## 1 Introduction
Given an observable \(\varphi(\mathbf{\Gamma})\), let us denote its time average and integral, between times \(t_{1}\) and \(t_{2}\), along the phase space trajectory segment with initial condition \(\mathbf{\Gamma}\) as  
\[\bar{\varphi}_{t_{1},t_{2}}\left(\mathbf{\Gamma}\right)\equiv\frac{1}{\left( t_{2}-t_{1}\right)}\int_{t_{1}}^{t_{2}}\varphi\left(S^{s}\mathbf{\Gamma} \right)ds\equiv\frac{\varphi_{t_{1},t_{2}}\left(\mathbf{\Gamma}\right)}{ \left(t_{2}-t_{1}\right)} \tag{3}\]As the time average \(\bar{\Lambda}_{0,\tau}\) over the period \(\tau\), equals the sum of the Lyapunov exponents \(\sum_{n}\lambda_{n}\) for the system in the long \(\tau\) limit, [PERSON], [PERSON] and [PERSON] [1] proposed and tested the asymptotic steady state relation (\(\Lambda-FR\))  
\[\lim_{\tau\rightarrow\infty}\frac{\mu_{i}}{\mu_{i^{*}}}=\frac{ \exp\left[-\sum_{n}^{+}\lambda_{i,n}\tau\right]}{\exp\left[-\sum_{n}^{+} \lambda_{i^{*},n}\tau\right]} \tag{4}\] \[= \exp\left[-\sum_{n}\lambda_{i,n}\tau\right]=\exp\left[Nd\bar{ \alpha}_{IE;0,\tau;i}\tau\right]\]  
Here \(i,i^{*}\) label trajectory segments generated over the period \(\tau\) and related by time reversal symmetry so that the averages of \(\Lambda\) over the segments \(i,i^{*}\) obey \(\bar{\Lambda}_{0,\tau;i^{*}}=-\bar{\Lambda}_{0,\tau;i}\). The \(n\)-th phase space expansion rate along trajectory segment \(i\) is given by \(\lambda_{i,n}\), and the sums run over all the positive \(\lambda\)'s. The values of \(\mu_{i}\) and \(\mu_{i^{*}}\) are the probabilities of observing trajectories initially within a small volume \(d\mathbf{\Gamma}\) about \(i\) or \(i^{*}\)[3]. As in periodic orbit expansions [4, 5], Equation (4) can then be predicted to hold within a bounded range \(\left|\bar{\Lambda}_{0,\tau;i}\right|\equiv|Nd\bar{\alpha}_{IE;0,\tau;i}|<B\), \(B>0\). Later it was discovered that for non-isoenergetic systems, the physically relevant asymptotic limit in (4) is really \(\lim(\tau\dot{\gamma}^{2}\rightarrow\infty)\) while \(B\) was estimated to be \(B=O(\dot{\gamma}^{2})\)[6]. These two facts can make the observability of the steady state \(\Lambda-FR\) difficult close to equilibrium. [PERSON] _et al._[1] show that if one considers Equation (4) and sums over all \(i,i^{*}\) for which \(\bar{\alpha}_{IE;0,\tau;i}\) (or equivalently the entropy production) takes on a particular value within some tolerance, the fluctuation relation (FR) for the entropy production rate can be obtained. However, the proportionality between \(\Lambda\), the energy dissipation and the entropy production given in Equation (2) holds only for Gaussian isoenergetic systems. For instance, systems with constant kinetic energy differ from isoenergetic systems, because,  
\[\alpha_{IK}=\frac{\sum_{i=1}^{N}\mathbf{F}_{i}\cdot\dot{\mathbf{q}}_{i}-\dot{ \gamma}P_{xy}V}{\sum_{i=1}^{N}\mathbf{p}_{i}{}^{2}/m} \tag{5}\]  
replaces \(\alpha_{IE}\) in Equation (1), and \(\Lambda\) becomes the sum of two terms, only one of which is related to energy dissipated by the strain rate:  
\[\Lambda=\frac{c_{1}\sum_{i=1}^{N}\mathbf{F}_{i}\cdot\dot{\mathbf{q}}_{i}}{ \sum_{i=1}^{N}\mathbf{p}_{i}{}^{2}/m}-\frac{c_{2}\dot{\gamma}P_{xy}V}{\sum_{i= 1}^{N}\mathbf{p}_{i}{}^{2}/m} \tag{6}\]  
Here, the constants \(c_{1},c_{2}\) depend on \(N\) and \(d\). In 1994, [PERSON] and [PERSON] [7, 8] developed an alternative approach for deriving fluctuation relations for the energy dissipation that could be generalised to non-isoenergetic systems, through the introduction of the dissipation function \(\Omega\)[9]. For field-driven systems, \(\Omega\) equals \(-JVF_{e}/k_{B}T\)[9], which, close to equilibrium, _is the rate of entropy production_[2]. Here, \(J\) is the dissipative flux due to the force \(F_{e}\), \(V\) is the volume, \(k_{B}\) is Boltzmann's constant and \(T\) is the temperature of the equilibrium state that the system would relax to if the driving force (strain rate) was set to zero. The numerical value of this temperature is, for this system, the kinetic temperature [7, 8] of the nonequilibrium driven system.