The question of how efficient self-organization is has long occupied biophysicists: how do cells, flocks of birds, or bacterial colonies turn chaotic energy consumption into ordered structures? Alongside this paper in the archive sit studies of heat engines, quantum batteries, and Maxwell's demon, but most of them either stay at equilibrium or measure dissipation rather than useful output. This article stands apart: it is the first to carry an information-theoretic measure of efficiency, previously tested only at equilibrium, over to two different mechanisms for driving a system out of equilibrium. That makes it possible to compare how a constant external drive or the self-propulsion of particles changes a system's ability to turn work into order.
The authors take two lattice models with different mechanisms for breaking detailed balance. In the persistent Ising model, a constant bias E0 is added to the standard Glauber spin-flip rule, either suppressing or stimulating transitions. In the active Ising model, particles with spins hop across the lattice with an asymmetry that depends on spin direction, which models self-propulsion. For each model, thermodynamic efficiency η is computed as the ratio of the change in Shannon entropy to the generalized work under a small perturbation of the control parameter (the coupling strength J). In parallel, inferential efficiency η▷◁ is calculated, which uses equal-time covariances of observables instead of the response to a perturbation. Numerical Monte Carlo simulations on lattices up to 200×200 give critical values Jc from the crossing of the Binder cumulant, and entropy is estimated via the Kikuchi approximation.
In both models, thermodynamic efficiency η peaks near the critical point, even when the system sits in a nonequilibrium steady state. For the persistent Ising model at E0 = −4, the critical value is Jc ≈ 1.2315, and the peak of η falls exactly there, just as in the equilibrium case (E0 = 0, Jc ≈ 0.4407). Meanwhile inferential efficiency η▷◁ behaves asymmetrically: for E0 > 0 (suppressed fluctuations) it lies below η, while for E0 < 0 (stimulated fluctuations) it lies above. The ratio of the maxima η▷◁/η grows as the drive strengthens, which serves as a phenomenological signature of the violation of the fluctuation-dissipation theorem. In the active Ising model at zero external field, η and η▷◁ also peak at the transition from the homogeneous disordered state to the coexistence state (Jc ≈ 1.5922), and η▷◁ shows an additional peak at the second transition into the homogeneous bistable state, one that the refined mean-field approximation fails to capture. The entropy production rate, unlike efficiency, is not maximal at the critical point: its derivative with respect to J has a sharp minimum at the transition, which points to an abrupt drop in dissipation as order emerges.
This work differs from its neighbors in the archive by applying 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 η and η▷◁ offers a simple way to gauge how far a system has been pushed from equilibrium without resorting to complicated dynamical correlators. Such an approach could prove useful for analyzing experimental data on active media, where the fluctuation-dissipation relation is often violated.
The natural next step is to consider time-dependent drives, in order to move beyond steady states and study transient processes, hysteresis, and path dependence. Another direction is to introduce heterogeneity among agents: different coupling strengths, local rules, or individual responses to the drive. That will require redefining generalized work through energy changes along a trajectory, since no global Hamiltonian exists for such systems. It would also be interesting to check whether the efficiency peak at the critical point survives in models with nonreciprocal interactions, which are currently a hot topic in statistical physics.
The results could be useful in biophysics for estimating the energetic cost of morphogenesis or collective motion, where systems operate in a critical regime. In materials science, they suggest how to control active media so as to achieve maximum structural order at minimum cost.
In the near term, the analysis should be extended to models with time-dependent drives and to systems with heterogeneous agents, in order to test how robust the discovered efficiency peak is. It is also important to develop methods for estimating entropy and work in transient regimes, where steady-state assumptions do not hold.
For the statistical physics of nonequilibrium systems, the question of which universal principles govern the efficiency of converting energy into order far from equilibrium remains open. This work links that question to the violation of the fluctuation-dissipation theorem and shows that criticality may be a consequence of optimal conversion of work into structure, not merely a singularity in dissipative fluxes.
At E0 = −4 in the persistent Ising model, the system behaves as if it were under the influence of an infinitely high temperature: the spin-flip rate approaches one, and fluctuations surge sharply, even though thermodynamic efficiency remains low.
The idea that complex systems self-organize into a critical state echoes the concept of "self-organized criticality" from Stanisław Lem's novel Solaris, where an ocean-planet constantly balances on the edge of order and chaos, spending energy to create fleeting structures.