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REVIEW 4 major objections 4 minor 77 references

Large Interconnected Thermodynamic Systems Nearly Minimize Entropy Production

T0 review · 4 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Large driven nonequilibrium systems typically settle close to the minimum entropy production state, though neither MINEP nor MAXEP holds in general.

desk verdict A clean MEPS derivation plus a suggestive but under-characterized numerical scaling; the title outruns the evidence. read the letter →

arxiv 2507.10476 v2 pith:FH2YEFPM submitted 2025-07-14 cond-mat.stat-mech

Kyle J. Ray , Alexander B. Boyd This is my paper · ORCID
classification cond-mat.stat-mech MSC 60J2782C0582C31 PACS 05.70.Ln
keywords minimumentropyproductionstochasticthermodynamicscontinuous-timeMarkovchainsnonequilibriumsteadystaterateArrheniuskineticsthermodynamicefficiencyself-organization
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper asks whether nonequilibrium systems maximize or minimize entropy production, a question with a century-long history. It derives, for any continuous-time Markov chain, the distribution that minimizes entropy production—the minimum entropy production state (MEPS)—even far from equilibrium, and shows that the actual nonequilibrium steady state generally violates both classical MINEP and MAXEP. The central numerical result is that for randomly generated pumped Arrhenius networks, the steady-state entropy production rate moves toward the MEPS as the number of states grows: at $N=729$, the scaled excess over the minimum is about $5\times 10^{-2}\,k_B$ even when the minimum itself is around $k_B$. The authors read this as evidence that large interconnected thermodynamic systems self-organize toward efficient use of thermodynamic resources, a 'soft' form of MINEP.

What carries the argument

The central object is the minimum entropy production state (MEPS), $m_R \equiv \arg\min_p \sigma(R,p)$, obtained by Lagrange multipliers on the CTMC entropy production rate $\sigma(R,p)=\sum_{s,s'}p(s)R_{s\to s'} \ln\big(p(s)R_{s\to s'}/p(s')R_{s'\to s}\big)$. The variational condition, $\partial_t\ln m_R(s)=\sigma(R,m_R,s)-\sigma(R,m_R)$, defines the minimizer and also gives a relaxation dynamics that can be simulated. The paper uses this state as the benchmark: comparing the steady state's entropy production to the MEPS through the scaled excess EPR turns the MAXEP/MINEP question into a quantitative, finite-size question.

What would settle it

Generate large ($N\ge 243$) pumped Arrhenius networks from a different physically motivated ensemble—for example, sparse connectivity, correlated energy landscapes, or heavy-tailed barriers—and compute the scaled excess $(\sigma_{\mathrm{NESS}}-\sigma_{\mathrm{MEPS}})/\sigma_{\mathrm{MEPS}}$; if it stays of order one rather than dropping to $\sim 10^{-2}$ as $N$ grows, the claimed typicality of near-minimum entropy production is an artifact of the original sampling scheme.

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Extended reading notes

Core claim

Using stochastic thermodynamics, the authors derive the condition for the minimum entropy production state $m_R$ of a rate matrix $R$: at the minimizer, $\partial_t \ln m_R(s) = \sigma(R, m_R, s) - \sigma(R, m_R)$, where $\sigma(R, p, s)$ is the state-wise contribution to the entropy production rate. The nonequilibrium steady state $\pi$ equals $m_R$ only if every state dissipates the same entropy; otherwise the steady state dissipates more than the minimum. The paper also shows that MAXEP is generically a trivial and unreachable limit, since placing zero probability on a state makes the system-entropy term diverge. The main discovery, supported by numerical sampling of ensembles with $N$ up to 729 states, pump strengths from 25% to 500%, and pump fractions from 5% to 80%, is that the scaled excess $(\sigma_{\mathrm{NESS}}-\sigma_{\mathrm{MEPS}})/\sigma_{\mathrm{MEPS}}$ shrinks as the interconnected system grows, so large driven systems nearly achieve the minimum even when driven far from equilibrium.

