# Broken versus Non-Broken Time Reversal Symmetry: Irreversibility and Response
## 3 Time Reversible Dissipative Dynamics  
As noted above, the picture of relaxation to equilibrium based on the concept of _typicality_ is justified for Hamiltonian systems, made of weakly interacting microscopic constituents. Being subjected to limitations is of course a general aspect of all theories, since they are developed having in mind certain phenomena or just aspects of a given phenomenon, not others. In physics, aspects related to space and time scales are particularly relevant, since they strongly affect the empirical observations [14]. In particular, the case of irreversibility is based on the empirical observation that objects observed on the space-time scales of our daily experience follow a well established arrow of time. Unfortunately, many different microscopic hypotheses imply some form of macroscopic irreversibility, hence they are consistent with this minimal observation, making it hard to identify the correct microscopic description or the most appropriate for what happens in nature. Let us consider NEMD models. Their original purpose is to compute the transport coefficients of fluids in nonequilibrium conditions, particularly in nonequilibrium steady states [13]. The goal is achieved describing in purely mechanical terms internal forces, external driving forces and the various kinds of couplings with the environment. In particular, artificial forces are introduced in the dynamical equations of the particles constituting the system, in order to account for thermostats, barostats, etc. The dynamics remain TRI, but they turn dissipative as explained below [15]. To clarify these facts, consider, for example, a copper wire under an electric potential difference: electrons are forced to move, on average, in a certain direction by the external field, resulting in a net electric current. The dissipative field acts like a constraint on the dynamics that reduces the region of phase space available to the electrons. Indeed, the corresponding non-equilibrium steady state, which is reached because the energy given by the driving forces is eventually dissipated as heat, passes from being an extremely rare statistical fluctuation of an equilibrium state (like in shot noise) to becoming the most likely state. In other words, while in equilibrium, the phase space trajectory spends almost all of its time in the vastly dominant region of vanishing current, the presence of external driving and dissipation constrain the trajectories within the set of positive current, which is a tiny part of the phase space. The system is no longer free to explore the entire phase space and \"counting\" states, to identify those in which the system spends most of its time, become pointless. Although nonequilibrium steady states live within tiny fractions of the totality of states, the corresponding trajectories are the relevant ones. At the same time, if the microscopic dynamics are TRI in the absence of an electric field, there is no reason to believe that this symmetry is broken by, e.g., an electric field. This kind of behaviour is captured by the standard models of NEMD, in which a reduced mobility due to phase space contraction is associated with dissipation without breaking the microscopic time symmetry, although at times this symmetry takes unexpected forms. Dissipation and TRI lead to one of the most interesting results in nonequilibrium statistical physics, which is known as steady-state Fluctuation Relation (FR). This relation was first derived by [PERSON], [PERSON] and [PERSON] in 1993 [16], for a model called SLLOD, i.e., the constant-energy sheared \(N\)-particles system following these equations of motion:  
\[\left\{ \begin{array}{l}\dot{\mathbf{q}}_{k}=\mathbf{p}_{k}/m+\mathbf{n}_{x} \gamma y_{k},\\ \dot{\mathbf{p}}_{k}=\mathbf{F}_{k}-\mathbf{n}_{x}\gamma p_{y_{k}}-\alpha \mathbf{p}_{k},\end{array}