## 1 Introduction  
Fluctuation relations (FRs) have been studied intensively over the past twenty years. Transient Fluctuation relations are the best understood. They are known to be exact for systems of arbitrary size, arbitrarily near or far from equilibrium. Previously, very few exact results were known in nonequilibrium statistical mechanics. Their application to small systems coincided with an upsurge of interest in nanotechnology and nano-biology and the study of small bio-engines. The applicability of the transient FRs far from equilibrium meant that they were an important touchstone for large deviation theory. Asymptotic steady state fluctuation relations are far less well understood. Some such relations were known to have convergence problems close to equilibrium. In this paper we prove that for the so-called T-mixing systems, the Evans-Searles steady state fluctuation relation is free of convergence difficulties, _i.e._, it applies to systems of arbitrary size and applies arbitrarily near, or far from, equilibrium. This paper identifies the conditions required for the steady state FR involving the dissipation function, to be derived from the exact Evans-Searles transient fluctuation theorem, and provides an illustration of the results using numerical simulations. In 1993 [1], [PERSON], [PERSON] and [PERSON] studied the steady state fluctuations of the entropy production rate in isoenergetic SLLOD (SLLOD is the term used to refer to the standard algorithm used to model homogeneous flow [2]) shearing systems, defined by:  
\[\dot{\mathbf{q}}_{i}=\mathbf{p}_{i}/m+\mathbf{n}_{x}\dot{\gamma}y_{i},\quad \dot{\mathbf{p}}_{i}=\mathbf{F}_{i}-\mathbf{n}_{x}\dot{\gamma}p_{yi}-\alpha_{ IE}\mathbf{p}_{i} \tag{1}\]  
where \(i=1,...,N\); \(N\) is the number of particles in a periodic system with a unit cell of volume \(V\); \(m\) is the mass of each particle; \(\dot{\gamma}\) is the shear rate; \(\mathbf{n}_{x}\) is the unit vector in the \(x\)-direction; \(\mathbf{F}_{i}\) is the internal force on particle \(i\); \(\mathbf{q}_{i}=(x_{i},y_{i})\) is the position of each particle; \(\mathbf{p}_{i}=(p_{xi},p_{yi})\) is the momentum of each particle; and \(\alpha_{IE}\) is a Lagrange multiplier that fixes the internal energy of the system, \(H_{0}=\sum\limits_{i=1}^{N}\mathbf{p}_{i}\cdot\mathbf{p}_{i}/(2m)+\Phi( \mathbf{q})\), where \(\Phi(\mathbf{q})\) is the interparticle potential. The value of \(\alpha_{IE}\) is related to: the rate of energy dissipation under unthermostatted (adiabatic) conditions, \(\dot{H}_{0}^{ad}\); the entropy production rate close to equilibrium, \(\Sigma\); and the phase space expansion rate, \(\Lambda=\partial/\partial\mathbf{\Gamma}\cdot\dot{\mathbf{\Gamma}}\) where \(\mathbf{\Gamma}=(\mathbf{q}_{1},...,\mathbf{q}_{N},\mathbf{p}_{1},...,\mathbf{ p}_{N})=(\mathbf{q},\mathbf{p})\). This relationship is given by,  
\[\alpha_{IE}=\frac{\dot{H}_{0}^{ad}}{\sum\limits_{i=1}^{N}p_{i}^{2}/m}=\frac{ \Sigma}{N_{df}}=-\frac{\Lambda}{N_{df}}=\frac{-\dot{\gamma}P_{xy}V}{\sum \limits_{i=1}^{N}p_{i}^{2}/m} \tag{2}\]  
Here \(P_{xy}\) is the off-diagonal part of the pressure tensor and \(N_{df}=dN-d-1\) is the number of degrees of freedom. In [1] the spatial dimension was \(d=2\). In this and similar molecular dynamics models, unphysical thermostatting terms like \(\alpha_{IE}\) may be confined to arbitrarily remote boundaries. Then, the bulk properties of the systems of interest should not be affected by the thermostatting details. To set our notation, let \(S^{s}\mathbf{\Gamma}\) be a point in phase space evolved via the time evolution operator, \(S\), (given by the equations of motion) for a period \(s\) from initial condition \(\mathbf{\Gamma}\).