Effective estimation of entropy production with lacking data

Abstract

Observing stochastic trajectories with rare transitions between states, practically undetectable on time scales accessible to experiments, makes it impossible to directly quantify the entropy production and thus infer whether and how far systems are from equilibrium. To solve this issue for Markovian jump dynamics, we show a lower bound that outperforms any other estimation of entropy production (including Bayesian approaches) in regimes lacking data due to the strong irreversibility of state transitions. Moreover, in the limit of complete irreversibility, our new effective version of the thermodynamic uncertainty relation sets a lower bound to entropy production that depends only on nondissipative aspects of the dynamics. Such an approach is also valuable when dealing with jump dynamics with a deterministic limit, such as irreversible chemical reactions.

Figures (4)

• Figure 1: Estimation of the entropy production in a 4-state model. For the 4-states model (inset and description in the text), estimates of the entropy production rate and theoretical value $\sigma_{\rm th}$ (left axis), and the fraction of trajectories displaying only irreversible transitions (green curve, right axis) as a function of the nonequilibrium strength $\alpha$. The sampling time is $t=10^3$, and bands show one standard deviation variability over trajectories. Highlighted regions: (i) the empirical estimate works well while lower bounds progressively depart from $\sigma_{\rm th}$; (ii) ${\sigma}_{\rm emp}$ deviates from $\sigma_{\rm th}$ but remains the best estimator; (iii) the new lower bound $\sigma_{\tanh}^{\rm hyp}$ is the best estimator.
• Figure 2: Estimation of the entropy production as a function of the sampling time.a. For three values of the nonequilibrium parameter $\alpha$ (see legend) in the model of Fig. \ref{['fig:4states']}, scaling with the sampling time $t$ of the estimators ${\sigma}_{\rm emp}$ and $\sigma_{\tanh}^{\rm hyp}$, and theoretical values (horizontal thick lines). Bands show one standard deviation variability over trajectories. b. Time when ${\sigma}_{\rm emp}$ becomes larger than $\sigma_{\tanh}^{\rm hyp}$, as a function of $\sigma_{\rm th}$.
• Figure 3: Estimation of the entropy production for a ring model. For the ring model described in the text, we show various estimates of the entropy production rate divided by the theoretical value, for $t=10^4$, as a function of the nonequilibrium strength $\alpha/N$, for ring lengths $N=10$ and $N=20$. The inset shows $\alpha^*/N$. For $\alpha\to 0$, the standard deviation of $\sigma/\sigma_{\rm th}$ (shaded bands) are amplified because also $\sigma_{\rm th}\to 0$.
• Figure 4: Estimation of the entropy production in a deterministic chemical reaction network.a. Estimates \ref{['tilde-emp']} and \ref{['tilde-hyp']} normalized to the true $\sigma_{\rm th}$ as a function of macroscopic time $\tilde{t}$ up to the inverse of the maximum flux, for the chemical reaction network \ref{['eq:CRN']} with forward rate constants $k_{+r}= \{5,2,1,0.2\}$ and backward $k_{-r} =10^{-5}$. b. With $2000$ random realizations of the chemical reaction network (uniformly sampled rate constants $k_{+r}\in [10^{-2},10^2]$, and $k_{-r} =10^{-5}$), estimate \ref{['tilde-emp']} vs. \ref{['tilde-hyp']}, both normalized to the true $\tilde{\sigma}_\text{th}$, for a fixed time $\tilde{t}=0.1/ \max(\omega_r(x^*))$; c. probability distribution of $\tilde{t}_q$ for $q=0.6$, and d$q=0.8$.