# Broken versus Non-Broken Time Reversal Symmetry: Irreversibility and Response
## 5 Conclusions  
In this contribution we have outlined a physical interpretation of the problem of macroscopic irreversibility, the temporal symmetry breaking of our daily life space and time scales, for relaxation of TRI dissipative systems, such as those of NEMD.  
In the case of conservative dynamics, the standard approach is represented by [PERSON]'s argument. This argument rests on the notion of _typicality_, which implies that a system starting in a non-equilibrium state with dynamics exploring the phase space without constraints irreversibly enters its proper equilibrium state. In our view, one difficulty of this approach is that it subordinates the irreversibility that we experience on our space and time scales to the evolution of the whole universe and its initial conditions: the \"past hypothesis\". Although the argument is fascinating and quite compelling within a given rational framework, it embeds in a single theory such remote scales as the microscopic, the macroscopic (or thermodynamic), and the cosmological scales. The purpose of explanation may therefore be better served by an approach that appears less fundamental but avoids numerous steps not amenable to experimental tests.  
Other recent studies [30] have shown that, at least for simple Newtonian \(N-\)body systems, a gravitational arrow of time seems to naturally arise out of simple physical laws without the need for special initial conditions, in contrast with the standard [PERSON]'s view. The approach of [PERSON]. [30] has been, however, criticized [31]. Former approaches, such as [PERSON]'s recourse to microscopically irreversible dynamics, also appear unnecessary, as we have no evidence of this lack of reversibility at the atomic/molecular scale, despite violations of the time parity being observed in high energy physics.  
Our approach is based on the theory of TRI dissipative dynamical systems, which successfully describe transport properties of nonequilibrium systems, focusing on the molecular level of description. Indeed, one may argue that dissipation phenomena fit human space and time scales much better than the universe as a whole does. The key hypothesis of this model is the microscopic _time reversal invariance_ of the dynamics.  
The technical tool that we have used to achieve our result is the dissipation function:  
\[\Omega^{(0)}=-G\cdot\partial_{\Gamma}\ln f^{(0)}-\Lambda,\]  
which we considered as a nonequilibrium thermodynamic potential, driving the relaxation of an ensemble of systems arbitrarly far from equilibrium. We extended the corresponding formalism to the case of time-dependent perturbations acting on the system. Furthermore, we have illustrated how weak t-mixing turns ensemble results into single system ones. Under the assumption of the existence of time averages for \(\mu_{0}\)--almost every \(\Gamma\in\mathcal{M}\)--we have thus extended the \"counting\" argument even to TRI dissipative systems.  
For a single system, arbitrarly far from equilibrium, we have proved that the condition to observe irreversible relaxation to a nonequilibrium steady state is the decay of auto-correlation of the dissipation function.  
Loss of memory again is connected with the presence of different time scales, i.e., the fact that any measurement of a system takes a certain characteristic time to be made, which is very long comparedto the microscopic characteristic times. For future investigations, it will be important to understand the role played by the system size \(N\) in the t-mixing conditions.  
**Acknowledgments:** The authors are grateful to [PERSON] and [PERSON] for enlightening discussions.  
**Author Contributions:** The contributions of the two authors are half and half on all parts of this paper.  
**Conflicts of Interest:** The authors declare no conflict of interest. The founding sponsors had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, and in the decision to publish the results.