Load-bearing premise

The numerical typicality claim rests on the assumption that the random ensemble of uniformly sampled energies, barriers, and pump forces represents thermodynamic systems in general; the authors concede that the sampling algorithm indirectly determines the statistics of entropy production, so a different physically motivated ensemble could weaken or eliminate the near-minimum trend.

Editorial extensions

If this is right

  • For large pumped CTMCs, the steady state typically dissipates roughly an order of magnitude less entropy than uniform or randomly sampled distributions, and the gap grows with $N$.
  • Classical MINEP fails exactly: the steady state is not the MEPS unless state-wise dissipation is uniform. Classical MAXEP fails even more severely, typically giving infinite entropy production.
  • Near-minimality holds across pump strengths from 25% to 500% and pump fractions from 5% to 80%, so it is not an artifact of weakly driven systems.
  • The results suggest a 'soft' MINEP principle: large nonequilibrium systems spontaneously lower dissipation rather than maximize it, which the authors connect to complexity and thermodynamic efficiency.
  • The MEPS relaxation dynamics provides a practical numerical method for locating thermodynamically optimal distributions of a given rate matrix.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper leaves open whether the near-minimality is a general mathematical property of high-dimensional rate matrices; a natural next step is to derive the scaling of $(\sigma_{\mathrm{NESS}}-\sigma_{\mathrm{MEPS}})/\sigma_{\mathrm{MEPS}}$ with $N$ from random matrix theory, which the authors list as future work.
  • If the typicality claim transfers to physical settings, engineered or biological networks with many states should generically operate close to minimal dissipation; one could test this in synthetic gene circuits or multi-level quantum heat engines by measuring steady-state heat currents.
  • The comparison to uniform and Dirichlet-random distributions suggests that the volume of the probability simplex is dominated by high-dissipation states, so the NESS sitting near the MEPS is a concentration phenomenon rather than a trivial bound.
  • The MEPS equation has the form of a replicator-style learning dynamics; the paper's size trend hints that large systems look more like adaptive learners, but connecting this to specific learning or inference algorithms remains speculative.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The paper studies whether nonequilibrium steady states (NESS) of continuous-time Markov chains tend to maximize or minimize entropy production. It derives a first-order condition for the minimum-entropy-production state (MEPS) via Lagrange multipliers, argues that MAXEP is generally ill-posed because distributions with zero-probability states can produce infinite instantaneous entropy, and compares NESS entropy production with the MEPS value for a three-state pumped Arrhenius model and for randomly generated pumped networks of increasing size. The central empirical claim is that, for large interconnected random networks, the steady-state entropy production is only slightly above the minimum, suggesting a form of 'soft MINEP' or typical near-minimization in large systems.

Significance. If the numerical typicality claim is statistically robust, the paper would be a useful contribution to the long-running MINEP/MAXEP debate: it gives a clean analytical characterization of the MEPS for general CTMCs, a simple explicit counterexample to exact MINEP, and a concrete scaling observation for large networks. The derivation in Section IV and Appendix A is algebraically clean, the MAXEP discussion is conceptually clear, the code is publicly available, and the paper is appropriately cautious in framing the result as typicality rather than a theorem. However, the headline claim rests on simulations that are currently under-specified statistically, and the manuscript itself acknowledges that the sampling protocol shapes the entropy-production statistics; the significance of the paper therefore depends on whether the scaling trend survives more thorough numerical analysis and alternative ensembles.

