3 GLYPH<LParen1> b GLYPH<RParen1> EQ GLYPH<Slash1> NESS OSC NESS NESS 0 1 2 3 4 5 6 0.0 0.5 1.0 1.5 2.0 GLYPH<LParen1> a GLYPH<RParen1> EQ GLYPH<Slash1> NESS NESS 0.0 0.1 0.2 0.3 0.4 0.000 0.002 0.004 0.006 0.008 0.010 GLYPH<LParen1> d GLYPH<RParen1> EQ GLYPH<Slash1> NESS OSC NESS 0.0 0.5 1.0 1.5 2.0 2.5 0.5 1.0 1.5 2.0 2.5 k$_{23}$ k 31 GLYPH<LParen1> c GLYPH<RParen1> EQ GLYPH<Slash1> NESS OSC NESS 0 10 20 30 40 0 20 40 60 80 100 Fig. 2: (color online) (a)-(c): Phase diagram in the ( ̂ M, ̂ T ) plane for different values of ̂ S with (a) ̂ S = 0 . 1, (b) ̂ S = 1 . 5 and (c) ̂ S = 10. The white area EQ/NESS allows both equilibrium states and NESSs, the purple area NESS can contain only NESSs and the brown area OSC corresponds to NESSs with oscillatory correlation function. The OSC regime touches the EQ/NESS regime at points ( 2 + 2 ̂ S,$_{−}$ 1 + ̂ S 2 ) . The checkered area is excluded since no choice of non-negative rate constants will lead to such an ( ̂ M, ̂ T ) value. The green line represents equilibrium states with some of the rate constants equal to zero. The black lines are the projection of the subdomain of the phase diagram shown in (d) that corresponds to the given ̂ S value. (d): Phase diagram in the ( k 23 , k 31 ) plane for fixed rate constants k 13 = 1, k 21 = 1, k 32 = 1 and k 12 = 4. For each point in the diagram, the parameters L, S, T and M are calculated and the three regimes accordingly identified. The blue hyperbola comprises the genuine equilibrium state according to eq. (13). This regime is denoted by OSC in fig. 2. It is easy to check that ̂ T$_{2}$ ( ̂ M ) ⩾ ̂ T$_{1}$ ( ̂ M ) (with equality only for ̂ M = 2( ̂ S + 1)). For all measured ̂ T with ̂ T$_{2}$ ⩾ ̂ T > ̂ T$_{1}$ , denoted by NESS in fig. 2, our new criterion establishes a NESS on the basis of two-state trajectories which do not exhibit oscillatory behaviour. While for ̂ S ≃ 1 the newly established region of NESSs beyond the oscillatory regime is rather small, see fig. 2(b), both for S ≪ 1 and for ̂ S ≫ 1 the present analysis shows a large regime of newly ascertained NESSs. In the latter two cases, the oscillatory regime is, for ̂ S ⩽ 0 . 5, totally absent or rather small, see figs. 2(a) and (c), respectively. For fixed ̂ S the phase diagrams of physically admissi- ble ( ̂ M, ̂ T ) values are restricted to ̂ M ⩽ 4 ̂ S and ̂ T ⩽ ̂ S $^{2}$, since values outside this range would require negative k$_{ij}$ values. Moreover, for ̂ M ⩽ 2 ̂ S , only T ⩽ T$_{1}$ can be re-