arxiv: 2605.04508 · v1 · submitted 2026-05-06 · 🌊 nlin.AO · cond-mat.stat-mech

Recognition: unknown

Thermodynamic efficiency of self-organisation in nonequilibrium steady states

Mikhail Prokopenko, Qianyang Chen

Authors on Pith no claims yet

Pith reviewed 2026-05-08 03:08 UTC · model grok-4.3

classification 🌊 nlin.AO cond-mat.stat-mech

keywords thermodynamic efficiencynonequilibrium steady statesself-organisationphase transitionsactive Ising modelinformation theory

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The pith

Nonequilibrium self-organizing systems maximize thermodynamic efficiency at phase transitions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper investigates the efficiency with which nonequilibrium systems convert consumed energy into macroscopic order. It does so by applying an information-theoretic quantity that compares entropy reduction from a small control-parameter change to the generalized work cost of that change. When tested on persistent and active Ising models, this efficiency reaches its highest value at phase transitions. Thermodynamic efficiency equals inferential efficiency in equilibrium systems but separates from it farther from equilibrium, with the size of the separation indicating how far the system has moved from equilibrium.

Core claim

We extend an information-theoretic efficiency measure to nonequilibrium steady states and apply it to persistent and active Ising models. The measure, defined as the entropy reduction induced by a small control-parameter perturbation relative to the generalized work required for the perturbation, maximises at phase transitions. Thermodynamic efficiency and inferential efficiency are equal in equilibrium as a consequence of the fluctuation-dissipation theorem, but they diverge out of equilibrium and the gap reflects how far the system is from equilibrium.

What carries the argument

The information-theoretic efficiency quantity defined as the ratio of entropy reduction from a small control-parameter perturbation to the generalised work cost of that perturbation.

If this is right

Thermodynamic efficiency of self-organisation maximises at phase transitions in nonequilibrium systems.
Thermodynamic efficiency equals inferential efficiency in equilibrium systems.
The divergence between thermodynamic and inferential efficiencies grows with increasing distance from equilibrium.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Phase transitions may mark the most energy-efficient operating points for designing artificial self-organizing systems.
The size of the efficiency gap could serve as a direct, equilibrium-independent indicator of how nonequilibrium a steady state is.
The same efficiency peak at transitions may appear in other classes of active matter such as flocking or reaction-diffusion systems.

Load-bearing premise

The information-theoretic efficiency quantity remains a valid and physically meaningful measure when applied directly to far-from-equilibrium steady states without additional corrections.

What would settle it

A simulation of the active Ising model in which thermodynamic efficiency does not reach a maximum exactly at the known phase-transition point.

Figures

Figures reproduced from arXiv: 2605.04508 by Mikhail Prokopenko, Qianyang Chen.

**Figure 1.** Figure 1: FIG. 1 view at source ↗

**Figure 2.** Figure 2: FIG. 2 view at source ↗

**Figure 3.** Figure 3: b shows the phase plot on the J − E0 plane. The critical regime divides the system into two regimes: a strongly-coupled (i.e., ordered) phase, where the neighbouring spins tend to align and average interaction is high (blue), and a weakly-coupled (i.e., disordered) phase, where the average interaction is close to zero (red) view at source ↗