major comments (4)
  1. [§VI.C, Figs. 5–6, Eq. (25)] The central claim that large systems nearly minimize entropy production is supported only by numerical clusters with no stated sample counts, no error bars, and no quantiles. The text reports that for N=729 the scaled excess is about 5e-2 kB, but it does not say how many rate matrices were drawn, whether the plotted points are individual realizations or aggregated statistics, or how the spread varies across N. Since the conclusion is explicitly a typicality statement ('According to these samples...'), the authors should report sample sizes, interval estimates, and, ideally, a convergence diagnostic as N increases. Without this, the apparent sharpening of the distribution at N=729 cannot be distinguished from sampling noise.
  2. [§VI.C sampling protocol and §VII item 2] The numerical ensemble is a single generative model: complete graphs, energies uniform in [-1,1] k_B T, barrier offsets uniform in [0,1] k_B T, pump forces uniform in [-alpha,alpha] k_B T, and a few pump fractions. The authors themselves write in §VII, item 2, that the sampling algorithm 'indirectly determine[s] the statistics of entropy production.' Because the title generalizes to thermodynamic processes, the paper should test whether the near-minimum trend is robust across physically motivated ensembles—for example, sparse or structured topologies, fixed cycle affinities, correlated energy barriers, or different rules for placing pumps. If an alternative ensemble yields larger scaled excess entropy production at large N, the headline claim would not survive; a robustness check is therefore load-bearing, not cosmetic.
  3. [Fig. 6 and Eq. (25)] The scaled excess EPR defined in Eq. (25) is normalized by sigma_MEPS, and Fig. 6 shows that sigma_MEPS itself grows with N. A decrease in sigma_NESS/sigma_MEPS - 1 can therefore reflect growth of the denominator rather than an absolute approach to the minimum. To support the wording 'converges toward the minimum' and 'nearly minimize entropy production,' the authors should also report the distribution of the absolute gap sigma_NESS - sigma_MEPS, or otherwise argue that the scaling is governed by the numerator and not by the normalization. As written, Fig. 6 mixes the two effects.
  4. [Appendix A, Eqs. (A1)–(A10)] The Lagrange-multiplier derivation yields a stationarity condition for sigma(R,p) in p, but the paper does not establish that sigma(R,p) is convex in p, that the stationary point is the global minimizer, or that the fixed point of Eq. (9) is attracting. The name MEPS, and the use of sigma_NESS - sigma_MEPS as a non-negative excess in Eq. (25), presuppose that the computed state is the actual minimum. If convexity is a standard result, the authors should state it and provide a citation or a short argument; otherwise the derivation needs a separate justification that the first-order condition is sufficient.
minor comments (4)
  1. [Eq. (23)] The second force in Eq. (23) is written as F_{s'->s'}, which appears to be a typo for F_{s'->s}; the logarithmic ratio of forward and reverse rates in Eq. (22) involves the force on the reverse transition, not a force on the same state.
  2. [Author footnote and abstract] There are typographical errors in the front matter: 'Corresponding Authoor' in the author footnote and 'entr´ee' in Section VIII; a spellcheck pass would catch these.
  3. [§VI.C and Fig. 5 caption] The text says pump strength alpha ranges from 25% to 500%, while the Fig. 5 caption reports 'pump strength = 400%'; because alpha is defined through F_{s->s'} in units of k_B T, the percentage notation needs a clear definition, and the relationship between the percentage and alpha should be stated once in the text.
  4. [§VI.C and Figs. 5–6] The entropy production values are quoted in units of k_B, but sigma(R,p) has dimensions of inverse time; since the pre-factor K is not specified in the sampling protocol, the authors should state the time normalization (e.g., K=1) so that the reported numbers are reproducible.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the MEPS condition is derived directly from the definition of entropy production, the NESS is computed independently, and the numerical typicality claim is an acknowledged extrapolation from a stated sampling ensemble.

