}\mid_{(-A)_{\delta}})}{\mu_{0}( \overline{\mathcal{O}}_{0,\tau}\mid_{(A)_{\delta}})}=\frac{\int_{\overline{ \mathcal{O}}_{0,\tau}\mid_{(A)_{\delta}}}f_{0}(S^{\tau}X)e^{\Lambda_{0,\tau}(X )}\mathrm{d}X}{\int_{\overline{\mathcal{O}}_{0,\tau}\mid_{(A)_{\delta}}}f_{0} (\Gamma)\mathrm{d}\Gamma}:=\left\langle e^{-\Omega_{0,\tau}^{(0)}}\right\rangle _{\overline{\mathcal{O}}_{0,\tau}\in(A)_{\delta}}^{(0)}\, \tag{48}\]  
where the last average is computed with respect to \(\mu_{0}\) over the set of initial conditions such that \(\overline{\mathcal{O}}_{0,\tau}(\Gamma)\in(A)_{\delta}\), and we use the following definitions for the dissipation function:  
\[\overline{\Omega}_{t,t+\tau}^{(0)}(\Gamma){=}\frac{1}{\tau}\,\Omega_{t,t+\tau }^{(0)}=\frac{1}{\tau}\ln\frac{f_{0}(S^{t}\Gamma)}{f_{0}(S^{t+\Gamma})}- \overline{\Lambda}_{t,t+\tau}(\Gamma), \tag{49}\]  
and we have assumed \(f_{0}\) to be even under \(i\), which means that \(f_{0}\) could be an equilibrium probability density. At the same time, the phase space expansion rate \(\Lambda\) is odd with respect to \(i\). If \(\mathcal{O}=\Omega^{(0)}\), we obtain:  
**Transient \(\Omega\)-FR:**  
\[\frac{\mu_{0}(\overline{\mathcal{O}}_{0,\tau}^{(0)}\mid_{(-A)_{\delta}})}{\mu_ {0}(\overline{\Omega}_{0,\tau}^{(0)}\mid_{(A)_{\delta}})}=\left\langle\exp \left(-\Omega_{0,\tau}^{(0)}\right)\right\rangle_{\overline{\Omega}_{0,\tau}^ {(0)}\in(A)_{\delta}}^{(0)}=e^{-[A+e(\delta,A,\tau)]\tau}\, \tag{50}\]  
where the magnitude of \(e(\delta,A,\tau)\) is at most \(\delta\), because the average in Equation (50) is computed over the initial conditions, which lead to a value for \(\overline{\Omega}_{0,\tau}^{(0)}\) within a distance \(\delta\) from \(A\). Remarkably, the only conditions for these results are that the system starts from a state defined by an even \(f_{0}\), and that its dynamics obeys TRI. These transient relations describe the statistics of repeated measurements of the observable \(\mathcal{O}\) in a time \(\tau\), starting in the same equilibrium state followed by some work for a time \(\tau\). This allows us to gain information on the initial equilibrium state, thanks to some nonequilibrium process.