The engine of order runs on the edge of chaos: nudge it a little harder and instead of neat structures you get a whirlwind of randomness. Physicists have long suspected that living systems, from flocks of birds to bacterial colonies, choose exactly this edge, but they could not measure how advantageous it is to spend energy on maintaining order away from equilibrium. A new study is the first to bring an information-based measure of efficiency to systems that are constantly being pushed.

The authors took two lattice models with different mechanisms for driving them out of equilibrium. In the persistent Ising model, a constant bias is added to the standard spin-flip rule, either suppressing or stimulating transitions. In the active model, spins hop across the lattice with an asymmetry that depends on spin direction, mimicking self-propulsion. For each model, thermodynamic efficiency is computed as the ratio of the change in Shannon entropy to the generalized work under a small perturbation of the coupling strength. In parallel, inferential efficiency is calculated, which uses equal-time covariances of observables instead of the response to a perturbation.

The result turned out to be strikingly clear: in both models, thermodynamic efficiency reaches a maximum near the critical point, even when the system sits in a nonequilibrium steady state. For the persistent model under strong bias, the critical coupling strength is 1.2315, and the efficiency peak falls exactly there, just as in the equilibrium case. Inferential efficiency, however, behaves asymmetrically: when fluctuations are suppressed it is lower than thermodynamic efficiency, and when they are stimulated it is higher. The ratio of the maxima grows with drive strength, which serves as a phenomenological signature of the violation of the fluctuation-dissipation theorem.

Until now it was unknown whether the maximum of thermodynamic efficiency at the critical point survives in systems driven out of equilibrium by a constant drive or self-propulsion. Now it has been shown that in two different nonequilibrium models efficiency really does peak near the phase transition, and the gap between thermodynamic and inferential efficiency quantitatively reflects the degree of broken detailed balance. This was achieved by comparing two measures of efficiency, the response to a perturbation and equal-time covariances, in numerical simulations of nonequilibrium spin models. This approach can be used to estimate the energetic cost of self-organization in biological and active systems, and also to diagnose departure from equilibrium from static observables. The engine of order is again running on the edge of chaos, but now we know how to measure its efficiency.