full rationale

The derivation chain is self-contained. The MEPS condition (Eq. 8, Appendix A) follows from the definition of σ(R,p) via Lagrange multipliers, while the NESS is independently defined by Rπ=0 and Eq. 4. Equation 25 directly compares two quantities computed from the same rate matrix, with no parameter fitted to the near-minimum behavior. The Section VI.C claim that large systems nearly minimize entropy production is presented as a numerical observation from a specified random ensemble (uniform energies, barriers, and pump forces), and the authors explicitly flag in Section VII item 2 that their sampling algorithm 'indirectly determine[s] the statistics of entropy production.' That is a limitation on typicality and generalization, not a circular reduction: the NESS and MEPS are not defined in terms of each other, and no fitted quantity is renamed as a prediction. Self-citations (e.g., Refs. [34], [35], [43], [46], [65]) appear in contextual, motivational, or future-work passages and are not load-bearing for the central derivation or the numerical claim. No circular step meets the evidentiary bar.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

These are the hand-chosen ensemble and modeling assumptions the central scaling claim depends on; none are fitted to the target result.

free parameters (4)
  • Energy distribution range for E_s = uniform in [-1,1] kBT
    State energies are sampled uniformly in this range; the range defines the ensemble of typical systems and affects the resulting entropy production statistics.
  • Barrier offset distribution E_{s<->s'} = uniform in [0,1] kBT
    Barrier offsets are sampled uniformly, setting the kinetic asymmetry of the rate matrix in the ensemble.
  • Pump strength alpha = varied from 25% to 500% of kBT
    The maximum pump force F in [-alpha, alpha] kBT is hand-chosen and the reported scaling behavior depends on it.
  • Pump fraction = 5%, 20%, and 80% of transitions
    The fraction of pumped transitions is chosen by hand in the simulations and is not fitted to a target.
assumptions (4)
  • domain assumption Continuous-time Markov chain rate matrix R has zero row sums and nonnegative off-diagonal rates, with time-reversal symmetric states, and the entropy production is given by Eq. (3).
    Needed for every formula and for the thermodynamic interpretation of the rate equations.
  • standard math The minimum entropy production state m_R is attained in the interior of the probability simplex, so the Lagrange multiplier method of Eq. (7) is valid.
    Section IV uses this without proving interiority; if the minimizer lay on the boundary, a different argument would be needed.
  • ad hoc to paper The random ensemble of energies, barriers, pump forces, pump fractions, and pump strengths represents typical nonequilibrium thermodynamic systems.
    This is the load-bearing assumption behind the scaling claim; the paper itself flags it as a limitation in Discussion item 2.
  • domain assumption Arrhenius rates with local detailed balance and pumped transitions, Eq. (21), capture the relevant class of nonequilibrium systems.
    Used to generate the numerical models and to guarantee the heat and entropy production interpretation of the rates.

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Pith. "Pith review of Large Interconnected Thermodynamic Systems Nearly Minimize Entropy Production." pith.science (2026). https://pith.science/paper/FH2YEFPM

@misc{pith2026250710476,
  author       = {Pith},
  title        = {Pith review of: Large Interconnected Thermodynamic Systems Nearly Minimize Entropy Production},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FH2YEFPM}},
  note         = {Machine review of arXiv:2507.10476}
}
read the original abstract

Many have speculated whether nonequilibrium systems obey principles of maximum or minimum entropy production. In this work, we use stochastic thermodynamics to derive the condition for the minimum entropy production state (MEPS) for continuous-time Markov chains (CTMCs), even far from equilibrium. We show that real nonequilibrium steady states (NESS) generally violate both the MINEP and MAXEP principles. However, through numerical sampling of large interconnected CTMCs, we find that as system size increases, the steady-state entropy production tends to converge toward the minimum. This suggests that large nonequilibrium systems may self-organize to make efficient use of thermodynamic resources, offering a nuanced perspective on the longstanding debate between MAXEP and MINEP.

Figures

Figures reproduced from arXiv: 2507.10476 by the authors.

