4 values would increase linearly on straight lines through the origin in fig. 2(a)-(c). Since the maximum value of ̂ T increases quadratically with the maximum value of M , the uncertainty on the phase boundary between NESS and EQ/NESS in fig. 2(a)-(c) increases with increasing ̂ S . Hence, a clear discrimination between the two regions would be affected by errors less in the small ̂ S regime than for large ̂ S . For the data shown in fig. 2(a) and ε = 0 . 05, the phase boundary effectively broadens to about 11% on the top and 10% at the bottom. alized with non-negative k$_{ij}$ . The region ̂ S 2 ⩾ T > ̂ T$_{1}$ , which is excluded by the latter condition, is shown check- ered in figs. 2(a)-(c). The equalities in these constraints of physically admissible values hold only if some of the rates are zero. Finally, an analysis of the boundary ̂ T$_{1}$ ( ̂ M ), which re- quires some rate constants k$_{ij}$ to be zero, shows that for 0 ⩽ ̂ M ⩽ 2 ̂ S the system is definitely in equilibrium on this boundary. For ̂ T = 0 the system is also definitely in equilibrium. For representative examples of these results in the orig- inal space of the six rates k$_{ij}$ , consider fig. 2(d). There we show the three regimes EQ/NESS , NESS and OSC in the ( k$_{23}$ , k$_{31}$ ) plane holding the other four rate constants fixed. The line of genuine equilibrium (13) is embedded into region EQ/NESS where information from the state trajectory is not sufficient to uniquely identify equilib- rium. However for ( k$_{23}$ , k$_{31}$ ) parameter values in the region NESS our analysis predicts correctly a NESS be- yond the previously established OSC region. In summary, we have derived a criterion sufficient to identify a non-equilibrium steady state in a three-state system based on information extracted from sufficently long two-state trajectories. While the new criterion ex- tends the region of ascertained NESSs significantly be- yond the previously identified region based on oscillation in the autocorrelation function, there remain parameter values for which two-state trajectories contain principally not enough information to discriminate NESS from an equilibrium state. It will be interesting to perform a similar analysis for four-state systems or even more com- plex networks, where one would expect that the region of ascertained NESSs on the basis of two-state trajec- tory information becomes relatively smaller than in the three-state case. Generalization of our approach to cases where three-state trajectories, obtained, e.g., via dual- colour fluorescence, are available on four or more-state networks is feasible in principle, but may, in practice, become algorithmically challenging. Experimental measurements of the waiting time distri- bution w$_{off}$ ( t ) (5) and autocorrelation function p$_{1}$ ( t ) (9) will cause some uncertainty due to finite data. For a rough estimate, we assume a relative error of ± ε in all eigenvalues ˜ λ$_{i}$ (6) and λ$_{i}$ (10), while it should be possible to determine the decay time L (3) with better precision. A linear error propagation then yields a constant relative error for ̂ S and ̂ T of ± ε and ± 2 ε respectively. The ef- fect on M is a relative error of ± (2 ε + 4 ε ̂ T/ ̂ M ). Thus error bars of experimentally determined pairs of ( ̂ M, ̂ T ) Chem. Phys. 128 , 074102 (2008). 1 C. Bustamante, J. Liphardt, and F. Ritort, Physics Today 58(7) , 43 (2005). 2 U. Seifert, Eur. Phys. J. B 64 , 423 (2008). 3 F. Ritort, Adv. Chem. Phys. 137 , 31 (2008). 4 P. Selvin and T. Ha, Single Molecule Techniques (Cold Spring Harbor Laboratory, 2007). 15 O. Flomenbom, J. Klafter, and A. Szabo, Biophys. J. 88 , 3780 (2005). 16 O. Flomenbom and J. Klafter, J. Chem. Phys. 123 , 064903 (2005). 17 O. Flomenbom and R. J. Silbey, Proc. Natl. Acad. Sci. U.S.A. 103 , 10907 (2006). 18 O. Flomenbom and R. J. Silbey, J. Chem. Phys. 128 , 114902 (2008). 5 H. P. Lu, L. Y. Xun, and X. S. Xie, Science 282 , 1877 (1998). 6 M. E. Fisher and A. B. Kolomeisky, Proc. Natl. Acad. Sci. U.S.A. 98 , 7748 (2001). 7 H. Qian and E. L. Elson, Biophys. Chem. 101 , 565 (2002). 8 U. Seifert, Europhys. Lett. 70 , 36 (2005). 9 W. Min, L. Jiang, J. Yu, S. C. Kou, H. Qian, and X. S. Xie, Nano Lett. 5 , 2373 (2005). 10 T. Schmiedl, T. Speck, and U. Seifert, J. Stat. Phys. 128 , 77 (2007). 19 O. Flomenbom and R. J. Silbey, Phys. Rev. E 78 , 066105 (2008). 20 C. V. Rao and A. P. Arkin, J. Chem. Phys. 118 , 4999 (2003). 21 S. C. Kou, B. J. Cherayil, W. Min, B. P. English, and X. S. Xie, J. Phys. Chem. B 109 , 19068 (2005). 22 H. Qian and M. Qian, Phys. Rev. Lett. 84 , 2271 (2000). 23 P. Schwille, F. J. MeyerAlmes, and R. Rigler, Biophys. J. 72 , 1878 (1997). 24 H. Qian and E. L. Elson, Proc. Natl. Acad. Sci. U.S.A. 101 , 2828 (2004). 11 P. Gaspard and E. Gerritsma, J. Theor. Biol. 247 , 672 (2007). 12 R. Lipowsky and S. Liepelt, J. Stat. Phys. 135 , 777 (2009). 13 S. Rahav and C. Jarzynski, J. Stat. Mech.: Theor. Exp. p. P09012 (2007). 14 Y. Li, T. Zhao, P. Bhimalapuram, and A. R. Dinner, J.