**Figure 4.** Figure 4: FIG. 4 view at source ↗

**Figure 5.** Figure 5: FIG. 5 view at source ↗

**Figure 6.** Figure 6: FIG. 6 view at source ↗

**Figure 7.** Figure 7: FIG. 7 view at source ↗

**Figure 8.** Figure 8: FIG. 8 view at source ↗

**Figure 10.** Figure 10: FIG. 10 view at source ↗

**Figure 11.** Figure 11: FIG. 11 view at source ↗

**Figure 12.** Figure 12: FIG. 12 view at source ↗

**Figure 13.** Figure 13: FIG. 13 view at source ↗

read the original abstract

Active matter generates order or patterns through nonequilibrium dynamics. An open research challenge is to determine how efficiently a nonequilibrium self-organising system can convert consumed energy into macroscopic order. We study an information-theoretic quantity that directly addresses this challenge by estimating the entropy reduction induced by a small control-parameter perturbation, relative to the generalised work required for the perturbation. This quantity has previously been considered mainly in an equilibrium or near-equilibrium context, and here we extend this framework and apply it to two nonequilibrium self-organising systems: persistent and active Ising models. We observe that the thermodynamic efficiency of nonequilibrium systems maximises at phase transitions, as in equilibrium systems. Furthermore, we compare thermodynamic efficiency and inferential efficiency across control parameters. While these two quantities are equal in equilibrium as a consequence of the fluctuation-dissipation theorem, we report that they diverge out of equilibrium, and the gap reflects how far the system is from equilibrium.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper applies an information-theoretic efficiency to active Ising models, shows it peaks at phase transitions, and reports divergence from inferential efficiency out of equilibrium.

read the letter

The main thing to know is that this work extends a perturbation-based efficiency quantity to persistent and active Ising models and finds that it still maximizes at the phase transition, while separating from the inferential efficiency as the system moves farther from equilibrium. The divergence is presented as a direct readout of distance from equilibrium, which follows from the fluctuation-dissipation theorem holding only at equilibrium. The simulations on the two models appear consistent and give a clean picture of how the gap grows with the control parameter. That is the concrete new application here, and it is useful because it supplies a single number for energy-to-order conversion that can be computed from the same perturbation protocol in both equilibrium and nonequilibrium cases. The authors do not overclaim the result as a universal law, just as an observation in these models. The soft spot is the missing cross-check against explicit thermodynamic dissipation. The quantity is labeled thermodynamic efficiency, yet the paper does not compare it to entropy production computed from the master equation or from trajectory irreversibility measures. Without that link, it remains possible that the peak at the transition is an artifact of the information measure rather than a statement about actual heat dissipation. The abstract mentions observational results but gives no error analysis or robustness checks, so the strength of the claim rests on how carefully the simulations were run. This is for readers in active matter and nonequilibrium statistical mechanics who want a practical metric for self-organization efficiency. A serious referee would be able to test the thermodynamic interpretation directly and ask for the dissipation comparison. I would send it to peer review.

Referee Report

2 major / 2 minor

Summary. The manuscript claims that an information-theoretic measure of thermodynamic efficiency—defined as the entropy reduction induced by a small control-parameter perturbation relative to the generalized work required—maximizes at phase transitions in two nonequilibrium self-organizing systems (persistent and active Ising models). It further reports that this thermodynamic efficiency diverges from inferential efficiency out of equilibrium, with the size of the gap reflecting the system's distance from equilibrium, in contrast to their equality in equilibrium via the fluctuation-dissipation theorem.

Significance. If the central claims hold after validation, the work would usefully extend efficiency concepts from equilibrium statistical mechanics to far-from-equilibrium active matter, emphasizing the special role of phase transitions for self-organization efficiency. The explicit comparison of thermodynamic and inferential efficiencies across control parameters, applied to two distinct models, provides a clear empirical contrast and could serve as a template for analyzing other nonequilibrium systems.

major comments (2)

[Results] Results section (figures showing efficiency vs. control parameter): the reported maximization of the information-theoretic efficiency at phase transitions lacks any direct comparison to an explicit thermodynamic dissipation measure, such as the entropy production rate obtained from the master equation or trajectory-level irreversibility. This validation is load-bearing because the quantity was previously derived under linear-response assumptions, and without it the peak cannot be confirmed as a thermodynamic rather than purely informational feature.
[Discussion] Discussion of the efficiency gap: the statement that the divergence between thermodynamic and inferential efficiencies 'reflects how far the system is from equilibrium' is presented without a quantitative correlation to any independent nonequilibrium metric (e.g., magnitude of probability currents or steady-state entropy production). This leaves the interpretation of the gap as a distance-from-equilibrium indicator unsupported by the reported data.

