Compensating random transition-detection blackouts in Markov networks

Abstract

In Markov networks, measurement blackouts with unknown frequency compromise observations such that thermodynamic quantities can no longer be inferred reliably. In particular, the observed currents neither discern equilibrium from non-equilibrium nor can they be used in extant estimators of entropy production. Our strategy to eliminate these effects is based on formally attributing the blackouts to a second channel connecting states. The unknown frequency of blackouts and the true underlying transition rates can be determined from the short-time limit of observed waiting-time distributions. A post-modification of observed trajectory data yields a virtual effective dynamics from which the lower bound on entropy production based on thermodynamic uncertainty relations can be recovered fully. Moreover, the post-processed data can be used in waiting-time based estimators. Crucially, our strategy does neither require the blackouts to occur homogeneously nor symmetrically under time-reversal.

Figures (3)

• Figure 1: Scheme of observation and post-modification. In the fluctuating microscopic trajectory (gray) of the left 4-state Markov network only transitions $I_+ =(12)$ (indigo) and $I_-=(21)$ (green) with rates $k_{I_+}$ and $k_{I_-}$, respectively, are, in principle, visible to an observer as indicated by the vertical bars. The actually observed transitions indicated by the crosses miss some of these bars due to random blackouts. The detection probabilities $\eta_{I_+}<1$ and $\eta_{I_-}<1$ lead to decreased effective rates $\eta_{I_+}k_{I_+}$ and $\eta_{I_-}k_{I_-}$ and to a second channel (purple) representing the blackouts with rates $(1-\eta_{I_+})k_{I_+}$ and $(1-\eta_{I_-})k_{I_-}$ as indicated in the middle network. The post-modification randomly discards transitions from the observed trajectory such that it corresponds to a virtual dynamics with equal detection probability for forward and backward transitions given by $\eta_I\leq\eta_{I}^\ast=\min\{\eta_{I_+},\eta_{I_-}\}$ as indicated by the right network.
• Figure 2: Waiting-time distributions for one observable pair of transitions $I_+=(12)$ and $I_-=(21)$ in the 4-state Markov network from \ref{['fig:ps_traj_graphs']} with $k_{12} = k_{23} = 3$, $k_{31} = k_{14} = k_{43} = 2$ and $k_{13} = k_{21} = k_{32} = k_{34} = k_{41} = 1$ obtained via an analog of Gillespie's algorithm gill77 for semi-Markov processes with detection probabilities $\eta_{I_+}=0.8$ and $\eta_{I_-}=0.9$, trajectory length $T=7\cdot 10^9$, bin width $\Delta t=10^{-3}$ and cut-off time $t_\text{c}=20$ for the experimental WTDs. The post-modification illustrated in \ref{['fig:ps_traj_graphs']} leads to the distributions for $\eta_I=0.8$ as indicated by the arrows.
• Figure 3: Entropy estimators for observations with blackouts of the 4-state Markov network from \ref{['fig:ps_traj_graphs']}. (a) Entropy estimators $\sigma_\text{TUR}(\eta_I)$ and $\sigma_\text{WTD}(\eta_I)$ for the rates as given in the caption of \ref{['fig:ps_wtd']} as a function of the detection probabilities $\eta_{I_+}=\eta_{I_-}=\eta_I$ in units of the mean entropy production rate $\sigma_\text{M}$ of the underlying Markov network. The waiting-time distributions obtained from a simulated trajectory as presented in \ref{['fig:ps_wtd']} result in the highlighted values of $\sigma_\text{WTD}(\eta_I)$ (blue crosses). The theoretical values of $\sigma_\text{WTD}(\eta_I)$ for $\eta_I>\eta_{I}^\ast=0.8$ (dashed) are inaccessible and require full knowledge of the system, whereas $\sigma_\text{TUR}(\eta_I)$ can be inferred in this regime through \ref{['res:TUR']} (greenyellow). (b) Entropy estimator $\sigma_\text{MTUR}(\eta_I,\eta_J)$ as a function of the detection probabilities $\eta_I$ and $\eta_J$ in units of $\sigma_\text{M}$ of this Markov network with rates $k_{12} = k_{23} = k_{31} = 2$, $k_{13} = k_{21} = k_{32} = k_{34} = k_{41} = 1$ and $k_{14} = k_{43} = 3$ in which, besides $I_+=(12)$ and $I_-=(21)$, transitions $J_+=(34)$ and $J_-=(43)$ are observable.