# Fluctuation Relation for the Dissipative Flux: The Role of Dynamics, Correlations and Heat Baths
## 2 Steady-State Fluctuation Relation  
The authors of Ref. [1] considered the Gaussian iso-energetic SLLOD model of a shearing fluid, for \(N\) particles in \(d\) dimensions, cf. Appendix A, and proposed and tested the following fluctuation relation:  
\[\frac{\mu_{i}}{\mu_{i}}=\frac{\exp\left[-\sum_{n}^{+}\lambda_{i,n}\tau\right]} {\exp\left[-\sum_{n}^{+}\lambda_{i,n}\tau\right]}=\exp[Nd\tau\overline{\alpha }_{i,\tau}] \tag{1}\]  
inspired by the theory of Anosov dynamical systems, cf. Appendix B, which they adapted to the time-reversal invariant SLLOD equations of motion. In this formula, \(i,i^{*}\) represent conjugate phase space trajectory segments of length \(\tau\), \(\lambda_{i,n}\) is the \(n\)-th finite-time Lyapunov exponent computed along the segment \(i\) in a time \(\tau\), and \(\overline{\alpha}_{i,\tau}\propto-\sum_{n}\lambda_{i,n}\) is proportional to the average energy dissipation rate, along one trajectory segment of duration \(\tau\). Then, \(\mu_{i}\) is the steady-state probability of trajectory segments of length \(\tau\) yielding the same \(\overline{\alpha}_{i,\tau}\), and \(\mu_{i^{*}}\) is the probability of segments yielding the opposite value, \(-\overline{\alpha}_{i,\tau}\). The result, according to the theory of large deviations, is then verified by taking larger and larger averaging times \(\tau\)In local thermodynamic equilibrium, Equation (1) quantifies the second law for reversible dynamics, which is what made it initially very popular. Consequently, in near equilibrium situations it yields linear response with Onsager and Green-Kubo relations [26; 27]. Moreover, its derivation from pure dynamics makes the fluctuation relation suitable to go beyond the local thermodynamic equilibrium regime, and of particular interest in the field of small and low-dimensional systems. In that realm, unlike the macroscopic one, fluctuations are observable and comparable to observed average signals. The work on steady-state fluctuations led to the transient fluctuation relation, derived in 1994 by [PERSON] and [PERSON] [9], and that opened the way to a different understanding of the steady-state one, not based on the Anosov assumption [26], cf. Appendix C. Here, we concisely illustrate this approach, which explains our findings. One fundamental property of the molecular dynamics systems for which the fluctuation relations have been derived is that they are dissipative, which allows systems to reach steady states, but time-reversal invariant. This can be abstractly described as follows. Let \(\mathcal{M}\) be the phase space and \(S^{t}:\mathcal{M}\to\mathcal{M}\) evolution in \(\mathcal{M}\), so that \(S^{t}\Gamma\) is the phase \(\Gamma\) reaches at time \(t\). In the cases of our interest, \(S^{t}\) yields the solution of the equations of motion \(\dot{\Gamma}=G(\Gamma)\). For instance, given the one-dimensional harmonic oscillator equations:  
\[\dot{q}=p\,,\quad\dot{p}=-q \tag{2}\]  
one has \(\Gamma=(q,p)\), and:  
\[S^{t}\Gamma=S^{t}(q,p)=(q\cos t+p\sin t,-q\sin t+p\cos t). \tag{3}\]  
The dynamics are called dissipative if the time average of the divergence of the vector field  
\[\Lambda=\operatorname{div}G \tag{4}\]  
is negative. When this is the case, the system is not Hamiltonian. In particular, negative \(\Lambda\) means that phase space volumes contract. However, time reversibility may still hold. Indeed, it suffices that the equations of motion are invariant under the inversion of time \(t\mapsto-t\), combined with the transformation  
\[i(\mathbf{q},\mathbf{p})=(\mathbf{q},-\mathbf{p})\quad\text{so that}\quad i^{2}= \text{Identity}, \tag{5}\]  
where \(\Gamma=(\mathbf{q},\mathbf{p})\) is the collection of positions and momenta of the particles of the system of interest, cf. Appendix B. This also implies that the trajectories stemming from the initial conditions \(\Gamma\) and \(\dot{\Gamma}=iS^{t}\Gamma\) trace back each other in configuration space for a time \(t\), and lead to the following general definition.