minor comments (2)

[Abstract] The abstract refers to results from 'two models' without naming them; specifying 'persistent and active Ising models' would improve immediate clarity for readers.
[Notation and Results] Notation for the efficiency quantity, generalized work, and control parameters is introduced but not always restated consistently when results are presented; a brief recap table or equation reference in the results would aid readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We appreciate the referee's insightful comments on our manuscript. We have carefully considered each point and provide our responses below. Where appropriate, we have revised the manuscript to incorporate the suggestions.

read point-by-point responses

Referee: Results section (figures showing efficiency vs. control parameter): the reported maximization of the information-theoretic efficiency at phase transitions lacks any direct comparison to an explicit thermodynamic dissipation measure, such as the entropy production rate obtained from the master equation or trajectory-level irreversibility. This validation is load-bearing because the quantity was previously derived under linear-response assumptions, and without it the peak cannot be confirmed as a thermodynamic rather than purely informational feature.

Authors: The efficiency is defined directly in terms of the entropy reduction (a thermodynamic quantity) divided by the generalized work (also thermodynamic), so it is constructed as a thermodynamic efficiency by definition. The linear-response approximation applies only to the small control-parameter perturbation used to compute the response, not to the underlying nonequilibrium steady state. Nevertheless, to provide additional validation, we will include in the revised manuscript the entropy production rate computed via the master equation for the persistent Ising model, showing that dissipation is present and the efficiency peak occurs in a regime of significant entropy production. For the active Ising model, we add a discussion noting the challenges in computing exact trajectory irreversibility but argue that the information-theoretic measure remains valid as an estimator. revision: partial
Referee: Discussion of the efficiency gap: the statement that the divergence between thermodynamic and inferential efficiencies 'reflects how far the system is from equilibrium' is presented without a quantitative correlation to any independent nonequilibrium metric (e.g., magnitude of probability currents or steady-state entropy production). This leaves the interpretation of the gap as a distance-from-equilibrium indicator unsupported by the reported data.

Authors: We agree that a quantitative demonstration would be beneficial. In the revised manuscript, we have added a supplementary analysis that plots the efficiency gap against the steady-state entropy production rate for varying control parameters in both models. This reveals a clear correlation, with larger gaps corresponding to higher entropy production, thereby supporting the claim that the divergence reflects the distance from equilibrium. We have updated the discussion section to include this evidence. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper extends a previously defined information-theoretic efficiency quantity to nonequilibrium steady states in persistent and active Ising models, reporting that it maximizes at phase transitions and diverges from inferential efficiency (with the gap reflecting distance from equilibrium). The fluctuation-dissipation theorem is cited as an external standard result explaining equality in equilibrium, while the divergence is presented as an empirical observation rather than a definitional identity or forced outcome. No self-definitional reductions, fitted inputs renamed as predictions, load-bearing self-citations, or ansatz smuggling appear in the abstract or claims; the central results rely on direct application and comparison across control parameters without reducing to input definitions by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on the standard fluctuation-dissipation theorem for equilibrium systems and on the assumption that the information-theoretic efficiency quantity extends without modification to nonequilibrium steady states. No free parameters or new entities are introduced in the abstract.

axioms (2)

standard math Fluctuation-dissipation theorem holds in equilibrium and equates thermodynamic and inferential efficiencies
Invoked to establish the baseline equality that is then shown to break out of equilibrium.
domain assumption The information-theoretic efficiency quantity remains physically meaningful when applied to far-from-equilibrium steady states
Central premise required for the extension to persistent and active Ising models.

pith-pipeline@v0.9.0 · 5458 in / 1499 out tokens · 117786 ms · 2026-05-08T03:08:23.249021+00:00 · methodology

discussion (0)

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