## 3 Discussion and Conclusions  
In accord with our observations, the validity of the steady state \(\Omega\)-FR,  
\[\frac{1}{\tau}\mathrm{ln}\frac{\mathrm{Prob}_{t}\left(\beta \mathbf{\bar{J}}_{0,\tau}\cdot\mathbf{F}_{e}V\big{|}_{A_{\delta}^{+}}\right)}{ \mathrm{Prob}_{t}\left(\beta\mathbf{\bar{J}}_{0,\tau}\cdot\mathbf{F}_{e}V \big{|}_{A_{\delta}^{-}}\right)} \tag{23}\] \[= A+O\left(\frac{\tau_{M}}{\tau}\right)+\varepsilon\left(A,t,\tau, \delta\right),\quad|\varepsilon|\leq\delta\]  
can be attributed to the following conditions. Firstly we assume the system is time reversal invariant, ergodically consistent, and the initial distribution is even under time reversal. Then, the main technical assumptions are that the system is reversible and T-mixing. This implies that at a fixed number of standard deviations from the mean, the distribution of the dissipative flux \(\mathbf{\bar{J}}_{0,\tau}\) must be Gaussian and furthermore typical average values for the time averaged flux scale as \(A\approx O(F_{e})\pm O(\tau^{-1/2})\) near the mean for weak fields. For a fixed number of samples, the easily observable range of fields goes to zero as \(\tau^{-1/2}\). However, although the easily observable range of values shrinks to zero as \(\tau\rightarrow\infty\), the error induced by truncating the correlation function \(O(\tau_{M}/\tau)\) vanishes even faster, ensuring that the steady state \(\Omega\)-FR  
\[\lim_{\tau/\tau_{M}\rightarrow\infty}\frac{1}{\tau}\ln\frac{ \mathrm{Prob}_{t}\left(\left.\beta\mathbf{\bar{J}}_{0,\tau}\cdot\mathbf{F}_{e} V\right|_{A_{\delta}^{+}}\right)}{\mathrm{Prob}_{t}\left(\left.\beta \mathbf{\bar{J}}_{0,\tau}\cdot\mathbf{F}_{e}V\right|_{A_{\delta}^{-}}\right)} \tag{24}\] \[= A+\varepsilon\left(A,t,\tau,\delta\right),\quad\left|\varepsilon \right|\leq\delta\]  
can be easily verified to arbitrary accuracy. By going to larger \(\tau\), we can observe the steady state \(\Omega\)-FR to arbitrarily high accuracy by taking ever more samples. Of course if the correlation time \(\tau_{M}\) is zero, the asymptotic limit is unnecessary since \(\tau/\tau_{M}=\infty,\ \forall\tau\) and (24) is exact for all values of \(\tau\). The lack of bounds on the range of values for the argument is highly significant. It means that provided the system is T-mixing, the steady state \(\Omega\)-FR is valid and verifiable outside the region of Gaussian statistics near the mean, and actually describes a symmetry of rare events in nonequilibrium steady states. This goes to the root of the fundamental difference between the second law of thermodynamics and the fluctuation theorems. The second law says that certain probabilities are actually zero. In contradistinction, the fluctuation theorems say that certain probabilities may be exceedingly small--unobservably small for macroscopic bodies over accessible timescales--but they precisely quantify these probabilities. Thus the transient \(\Omega\)-FR and the steady state \(\Omega\)-FR for T-mixing systems are valid for all values of their argument that can be generated by trajectories originating in the ostensible phase space. Because of ergodic consistency, regardless of how strong or weak the field is, we can prove that the steady state \(\Omega\)-FR for T-mixing systems will eventually be satisfied. The asymptotic time required for the convergence of the steady state \(\Omega\)-FR does not scale with the applied field. T-mixing is to be contrasted with the mixing property of stationary measures for dynamical systems. T-mixing does not consider stationary averages but rather concerns itself with transient averages. Our approach is novel and, among other issues, identifies the physically relevant time scales (the decorrelation times with respect to \(f\left(\mathbf{\Gamma};0\right)\)). If, on the other hand, the system is not T-mixing and a steady state is not reached, the transient relations, as well as Equation (19), still hold for ensembles of systems at all finite times.