Inferring Entropy Production from Short Experiments

We provide a strategy for the exact inference of the average as well as the fluctuations of the entropy production in nonequilibrium systems in the steady state, from the measurements of arbitrary current fluctuations. Our results are built upon the finite-time generalization of the thermodynamic uncertainty relation, and require only very short time series data from experiments. We illustrate our results with exact and numerical solutions for two colloidal heat engines.

Figure 1

An illustration of the exact estimation of the entropy production rate $\sigma$, using the $\tau \to 0$ limit of Eq. (1) for two colloidal engine models in nonequilibrium steady states. Left: $\sigma$ inferred as a function of time, for the Brownian gyrator model, with analytic solutions for ${\sigma }_L$ in Eq. (1). The black horizontal line corresponds to the actual entropy production rate. The red solid and blue dotted lines correspond to arbitrary currents with and without boundary contributions (see the main text). The results show that the best inference of $\sigma$ is given by $\mathrm{\Delta }{S}_{\text{tot}}$ (green dashed line) itself, and that several of the currents having boundary contributions (red solid lines) infer $\sigma$ arbitrarily close to the actual value, in the $\tau \to 0$ limit. Right: Inferring $\sigma$, as the maximum of the measured ${\sigma }_L$’s of $N$ arbitrary currents [Eq. (6)], for the isothermal work-to-work converter engine, from numerical simulations. Here the black dashed line corresponds to the actual $\sigma$, obtainable from a large-time computation [45]. ${\sigma }_L^N$ corresponds to the maximum ${\sigma }_L$ inferred by the $N$ currents, at $\tau \to 0$. We see that as $N$ increases, ${\sigma }_L^N$ saturates to the known value of $\sigma$. The inset shows that the inference procedure makes a large error if the boundary terms are not included.

Figure 2

Entropy production rate inferred in the $\tau \to 0$ limit, for the Brownian gyrator model. Arbitrary currents are constructed as linear combinations of the basis currents $Q_1$, $W$, and $\mathrm{\Delta }{S}_{\mathrm{int}}$, with coefficients $c_1$, $c_2$, and $c_3$. The green line corresponds to the $\mathrm{\Delta }{S}_{\text{tot}}$ current direction. In the scale provided, red corresponds to a more accurate inference and blue to an inaccurate inference. It is found that, for nonzero values of $c_3$, the best inference is given by the $\mathrm{\Delta }{S}_{\text{tot}}$ current itself. However, the optimal current differs from the corresponding $\mathrm{\Delta }{S}_{\text{tot}}$, when we set $c_3=0$ (see Supplemental Material [46]).

Figure 3

Results for the isothermal work-to-work converter engine. (a) $\mathrm{\Phi }({\lambda }_Q,{\lambda }_W,{\lambda }_S,\tau )$ obtained analytically. $\mathrm{\Phi }$ has a limited domain of convergence (only a few ${\lambda }_S$ planes are shown for clarity), and also displays the fluctuation theorem symmetry around the point (0.5,0.5,0.5). In (b) and (c), the thick green curve is the actual ${\varphi }_{\mathrm{\Delta }{S}_{\text{tot}}}$. In (b), the blue solid and red dashed curves, respectively, correspond to ${\varphi }_J^{\sigma }$ with $c_3=0$ and $c_3\ne 0$ case. In (c), ${\varphi }_{\mathrm{\Delta }{S}_{\text{tot}}}$ is obtained using Eq. (8), where red squares and blue circles, respectively, correspond to $c_3\ne 0$ and $c_3=0$ at large $\tau$ [46]. (d) Inferring $\mathrm{Var}(\mathrm{\Delta }{S}_{\text{tot}})$, as the minimum of the measured variances of $N$ scaled, arbitrary currents, $\sigma \tau J/⟨J⟩$ according to Eq. (9). Here the black line corresponds to the variance of $\mathrm{\Delta }{S}_{\text{tot}}$, in the long time limit [45].