# Fluctuation Relation for the Dissipative Flux: The Role of Dynamics, Correlations and Heat Baths
## Appendix B Time-Reversal Invariance  
Time-independent Hamiltonian enjoys a symmetry that make the associated equations of motion time-reversal invariance. By this, one commonly means that the transformation of coordinates \(\mathbf{q}=(x,y,z)\) and momenta \(\mathbf{p}=(p_{x},p_{y},p_{z})\)  
\[\mathbf{q}\rightarrow\mathbf{q}\,,\;\mathbf{p}\rightarrow-\mathbf{p} \tag{15}\]  
preserves the Hamiltonian, and the transformation  
\[t\rightarrow-t\,,\;\mathbf{p}\rightarrow-\mathbf{p} \tag{16}\]  
preserves the equations of motion. For instance, consider the classical mechanics case for \(N\) particles:  
\[\mathcal{H}=\sum_{i=1}^{N}\frac{\mathbf{p}_{i}^{2}}{2m_{i}}+\frac{1}{2}\sum_{i \
eq j}^{1,N}U(|\mathbf{r}_{ij}|)\,,\quad\text{with}\;\;\mathbf{r}_{ij}= \mathbf{q}_{j}-\mathbf{q}_{i}, \tag{17}\]which yields the following equations of motion:  
\[\dot{x}_{i} =\frac{p_{i_{X}}}{m_{i}} \dot{p}_{i_{X}} =F_{i_{X}}\] \[\dot{y}_{i} =\frac{p_{i_{Y}}}{m_{i}} \dot{p}_{i_{Y}} =F_{i_{Y}} \text{with}\quad i=1,\dots,N \tag{111}\] \[\dot{z}_{i} =\frac{p_{i_{Z}}}{m_{i}} \dot{p}_{i_{Z}} =F_{i_{Z}}\]  
where  
\[F_{i_{X}}^{ij}=-\frac{\partial U(|\mathbf{r}_{ij}|)}{\partial x_{i}}=\frac{x_ {i}-x_{j}}{r_{ij}}\,\frac{\partial U(r_{ij})}{\partial r_{ij}}\,\quad F_{i_{X}}=\sum_{j\
eq i}F_{i_{X}}^{ij}. \tag{112}\]  
One immediately realizes that the transformation \((\mathbf{q}_{1},\dots\mathbf{q}_{N},\mathbf{p}_{1},\dots,\mathbf{p}_{N},t) \mapsto(\mathbf{q}_{1},\dots\mathbf{q}_{N},-\mathbf{p}_{1},\dots,-\mathbf{p}_ {N},-t)\) leaves these equations of motion invariant. Then, let us denote by \(\Gamma=(\mathbf{q}_{1},\dots\mathbf{q}_{N},\mathbf{p}_{1},\dots,\mathbf{p}_ {N})\in\mathcal{M}\) the phase space points, by \(i\) the mapping that preserves the coordinates and inverts the momenta:  
\[i(\mathbf{q}_{1},\dots\mathbf{q}_{N},\mathbf{p}_{1},\dots,\mathbf{p}_{N})=( \mathbf{q}_{1},\dots\mathbf{q}_{N},-\mathbf{p}_{1},\dots,-\mathbf{p}_{N}) \tag{113}\]  
and by \(S^{t}\Gamma\) the solution of the equations of motion at time \(t\), with initial condition \(\Gamma\). One realizes that the double application of \(i\) is the identity, \(\dot{i}^{2}\Gamma=\Gamma\), and that \(iS^{t}\Gamma=S^{-t}i\Gamma\) for all times \(t\) and all phases \(\Gamma\). For instance, consider a harmonic oscillator, i.e., one particle tethered to a very massive wall that does not move. Its equations of motion are \(\dot{q}=p\,,\quad\dot{p}=-q\), which yields \(S^{t}\Gamma=S^{t}(q,p)=(q\cos t+p\sin t,-q\sin t+p\cos t)\). Applying the transformation \(i\) yields:  
\[iS^{t}\Gamma=(q\cos t+p\sin t,q\sin t-p\cos t). \tag{114}\]  
At the same time, the solution of the equations of motion at time \(-t\), with initial condition \(i\Gamma=(q,-p)\), is given by:  
\[S^{-t}i\Gamma = (q\cos(-t)+(-p)\sin(-t),-q\sin(-t)-p\cos(-t)) \tag{115}\] \[= (q\cos t+p\sin t,q\sin t-p\cos t),\]  
which equals \(iS^{t}\Gamma\). It also follows that \(S^{-t}\Gamma=S^{-t}i(i\Gamma)=iS^{t}(i\Gamma)\), which is why \(i\) is called the time reversal operator. Then, one can speak of time-reversal invariance of the dynamics \(S^{t}\) whenever there exists one mapping \(i\) whose square is the identity, and \(iS^{t}\Gamma=S^{-t}i\Gamma\) holds.