# Broken versus Non-Broken Time Reversal Symmetry: Irreversibility and Response
## 3 Time Reversible Dissipative Dynamics
\right.\qquad\alpha=\frac{-\gamma P_{xy}V}{\sum_{k=1 }^{N}\mathbf{p}_{k}^{2}/m}\,\qquad i=1,...,N. \tag{8}\]Here, \(\gamma\) is the shear rate, \(\mathbf{n}_{x}\) is the unit vector along the \(x\) direction, and \(\mathbf{F}_{i}\) is the force exerted by all other particles on particle \(i\), and \(\alpha\) is the ergostatting term, obtained from [PERSON] principle of least constraint, in order to keep the internal energy constant. Equation (8) are TRI, in the sense that there exists a symmetry operator \(i\) acting on the phase space \(\mathcal{M}\) such that \(i^{2}\,\Gamma=i\circ i\,\Gamma=\Gamma\) for all \(\Gamma\in\mathcal{M}\) and that, combined with the inversion of time \(t\mapsto-t\), leaves Equation (8) invariant. In other words, replacing \(\Gamma\) with \(i\Gamma\) and \(t\) with \(-t\), Equation (8) do not change. Introducing the time evolution operator, \(S^{t}:\mathcal{M}\rightarrow\mathcal{M}\), meaning that \(S^{t}\Gamma\) is the point at time \(t\) along a phase space trajectory starting at \((\mathbf{q}_{1},...,\mathbf{p}_{N})=\Gamma\), TRI also amounts to state that  
\[S^{t}i\,\Gamma=iS^{-t}\,\Gamma\,,\qquad\text{for all points }\Gamma\in \mathcal{M}\;\text{ and all times }t\in\mathcal{R}. \tag{9}\]  
This property allows us to identify pairs of phase space trajectory segments enjoying opposite properties. Indeed, if TRI is preserved, a phase space trajectory with positive current and positive dissipation for a time \(t\) can be associated with another trajectory with opposite current and opposite dissipation within the same time \(t\). For a monatomic gas in equilibrium, one possible symmetry \(i\) is the usual operator \(i(\mathbf{q},\mathbf{p})=(\mathbf{q},-\mathbf{p})\), but it easy to check that this does not work for Equation (8). The SLLOD model is however invariant under the following operation:  
\[i(x,y,z,p^{x},p^{y},p^{z})=(x,-y,z,-p^{x},p^{y},-p^{z}). \tag{10}\]  
This means that, if the phase space trajectory \(\ell\) starting in \(\Gamma\in\mathcal{M}\) produces positive dissipation in a time \(t\), then the trajectory \(\ell^{*}\) starting in \(\Gamma^{*}=iS^{t}\Gamma\) will produce the opposite dissipation in the same time \(t\). One may then ask how these two events are related to each other in a steady state. The answer given in Ref. [16] is the following:  
\[\frac{\mu_{\ell}}{\mu_{\ell^{*}}}=\exp\left[Ndt\overline{\alpha}_{\ell,t} \right], \tag{11}\]  
where \(\mu_{\ell}\) and \(\mu_{\ell^{*}}\) are the steady state probabilities of \(\ell\) and \(\ell^{*}\), respectively, \(d\) is the dimension of space, and \(\overline{\alpha}_{\ell,t}\) is the time average of \(\alpha\) along segment \(\ell\). Equation (11) is parameter-free and exact in the long \(t\) limit for quite general systems. Although it is a direct consequence of the time reversibility of Equation (8), it constitutes a tool to characterize the temporal symmetry breaking that must take place in the macroscopic limit, as sanctioned by the second law of thermodynamics. The reason is that the argument of the exponential contains \(N\), which is very large for a macroscopic object, as well as the observation time \(t\), which is also very large in terms of microscopic time units. As a consequence, negative dissipations, possible in a reversible framework, have a totally negligible probability for macroscopic objects. Letting \(N\) grow without bounds, one approaches the situation in which one value of the entropy production has probability \(1\), and all others have vanishing probability, as irreversible thermodynamics requires. In other words, Equation (8) are TRI for any \(N\), and Equation (11) states that measurements lasting a finite time \(t\) may yield negative dissipations. At the same time, Equation (11) implies that the macroscopic limit \(N_{t}\rightarrow\infty\) makes unobservable such events, i.e. that the temporal symmetry is broken by the macroscopic limit for the macroscopic observables.