How Can We Measure Entropy When We See Only Part of the System?

Entropy production is the most universal gauge of how far a system is from equilibrium, especially when it interacts with a thermal bath. Imagine a cell or a moving gel droplet: entropy tells us how much "disorder" is inside. The catch is that, in tiny systems, fluctuations hide the true value and we cannot observe every variable that contributes. This is where coarse‑graining comes in – we watch only a few coordinates, like a particle's position, and try to infer entropy from these partial data. Over the past decade, stochastic thermodynamics has offered several inequalities that give lower bounds on entropy production without needing the full model. One approach uses transition probabilities between observed macrostates; another relies on currents that emerge from time series. We can also examine correlation functions of coarse‑grained observables or the waiting‑time distributions between "marked events," i.e., transitions between mesoscopic states. All these methods assume the underlying dynamics is Markovian, i.e., memoryless, even if it’s not directly observable. The review covers both steady non‑equilibrium states and systems that evolve over time or relax toward equilibrium. Besides average entropy, it also discusses estimating entropy along individual trajectories, opening new experimental possibilities in biology and soft matter.

The lower bound on entropy production derived from coarse‑graining is tightly linked to the thermodynamic uncertainty relation (TUR), which connects the variance of an observed current to the mean entropy value. Formally, for a current J measured over a time interval τ, one has \(\frac{\mathrm{Var}(J)}{\langle J\rangle^2} \ge \frac{2}{\langle \Sigma\rangle}\), where \(\langle \Sigma\rangle\) is the average entropy production. This inequality holds for continuous or discrete Markovian dynamics and offers a practical way to estimate a lower bound by measuring only the current's fluctuations. Another class of results stems from fluctuation theorems (FT), which relate the probability of observing a trajectory to that of its time‑reversed counterpart. When only coarse‑grained variables are observed, the FT becomes an "information FT" that includes an information‑loss term due to coarse‑graining. Moreover, waiting‑time distributions between marked events can be analyzed via first‑passage process theory, providing entropy estimates based on interval statistics. Recent studies have extended these approaches to non‑stationary systems, introducing conditioned path entropies and non‑equilibrium response functions. In practice, the method choice depends on experimental data: high‑resolution time series favor currents and FT analyses; sparse or discrete event data make waiting‑time distributions a robust alternative. Combining multiple techniques can further improve estimate precision, reducing coarse‑graining uncertainty and bringing experimental measurements closer to theoretical entropy production values.