# Fluctuation Relation for the Dissipative Flux: The Role of Dynamics, Correlations and Heat Baths
## 2 Steady-State Fluctuation Relation
With minimal algebra, Equation (21) yields another exact relation:  
\[\frac{1}{\tau}\ln\frac{\mu^{(t)}\left(\overline{\Omega}^{(0)}_{0, \tau}\in\mathcal{B}_{A,\delta}\right)}{\mu^{(t)}\left(\overline{\Omega}^{(0)} _{0,\tau}\in\mathcal{B}_{-A,\delta}\right)}=-\frac{1}{\tau}\ln\biggl{\langle} e^{-\Omega^{(0)}_{0,t}}\cdot e^{-\Omega^{(0)}_{t,t+\tau}}\cdot e^{-\Omega^{(0)}_{t+ \tau,2t+\tau}}\biggr{\rangle}^{(0)}_{\overline{\Omega}^{(0)}_{t,\tau}\in \mathcal{B}_{A,\delta}} \tag{24}\] \[=A+\epsilon(\delta,t,A,\tau)-\frac{1}{\tau}\ln\biggl{\langle}e^{ -\Omega^{(0)}_{0,t}}\cdot e^{-\Omega^{(0)}_{t+\tau,2t+\tau}}\biggr{\rangle}^{( 0)}_{\overline{\Omega}^{(0)}_{t,\tau+\tau}\in\mathcal{B}_{A,\delta}}\,.\]  
Here, the second line yields the expression of the fluctuation relation, in the \(t\to\infty\) limit, if such a limit exists, and the conditional average behaves properly when \(\tau\) becomes large. Therefore, although it is derived from an exact transient relation that practically always holds, the steady-state fluctuation relation does not always makes sense, and when it does it does not necessarily correspond to the exponential unbalance of probabilitiesexpressed by the original relation (1). Indeed, the \(t\to\infty\) of the conditional average may diverge, or it may yield a non-negligible contribution, even in the \(\tau\to\infty\) limit. When this term constitutes a correction that vanishes as \(O(1/\tau)\), we have a situation analogous to the one discussed within the Anosov formalism. In all circumstances, it is a relation that only becomes exact when \(\tau\) grows, unlike the transient fluctuation relation, which is exact for all \(\tau\). Another difference between the transient and the steady-state relations is that the second applies also to the fluctuations along a single trajectory, and does not need an ensemble. **Remark 1**. : _The last term of Equation (24) must vanish when the limit \(\tau\to\infty\) is taken after the limit \(t\to\infty\). This is not only a sufficient condition but it is also necessary, because Equation (24) is an exact relation. This says that correlations of the dissipation function with respect to the initial distribution follow a given rule, when reversible dynamical systems verify the steady-state fluctuation relation. This behavior of the correlations of the dissipation function constitutes the mechanism implying the validity of the steady-state fluctuation relation. The relation is violated under different conditions [8]._  
To understand the significance of the last term in Equation (24), consider the case in which the correlations with respect to the initial distribution \(\mu^{(0)}\) decay instantaneously. Considering that the conditional average then turns equal to the full average, and that the transient fluctuation relation, with some manipulation, also yields [8]:  
\[\left\langle e^{-\Omega_{0,t}^{(0)}}\right\rangle^{(0)}=1 \tag{25}\]  
one obtains:  
\[\left\langle e^{-\Omega_{0,t}^{(0)}}\cdot e^{-\Omega_{t+\tau,2 \mu+\tau}^{(0)}}\right\rangle^{(0)}_{\overline{\Omega}_{t+\tau}^{(0)}\in \mathcal{B}_{A,t}}=\left\langle e^{-\Omega_{0,t}^{(0)}}\cdot e^{-\Omega_{t+ \tau,2\mu+\tau}^{(0)}}\right\rangle^{(0)}\] \[\qquad=\left\langle e^{-\Omega_{0,t}^{(0)}}\cdot\left(e^{- \Omega_{0,t}^{(0)}}\circ S^{t+\tau}\right)\right\rangle^{(0)}=\left\langle e^ {-\Omega_{0,t}^{(0)}}\right\rangle^{(0)}\left\langle e^{-\Omega_{0,t}^{(0)} }\right\rangle^{(t+\tau)}=1,\]  
where \(\langle\rangle^{(t+\tau)}\) represents the full phase space average with respect to \(\mu^{(t+\tau)}\), cf. [8]. This implies that the steady state is verified at an arbitrarily short \(\tau\). If the decay of correlations is not immediate with a growing \(\tau\), the final result will depend on how they depend on \(\tau\). An investigation of these issues was performed in Ref. [29] for color diffusion molecular dynamics. In that case, it was found that the correlations decay, hence the steady-state fluctuation relation is verified.