# Broken versus Non-Broken Time Reversal Symmetry: Irreversibility and Response
## 4 The Dissipation Function
### Time-Dependent Perturbations  
Consider a time dependent perturbation of the equations of motion (14) that leads to non-autonomous equations such as:  
\[\dot{\Gamma}=\hat{G}(\Gamma,t)=G(\Gamma)+p(t)\quad\text{or}\quad\dot{\Gamma}= \hat{G}(\Gamma,t)=p(t)G(\Gamma). \tag{27}\]  
On the left hand side, \(p\) is a vector field of same dimension as \(\mathcal{M}\). On the right hand side, \(p\) is a scalar function. If the perturbation is periodic, we may restore the autonomous character of the dynamics introducing new variables \(\phi\) and \(\theta\), that evolve as:  
\[ \begin{cases}\dot{\theta}=\phi,\\ \dot{\phi}=-\theta,\end{cases}  \tag{28}\]  
and considering \(p\) as function of \(\theta\) and \(\phi\). Let \(\dot{\Gamma}=(\mathbf{q}_{1},...,\mathbf{q}_{N},\mathbf{p}_{1},...,\mathbf{p} _{N},\theta,\phi)=(\Gamma,\theta,\phi)\) be one point in the _augmented_ phase space \(\tilde{\mathcal{M}}\). We denote by a tilde the counterparts in \(\tilde{\mathcal{M}}\) of the quantities previously considered in \(\mathcal{M}\):  
\[\tilde{f}_{0}=\tilde{f}_{0}(\Gamma,\theta,\phi)\qquad\tilde{\Lambda}=\tilde{ \Lambda}(\Gamma,\theta,\phi)\qquad\tilde{\Omega}^{f_{0}}=\tilde{\Omega}^{f_{ 0}}(\Gamma,\theta,\phi). \tag{29}\]  
Then, our perturbation lives on an ellipse, determined by the initial conditions, and the dynamics in \(\tilde{\mathcal{M}}\) can be expressed by:  
\[\dot{\Gamma}=\tilde{G}(\Gamma,\theta,\phi)=\left( \begin{array}{c}G(\Gamma)+ p(\theta,\phi)\\ \phi\\ -\theta\end{array} \right), \tag{30}\]  
or  
\[\dot{\Gamma}=\tilde{G}(\Gamma,\theta,\phi)=\left( \begin{array}{c}p(\theta, \phi)G(\Gamma)\\ \phi\\ -\theta\end{array} \right), \tag{31}\]  
where \(p\) is a given function of \(\theta\) and \(\phi\). We may endow the ellipse of \(\theta\) and \(\phi\) with a uniform probability distribution. Then, the probability density function \(\tilde{f}_{0}\) in the augmented phase space takes the form:  
\[\tilde{f}_{0}(\Gamma,\theta,\phi)=\frac{1}{\sigma}f_{0}(\Gamma), \tag{32}\]where \(\sigma\) is the length of the one-dimensional torus. Then, in the case of Equation (30), the new dissipation function takes the form:  
\[ \begin{split}\tilde{\Omega}^{f_{0}}&(\Gamma,\theta, \phi)=-\tilde{\Lambda}(\Gamma,\theta,\phi)-\frac{d}{dt}\ln\tilde{f}_{0}(\Gamma,\theta,\phi)\\ &=-\tilde{\Lambda}(\Gamma,\theta,\phi)-\frac{1}{\tilde{f}_{0}( \Gamma,\theta,\phi)}\left[\big{(}G(\Gamma)+p(\theta,\phi)\big{)}\frac{\partial \tilde{f}_{0}(\Gamma,\theta,\phi)}{\partial\Gamma}+\phi\frac{\partial\tilde{f} _{0}(\Gamma,\theta,\phi)}{\partial\theta}-\theta\frac{\partial\tilde{f}_{0}( \Gamma,\theta,\phi)}{\partial\phi}\right]\\ &=-\Lambda_{old}(\Gamma)-\frac{1}{f_{0}(\Gamma)}\big{(}G(\Gamma)+ p(\theta,\phi)\big{)}\frac{\partial f_{0}(\Gamma)}{\partial\Gamma}\end{split}  \tag{33}\]  
because \(p\) does not depend on \(\Gamma\) and \(\tilde{f}_{0}\) does not depend on \(\theta\) and \(\phi\). Then:  
\[\tilde{\Omega}^{f_{0}}(\Gamma,\theta,\phi)=\underbrace{\Omega^{f_{0}}_{old}( \Gamma)}_{\text{old dissipation function}}-\underbrace{p(\theta,\phi)\frac{\partial\ln f_{0}}{ \partial\Gamma}(\Gamma)}_{\text{additional term}}. \tag{34}\]  
In the case of Equation (31), we find:  
\[\tilde{\Omega}^{(0)}(\Gamma,\theta,\phi)=p(\theta,\phi)\Big{(}-\Lambda_{old} (\Gamma)-G(\Gamma)\frac{\partial\ln f_{0}(\Gamma)}{\partial\Gamma}\Big{)}= \underbrace{p(\theta,\phi)}_{\text{additional term}}\tilde{\Omega}^{(0)}_{ old}(\Gamma). \tag{35}\]  
The above shows how the dissipation function can be defined in the presence of time dependent perturbations, and how it differs from the unperturbed case. In particular, this is done eliminating the explicit dependence on time, that makes problematic both the notion of TRI and the physical interpretation of the statistical quantities. Once the dissipation function has been re-introduced in such a way that TRI holds in the extended phase space, the results previously obtained for autonomous systems can be transferred to the time dependent case, simply using the modified dissipation function.