# Broken versus Non-Broken Time Reversal Symmetry: Irreversibility and Response
## 4 The Dissipation Function
### Time-Dependent Perturbations
In particular, observing that \(\theta\) and \(\phi\) do not depend on \(\Gamma\) and obey the TRI Equation (28) with an involution operator \(\hat{i}\) such that \(\hat{i}(\theta,\phi)=(\theta,-\phi)\), we may define \(\tilde{i}:\tilde{\mathcal{M}}\rightarrow\tilde{\mathcal{M}}\) as:  
\[\tilde{i}\Gamma=\tilde{i} \begin{pmatrix}\Gamma\\ \theta\\ \phi\end{pmatrix} = \begin{pmatrix}i\Gamma\\ \theta\\ -\phi\end{pmatrix} . \tag{36}\]  
To understand the conditions under which TRI holds in the perturbed case (30), observe that \(p(\theta,\phi)\) can be divided in four parts: the vector \(p_{q}(\theta,\phi)\) perturbing the evolution of the vector of generalized coordinates, the vector \(p_{p}(\theta,\phi)\) perturbing the evolution of the vector of momenta, and two zeros in the position of the variables \(\theta\) and \(\phi\), that are subjected to no perturbation. Then, TRI requires the existence of an involution \(\tilde{i}\) such that:  
\[\tilde{i}p(\theta,\phi)=\tilde{i} \begin{pmatrix}p_{q}(\theta,\phi)\\ p_{p}(\theta,\phi)\\ 0\\ 0\end{pmatrix} = \begin{pmatrix}-p_{q}(\theta,\phi)\\ p_{p}(\theta,\phi)\\ 0\\ 0\end{pmatrix} . \tag{37}\]  
In the case of Equation (31), it suffices instead to have an even function \(p\):  
\[p(\theta,-\phi)=p(\theta,\phi). \tag{38}\]This approach may be extended to more general time dependent perturbations, since any perturbation can be thought to be periodic of period \(T\), if \(T\) is conveniently larger than the observation times [28]. For instance, any periodic function \(p\in\mathcal{L}^{2}[0,T]\) can be expanded in Fourier components:  
\[p(t)=\sum_{n=-\infty}^{\infty}V_{n}e^{i\omega_{n}t}, \tag{39}\]  
with \(\omega_{n}=2\pi n/T\), and one may eliminate \(t\) introducing the representation:  
\[p(t)=p(\theta,\phi)=\sum_{n=-\infty}^{\infty}V_{n}p_{n}(\theta,\phi), \tag{40}\]  
where with a slight abuse of notation, \(\theta\) and \(\phi\) replace \(t\) and obey:  
\[ \begin{cases}\dot{\theta}=\phi,\\ \dot{\phi}=-\omega^{2}\theta,\end{cases} \quad\text{with}\quad\omega^{2}=\left( \frac{2\pi}{T}\right)^{2}, \tag{41}\]  
while the \(n\)-th Fourier basis element has been replaced by:  
\[p_{n}=\left(\theta-i\frac{\phi}{\omega}\right)^{n}\,,\qquad n\in\mathbb{Z}. \tag{42}\]  
Then, the equations of motion in the extended phase space \(\tilde{\mathcal{M}}\) can be written as:  
\[\left( \begin{array}{c}\dot{\Gamma}_{1}\\ \dot{\Gamma}_{2}\\ \dot{\Gamma}_{2}\\ \dot{\Gamma}_{2N}\\ \dot{\theta}\\ \dot{\phi}\end{array} \right)=\left( \begin{array}{c}G(\Gamma_{1})+\sum_{n=- \infty}^{\infty}V_{n1}\left(\theta-i\frac{\phi}{\omega}\right)^{n}\\ G(\Gamma_{2})+\sum_{n=-\infty}^{\infty}V_{n2}\left(\theta-i\frac{\phi}{\omega} \right)^{n}\\ \vdots\\ G(\Gamma_{2N})+\sum_{n=-\infty}^{\infty}V_{n2N}\left(\theta-i\frac{\phi}{ \omega}\right)^{n}\\ \phi\\ -\omega^{2}\theta\end{array} \right), \tag{43}\]  
while the dissipation function takes the form:  
\[\dot{\Omega}^{f_{0}}(\Gamma,\theta,\phi)=-\Lambda_{old}(\Gamma) -\sum_{k=0}^{2N}G_{k}(\Gamma)\frac{\partial lnf_{0}(\Gamma)}{\partial\Gamma_ {k}}-\sum_{k=0}^{2N}\sum_{n=-\infty}^{\infty}V_{nk}\left(\theta-i\frac{\phi} {\omega}\right)^{n}\frac{\partial lnf_{0}(\Gamma)}{\partial\Gamma_{k}}, \tag{44}\] \[=\underbrace{\Omega_{old}^{f_{0}}(\Gamma)}_{\text{old dissipation function}}-\underbrace{\sum_{k=0}^{2N}\sum_{n=-\infty}^{\infty}V_{nk}\left(\theta-i\frac{\phi}{ \omega}\right)^{n}\frac{\partial lnf_{0}(\Gamma)}{\partial\Gamma_{k}}}_{\text {additional term}}. \tag{45}\]  
This expression is more general than Equation (34) since it holds for all \(\mathcal{L}^{2}[0,T]\) perturbations given any observation interval \([0,T]\). It appears that, in general, TRI may be violated because of the time dependent perturbation \(p\). If that is the case, some of the results strictly related to TRI may fail, but a number of other results implied by the existence of a dissipation function still hold. Indeed, even the FR continues to hold in various cases in which TRI is absent, cf. Ref. [29].