Can one identify non-equilibrium in a three-state system by analyzing two-state trajectories? Christian P. Amann, Tim Schmiedl, and Udo Seifert II. Institut f¨ ur Theoretische Physik, Universit¨ at Stuttgart, 70550 Stuttgart, Germany For a three-state Markov system in a stationary state, we discuss whether, on the basis of data obtained from effective two-state (or on - off ) trajectories, it is possible to discriminate between an equilibrium state and a non-equilibrium steady state. By calculating the full phase diagram we identify a large region where such data will be consistent only with non-equilibrium conditions. This regime is considerably larger than the region with oscillatory relaxation, which has previously been identified as a sufficient criterion for non-equilibrium. PACS numbers: 05.40.-a, 05.70.Ln, 82.37.-j For the emerging field of thermodynamics of small systems, or stochastic thermodynamics $^{1–3}$, enzymes and proteins like molecular motors constitute paradigmatic systems since their conformational changes can be ob- served on a single molecule level using a variety of exper- imental techniques $^{4}$. The analysis of trajectories record- ing the transition between microstates of the enzyme can reveal insights into both the underlying kinetics and the thermodynamic implications of such changes $^{5–12}$. If the surrounding conditions like temperature or concentra- tions of other reactants are constant, the enzyme reaches a stationary state for which two classes must be distin- guished. First, in a genuine equilibrium state, no net flux between any two microstates of the enzyme occurs. Second, in a non-equilibrium steady state (NESS), the probability to find the enzyme in any microstate is still time-independent. However, non-zero net fluxes between microstates indicate that the system is externally driven with concomitant entropy production. If all microstates are experimentally accessible, in principle, distinguish- ing between these two alternatives is trivial, given long enough trajectories. It suffices to infer both the station- ary probabilities p s i for the system to be in state i and the rates k$_{ij}$ for a transition between i and j , and then to check for genuine equilibrium in which the detailed balance condition p s $_{i}$k$_{ij}$ = p s $_{j}$k$_{ji}$ for any two states must hold. In practice, rarely all relevant microstates are ex- perimentally accessible or distinguishable. The result- ing only coarse-grained information affects and aggra- vates a thermodynamic consistent analysis of individual trajectories $^{13,14}$. In standard applications, often only a two-state trajectory is obtained in which each observ- able state may in fact contain several microstates. In an ambitious program, Flomenbom and co-workers 15–19 have investigated which information about the underly- ing topology of the state space, or ‘network’, can be in- ferred from the observation of such two-state trajectories. But even if the underlying topology is known, two-state trajectories cannot reveal the full information like the value of all rates between microstates. For time-independent external conditions, arguably, the most relevant question is, whether or not such an enzyme acts in an equilibrium state or in a NESS. The simplest system for which this problem can be posed is a three-state system like the stochastic Michaelis- Menten scheme for an enzyme turning substrate into a product $^{7,20,21}$. Suppose that two of the three states are experimentally not distinguishable, can we decide in which of the two classes of stationary states the enzyme is in, if we are given a sufficiently long two-state trajec- tory? It is well-known that oscillations in the correla- tion function of such a two-state trajectory imply that the underlying three-state system is in a NESS rather than in equilibrium $^{22}$. Here, we show that the range of ascertained non-equilibrium can be extended much be- yond this oscillatory regime, i.e., even for decaying cor- relations it can be possible to infer an underlying NESS state. However, there will remain parameters for which the question posed in the title cannot be answered af- firmatively. Only if all three states are visible, like in dual-color flourescence correlation spectroscopy, one can discriminate between equilibrium and non-equilibrium steady states $^{23,24}$. We consider a three-state Markov system with states i = 1 , 2 , 3 and transition rates k$_{ij}$ . The dynamics of the probability p$_{i}$ ( t ) to be in state i at time t then is governed by the master equation ∂$_{t}$p$_{i}$ = 3 ∑ j =1 K$_{ij}$p$_{j}$ (1) K =   − k$_{12}$ − k$_{13}$ k$_{21}$ k$_{31}$ k$_{12}$ − k$_{21}$ − k$_{23}$ k$_{32}$ k$_{13}$ k$_{23}$ − k$_{31}$ − k$_{32}$   . (2) with rate matrix We assume that state 1 (the on -state) is fluorescently labelled and can be observed, but that it cannot be de- termined experimentally, whether the system is in state 2 or state 3 (together called the off -state), as sketched in fig. 1. The experiment then yields an effective two-state trajectory which no longer follows a Markovian dynam- ics. Four quantities can be extracted from such two-state trajectories and identified with their theoretical corre- spondences as follows: arXiv:1001.0528v1 [cond-mat.stat-mech] 4 Jan 2010