The engine of order runs right at the edge of chaos: push 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 worthwhile it is to spend energy maintaining order far from equilibrium. A new paper is the first to carry an information-based measure of efficiency over to systems that are constantly being pushed.

The authors took two lattice models with different mechanisms for pushing 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, particles with spins hop across the lattice with an asymmetry that depends on the spin direction, which models 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 unexpectedly clear: in both models, thermodynamic efficiency peaks near the critical point, even when the system sits in a nonequilibrium steady state. For the persistent model under strong bias, the critical value of the coupling strength is 1.2315, and the efficiency peak falls exactly there, just as in the equilibrium case. Inferential efficiency, meanwhile, 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 stronger driving, which serves as a phenomenological signature of the violation of the fluctuation-dissipation theorem.

This work stands out from its neighbors in the archive because it applies an information-theoretic measure of efficiency to systems with an explicit violation of detailed balance, rather than to equilibrium or quasi-equilibrium models. It is the first test of whether the principle of maximum efficiency at the critical point survives in nonequilibrium steady states as well. What is more, the gap between the two measures of efficiency offers a simple way to estimate how far a system has been driven from equilibrium without resorting to complicated dynamical correlators. Such an approach could be useful for analyzing experimental data on active media, where the fluctuation-dissipation relation is often violated.

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