## 3 Discussion and Conclusions
Previous work has also resulted in transient and steady state fluctuation relations for arbitrary odd phase variables, referred to as generalised fluctuation relations [14, 18, 21]. These can be applied to obtain a fluctuation relation for the phase space expansion. Combining Equations (12) and (10) gives a relationship between the instantaneous dissipation function and the phase space expansion rate for a thermostatted system. Then, using the generalised transient fluctuation relation, the transient fluctuation relation for the phase space expansion for the thermostatted system can be written as:  
\[\frac{\mathrm{Prob}(\mathbf{\bar{\Lambda}}_{0,\tau}\big{|}_{A_{ \delta}^{+}})}{\mathrm{Prob}(\mathbf{\bar{\Lambda}}_{0,\tau}\big{|}_{A_{ \delta}^{-}})} = \frac{\int_{\bar{\Lambda}_{0,\tau}\big{|}_{A_{\delta}^{+}}}f( \mathbf{\Gamma};0)d\mathbf{\Gamma}}{\int_{\bar{\Lambda}_{0,\tau}\big{|}_{A_{ \delta}^{+}}}\exp\left[-\Omega_{0,\tau}(\mathbf{\Gamma})\right]f(\mathbf{ \Gamma};0)d\mathbf{\Gamma}}\]\[= \frac{\int_{\bar{\Lambda}_{0,\tau}}\left|{}_{A_{\delta}^{+}}\right.}{ \int_{\bar{\Lambda}_{0,\tau}}\left|{}_{A_{\delta}^{+}}\right.}\exp\left[{\Lambda _{0,\tau}}(\mathbf{\Gamma})-\beta(H_{0}(S^{\tau}\mathbf{\Gamma})-H_{0}( \mathbf{\Gamma}))\right]f(\mathbf{\Gamma};0)d\mathbf{\Gamma} \tag{25}\] \[= e^{[-A+\varepsilon(\delta,A,\tau)]\tau}\left\langle e^{-\beta(H_ {0}(S^{\tau}\mathbf{\Gamma})-H_{0}(\mathbf{\Gamma}))}\right\rangle_{\bar{ \Lambda}_{0,\tau}\in A_{\delta}^{+}}^{-1}\]  
In the special case where there are no correlations between \(\bar{\Lambda}_{0,\tau}\) and \(H_{0}(S^{\tau}\mathbf{\Gamma})-H_{0}(\mathbf{\Gamma})\), this will simplify to:  
\[\frac{\mathrm{Prob}(\mathbf{\bar{\Lambda}}_{0,\tau}\left|{}_{A_ {\delta}^{+}})}{\mathrm{Prob}(\mathbf{\bar{\Lambda}}_{0,\tau}\left|{}_{A_{ \delta}^{-}})} = e^{[-A+\varepsilon(\delta,A,\tau)]\tau}\left\langle e^{-\beta(H_ {0}(S^{\tau}\mathbf{\Gamma})-H_{0}(\mathbf{\Gamma}))}\right\rangle^{-1} \tag{26}\] \[= e^{[-A+\varepsilon(\delta,A,\tau)]\tau}\]  
where the final equality is obtained by recognising that the probability ratio must be equal to unity when \(A=0\). We note that this can then be used to write a fluctuation relation for joint probability distributions of \(\bar{\Lambda}_{0,\tau}\) and \(H_{0}(S^{\tau}\mathbf{\Gamma})-H_{0}(\mathbf{\Gamma})\) considered for stochastic systems in [22; 23; 24]. In the deterministic systems studied here, these correlations will be significant, however limits under which these correlations become negligible are important to establish and will be considered in future work. Similarly, a steady state fluctuation relation for the phase space contraction over the period of length \(\tau\) can be obtained:  
\[\frac{\mathrm{Prob}\left(\bar{\Lambda}_{t,t+\tau}\left|{}_{A_{ \delta}^{+}}\right.\right)}{\mathrm{Prob}\left(\bar{\Lambda}_{t,t+\tau}\left| {}_{A_{\delta}^{-}}\right.\right)} = \frac{\mathrm{Prob}_{t}\left(\bar{\Lambda}_{0,\tau}\left|{}_{A_{ \delta}^{+}}\right.\right)}{\mathrm{Prob}_{t}\left(\bar{\Lambda}_{0,\tau} \left|{}_{A_{\delta}^{-}}\right.\right)} \tag{27}\] \[= A+O\left(\frac{\tau_{M}}{\tau}\right)+\varepsilon\left(A,t,\tau, \delta\right),\quad|\varepsilon|\leq\delta\]0 (28) \[= A+O\left(\frac{\tau_{M}}{\tau}\right)+\varepsilon\left(A,t,\tau, \delta\right),\quad|\varepsilon|\leq\delta\]1  
In the simplistic case where there are no correlations between \(\bar{\Lambda}_{t,t+\tau}\) and \(H_{0}(S^{t+\tau}\mathbf{\Gamma})-H_{0}(S^{t}\mathbf{\Gamma})\) and assuming that T-mixing holds would enable a simple form of the limiting steady state fluctuation relation for the phase space expansion to be obtained.