## 2 Calculations
If correlations do not decay instantaneously with respect to \(f\left(\mathbf{\Gamma};0\right)\), but vanish to some arbitrarily fine tolerance, for times greater than some cut-off time, \(c\tau_{M}\) (some multiple of the Maxwell time, \(\tau_{M}\), which depends on the tolerance required), one has \(\lim_{t\rightarrow\infty}\left(\Xi(t,\tau;A_{\delta}^{+})\right)=\mathrm{ const}=O\left(e^{\tau_{M}}\right)\)[14]. Then the steady state FR is equally obtained in the \(\tau\rightarrow\infty\) limit, but with corrections of order \(O(\tau_{M}/\tau)\), which also requires the averaging time \(\tau\) to be large. By expanding the exponential terms and noting \(\int_{s}^{t}\Omega(S^{u}\mathbf{\Gamma})du=\int_{0}^{t-s}\Omega(S^{s+z} \mathbf{\Gamma})dz\), it can be seen that the decay in correlation of the two exponential terms is a special instance of the condition,  
\[\Omega_{0,2t+\tau}\left(\mathbf{\Gamma}\right)\equiv\ln\left[\frac{f\left( \mathbf{\Gamma};0\right)}{f\left(S^{2t+\tau}\mathbf{\Gamma};0\right)}\right]- \int_{0}^{2t+\tau}\Lambda\left(S^{s}\mathbf{\Gamma}\right)ds \tag{9}\]5  
now known as T-mixing (first defined in [17]). This guarantees the time integrability of the transient time correlations function \(\left\langle\varphi(t)\Omega(0)\right\rangle\). Together with the dissipation theorem [14, 19], which states that beyond some correlation time \(t_{c}\) phase variables become constant and are given by \[\Omega_{0,2t+\tau}\left(\mathbf{\Gamma}\right)\equiv\ln\left[\frac{f\left( \mathbf{\Gamma};0\right)}{f\left(S^{2t+\tau}\mathbf{\Gamma};0\right)}\right]- \int_{0}^{2t+\tau}\Lambda\left(S^{s}\mathbf{\Gamma}\right)ds \tag{9}\]6 \[\Omega_{0,2t+\tau}\left(\mathbf{\Gamma}\right)\equiv\ln\left[\frac{f\left( \mathbf{\Gamma};0\right)}{f\left(S^{2t+\tau}\mathbf{\Gamma};0\right)}\right]- \int_{0}^{2t+\tau}\Lambda\left(S^{s}\mathbf{\Gamma}\right)ds \tag{9}\]7  
one has that \(\mathrm{Prob}_{t}\) tends to a steady state \(\mathrm{Prob}_{\infty}\) when \(t\rightarrow\infty\) and the averages of smooth phase functions according to \(\mathrm{Prob}_{\infty}\) are given by the asymptotic statistics of single dynamical trajectories. The physical conditions for the existence and uniqueness of \(\mathrm{Prob}_{\infty}\) are common. For example, the transport coefficients of fluids at low Reynolds or Rayleigh numbers are observed to be single valued functions of temperature, density and possibly the driving force, such as the shear rate or temperature gradient respectively. If they were not single valued functions of these state variables, then the system could not be T-mixing because correlation functions of phase space measures or order parameters for these different states would never decay in time. In T-mixing systems, the \(t\rightarrow\infty\) limit of Equation (19) turns from a statement about the time evolving ensemble \(f\left(\mathbf{\Gamma};t\right)\) to a statement concerning also the evolution of typical single dynamical trajectories. Then, for time reversal invariant, ergodically consistent, T-mixing systems, the steady state \(\Omega\)-FR holds with a correction that vanishes as \(O(\tau_{M}/\tau)\). It is now necessary to quantify this process. In Figure 1, we show the results of simulations of 8 WCA [20] particle systems with densities of 0.8 and 1.0 at a temperature of 1. The ensemble averages were over a set of at least 120 million trajectories with \(t=1.0\) and \(\tau\) varying between 0.1 and 4.0. The results in Figure 1 indicate that, indeed, the conditional ensemble average appearing in Equations (18)  
\[\Omega_{0,2t+\tau}\left(\mathbf{\Gamma}\right)\equiv\ln\left[\frac{f\left( \mathbf{\Gamma};0\right)}{f\left(S^{2t+\tau}\mathbf{\Gamma};0\right)}\right]- \int_{0}^{2t+\tau}\Lambda\left(S^{s}\mathbf{\Gamma}\right)ds \tag{9}\]8 Plots of the conditional ensemble average, \(\Xi(t,\tau;A)\), which is defined in Equation (19), as a function of \(\tau\) and \(A\). The systems consist of 8 particles in a periodic box undergoing colour diffusion.