Estimator of entropy production for partially accessible Markov networks based on the observation of blurred transitions

A central task in stochastic thermodynamics is the estimation of entropy production for partially accessible Markov networks. We establish an effective transition-based description for such networks with transitions that are not distinguishable and therefore blurred for an external observer. We demonstrate that, in contrast to a description based on fully resolved transitions, this effective description is typically non-Markovian at any point in time. Starting from an information-theoretic bound, we derive an operationally accessible entropy estimator for this observation scenario. We illustrate the operational relevance and the quality of this entropy estimator with a numerical analysis of various representative examples.

Figure 1

Blurred observation of the Schlögl model. (a) Reaction scheme: $X$ molecules can undergo chemical reactions with $A$ molecules and $B$ molecules provided by two different chemical reservoirs. (b) If an observer resolves the number of $X$ molecules $n_X$, each change in $n_X$ can be represented as transition in an effective Markov network with two different channels. (c) If the observer only registers forward and backward reactions along the A-reaction channel (dark color) or the B-reaction channel (bright color) but cannot resolve the changes in the number of $X$ molecules $n_X$, the corresponding transitions of the effective network are blurred for this observer.

Figure 2

Observation of a partially accessible five-state Markov network with four resolved transitions. The resolved observer registers the transitions (12), (21), (45), and (54) and the corresponding in-between waiting times.

Figure 3

Observation of a partially accessible five-state Markov network with two blurred transitions. (a) The blurred observer registers transitions from classes $I$ and $J$ and the corresponding in-between waiting times but cannot resolve the individual transitions within a class. (b) Breakdown of the renewal property, i.e., Eq. (15), for three out of eight possible sequences with three transitions. The deviations between the conditioned probabilities, e.g., $p(I\to J;t|I\to I)$, and the corresponding products, e.g., $p_{I\to I}{\mathrm{\Psi }}_{I\to J}\left(t\right)$, indicate that the observed transitions are not Markovian. The probabilities and waiting time distributions are calculated with an absorbing master equation for transition rates $k_{12}=0.4,k_{21}=8.9,k_{23}=4.2,k_{32}=1.1,k_{34}=1.8,k_{43}=0.01,k_{45}=9.4,k_{54}=0.1, k_{51}=8.8$, and $k_{15}=0.6$.

Figure 4

Quality of the entropy bound in Eq. (28) for a unicyclic five-state Markov network as a function of the cycle affinity for three different observation scenarios. (a) Observation of the odd transition classes $I=\left\{\right(12),(45\left)\right\}$ and $J=\left\{\right(21),(54\left)\right\}$. The mean quality factor is given by $Q\simeq 0.66$. (b) Observation of the even transition class $G=\left\{\right(12),(21),(34),(43\left)\right\}$ and of the odd transition classes $H=\left\{\right(23),(45\left)\right\}$ and $I=\left\{\right(32),(54\left)\right\}$. The mean quality factor is given by $Q\simeq 0.77$. (c) Observation of the odd transition classes $I$ and $J$ including all forward and backward transitions, respectively. The mean quality factor is given by $Q\simeq 0.8$. Each scenario includes more than 4 000 000 network realizations with transition rates that are randomly drawn from uniform distributions between 0.1 and 6. All waiting time distributions are calculated with an absorbing master equation and Eq. (13).

Figure 5

Quality of the entropy bound in Eq. (28) for the observation of three multicyclic Markov network as a function of the average cycle affinity $\overline{{\mathcal{A}}_{\mathcal{C}}}$. For each link, the forward and backward transitions are blurred into the same transition class. (a) Observation of a three channel network with even transition classes $I,J$ and $K$. The mean quality factor is given by $Q\simeq 0.06$. (b) Observation of a three state network with two channels and even transition classes $H,I,J$, and $K$. The mean quality factor is given by $Q\simeq 0.67$. (c) Observation of a four state network with even transition classes $G,H,I,J$, and $K$. The mean quality factor is given by $Q\simeq 1.0$. Each observation includes more than 1 300 000 network realizations with transition rates that are randomly drawn from uniform distributions between 0.1 and 6. All waiting time distributions are calculated with an absorbing master equation and Eq. (13).

Figure 6

Entropy production rate $\langle \sigma \rangle$ and entropy estimator ${\mathrm{\Sigma }}_{\mathrm{BT}}$ for the blurred observation of the Schlögl model as a function of $c_A$ at different system sizes $\mathrm{\Omega }$ with $\mathrm{\Delta }\mu =ln9$ and ${\kappa }_{\pm }^{A/B}=1$. The stationary distribution of $n_X$ needed for calculating the waiting time distributions via Eq. (13) is estimated numerically based on Gillespie simulation data. The slight deviations between $\langle \sigma \rangle$ and ${\mathrm{\Sigma }}_{\mathrm{BT}}$ emerge as a consequence of the numerical error of the estimation method for the stationary distribution and as a consequence of the finite sample size. The increase in $\langle \sigma \rangle$ at $c_A\simeq 1.73$ is caused by the phase transition of the Schlögl model at the critical point [52, 53].