## 2 Calculations
For our system, the instantaneous dissipation function is [14]  
\[\Omega(\mathbf{\Gamma})=-\beta\mathbf{J}(\mathbf{\Gamma})V\cdot\mathbf{F}_{e} \tag{10}\]  
where the dissipative flux, \(\mathbf{J}\), is obtained from the adiabatic time-derivative of the internal energy \(H_{0}\):  
\[\dot{H}_{0}^{ad}(\mathbf{\Gamma})=\sum_{i=1}^{N}c_{i}\frac{\mathbf{p}_{i}}{m} \cdot\mathbf{F}_{e}\equiv-\mathbf{J}(\mathbf{\Gamma})V\cdot\mathbf{F}_{e} \tag{11}\]  
while the fully thermostatted dynamics of Equation (7) yields  
\[\dot{H}_{0}(\mathbf{\Gamma}) = -\mathbf{J}(\mathbf{\Gamma})V\cdot\mathbf{F}_{e}-2K_{0}\alpha \tag{12}\] \[= -\mathbf{J}(\mathbf{\Gamma})V\cdot\mathbf{F}_{e}+\Lambda(\mathbf{ \Gamma})k_{B}T\]  
by equipartition, the phase space expansion rate is \(\Lambda=-\left(2N-3\right)\alpha\). Time reversibility implies that for every initial condition \(\mathbf{\Gamma}\) that yields a value \(\Omega_{0,2t+\tau}\left(\mathbf{\Gamma}\right)\), there is an initial condition \(\mathbf{\Gamma}^{*}\) yielding the value \(\Omega_{0,2t+\tau}\left(\mathbf{\Gamma}^{*}\right)=-\Omega_{0,2t+\tau}\left( \mathbf{\Gamma}\right)\). To obtain \(\mathbf{\Gamma}^{*}\) the time reversal map, \(M^{T}\), is applied to the time-evolved phase, \(S^{2t+\tau}\mathbf{\Gamma}\), _i.e._, \(\mathbf{\Gamma}^{*}=\mathrm{M}^{T}S^{2t+\tau}\mathbf{\Gamma}\). We also assume that \(f(M^{T}\mathbf{\Gamma};0)=f(\mathbf{\Gamma};0)\), which guarantees that we measure properties in a co-moving frame since the average velocity of particles in the system is zero. From (8):  
\[\frac{f\left(\mathbf{\Gamma};0\right)}{f\left(S^{2t+\tau}\mathbf{\Gamma};0 \right)}=\exp\left(\beta\int_{0}^{2t+\tau}\dot{H}_{0}\left(S^{s}\mathbf{ \Gamma}\right)ds\right) \tag{13}\]  
Substituting (13) into (9) using (11) gives,  
\[\bar{\Omega}_{0,2t+\tau}(\mathbf{\Gamma}) = \frac{\beta}{(2t+\tau)}\int_{0}^{2t+\tau}\dot{H}_{0}(S^{s}\mathbf{ \Gamma})ds-\bar{\Lambda}_{0,2t+\tau}(\mathbf{\Gamma}) \tag{14}\] \[= -\mathbf{\bar{J}}_{0,2t+\tau}(\mathbf{\Gamma})\cdot\mathbf{F}_{e }V\beta\]  
It is very important to note that in Equations (9) and (13), the phase point evolves but we only refer to the initial phase space distribution function. This is the Schrodinger picture. Consider firstly the transient \(\Omega\)-FR, which is just a consequence of time reversibility and ergodic consistency. In the case of the transient \(\Omega\)-FR, we wish to consider evolution of the trajectory from \(\mathbf{\Gamma}\), and therefore \(t\) in the equations above is \(0\). Time reversibility implies that the coordinate transformation \(\mathbf{\Gamma}^{*}=M^{T}S^{\tau}\mathbf{\Gamma}\) has Jacobian,  
\[f(\mathbf{\Gamma};0)=\frac{\exp[-\beta H_{0}(\mathbf{\Gamma})]\delta(\mathbf{ P})\delta(K(\mathbf{\Gamma})-K_{0})}{\int\exp[-\beta H_{0}(\mathbf{\Gamma})] \delta(\mathbf{P})\delta(K(\mathbf{\Gamma})-K_{0})d\mathbf{\Gamma}} \tag{8}\]0 \[f(\mathbf{\Gamma};0)=\frac{\exp[-\beta H_{0}(\mathbf{\Gamma})]\delta(\mathbf{ P})\delta(K(\mathbf{\Gamma})-K_{0})}{\int\exp[-\beta H_{0}(\mathbf{\Gamma})] \delta(\mathbf{P})\delta(K(\mathbf{\Gamma})-K_{0})d\mathbf{\Gamma}} \tag{8}\]1_i.e._, the exponent involves the change in the heat. Let \(A_{\delta}^{+}=\left(A-\delta,A+\delta\right)\) and \(A_{\delta}^{-}=\left(-A-\delta,-A+\delta\right)\) be small intervals of width \(2\delta\) centred in \(A\) and \(-A\), and denote by \(\left.\phi\right|_{E}\) the set of \(\mathbf{\Gamma}\) such that \(\phi(\mathbf{\Gamma})\in E\) where \(E=A_{\delta}^{+}\) or \(A_{\delta}^{-}\).