Figure 1
Figure 1. A nonequilibrium network of thermal transitions (arrows) between states (blue circles): The states each have an energy. The energy difference between two states is the heat flow into the environment when a thermal transition is made between those states. A pumped transition, by contrast, has an additional external force that contributes to the heat production and the transition rates. Polettini proves that MINEP hol… view at source ↗
Figure 2
Figure 2. Three-state pumped nonequilibrium system: The en￾ergy landscape descends by a value ∆E from A to B, then by the same amount from B to C. This sets the energy landscape up to a constant factor, which would make the transition from C to A highly unlikely in equilibrium. The transition from C to A is pumped by an external force to create persistent nonequilibrium currents. This results in the heat dissipation ∆E ′ in t… view at source ↗
Figure 3
Figure 3. The entropy production rate of the rate equation [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: As we change the pumping parameter ∆E ′ , we see that the relationship between entropy production σ of the MEPS (blue) and NESS (green) changes. They are often close, for low and negative values of ∆E ′ , but the steady state entropy production diverges from the MEPS f…
Figure 5
Figure 5. Figure 5: As the number of states N of a nonequilibrium system increases, it is increasingly typical for the NESS to closely match the MEPS. Interconnected systems with many degrees of freedom appear to self-organize to minimize entropy production. We compare the scaled excess e…
Figure 6
Figure 6. Figure 6: We see that the minimum entropy production increases with the number of states (complexity) of the pumped model, [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

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Reference graph

Works this paper leans on

77 extracted references · 67 canonical work pages

  1. [28]

    Jarzynski, Journal of Statistical Physics 98, 77 (2000)

    C. Jarzynski, Journal of Statistical Physics 98, 77 (2000)

    work page 2000

  2. [1]

    Thus, highly symmetric nonequilibrium systems, for which every state produces the same average entropy in steady state, satisfy the MINEP condi- tion

    The MEPS evolves towards states that dissipate greater entropy than the average ( σ(R, mR, s) − σ(R, mR) > 0) and away from states that dissipate less than the average (σ(R, mR, s)−σ(R, mR) < 0). Thus, highly symmetric nonequilibrium systems, for which every state produces the same average entropy in steady state, satisfy the MINEP condi- tion

    work page

  3. [2]

    The MEPS and the stationary distribution ( ˙p = 0) are the same iff every state s produces the same amount of entropy in steady state σ(R, π, s) = σ(R, π)

    work page

  4. [3]

    local equilibrium

    We can discover the MEPS by setting a dynami- cal equation for the probability with computational time τ ∂τ p(s) = ∂tp(s) − p(s)(σ(R, p, s) − σ(R, p)). (9) When this equation of motion reaches steady state ∂τ p(s) = 0, the condition for minimum entropy production is reached. Applying this dynamic is a strategy for discovering thermodynamically opti- mal s...

    work page

  5. [4]

    qXm/WSCdvBMo7ye8ndTUxrbLlX4=

    Decomposing nonequilibrium systems into agent and environment: If a living agent is learning about its environment in a perception-action loop [64], 8 MEPS kB <latexit sha1_base64="qXm/WSCdvBMo7ye8ndTUxrbLlX4=">AAACB3icbVBNS8NAEN34WetX1KMgwSJ4KkkV9CRFEbwIFe0HNCVstpt26W4SdidiCbl58a948aCIV/+CN/+N2zYHbX0w8Hhvhpl5fsyZAtv+NubmFxaXlgsrxdW19Y1Nc2u7oaJEElonEY9ky8...

    work page

  6. [5]

    ZPJOUf+IcKX3+pAJ6B7YozoqK4k=

    Nonlinearity: With these definitions, fluxes can still be seen as direct responses to forces, but they are nonlinear Js→s′ = eXs→s′ − eXs′ →s . (19) Thermal Transitions: <latexit sha1_base64="ZPJOUf+IcKX3+pAJ6B7YozoqK4k=">AAACBHicbVC7SgNBFJ31GeNr1TLNYBCswm4UFKugjWWEvCAJYXZykwyZnV1m7ophSWHjr9hYKGLrR9j5N04ehSYeuHA4514u5wSxFAY979tZWV1b39jMbGW3d3b39t2Dw5qJEs2...

    work page

  7. [6]

    The state probability p(s) is normalized, meaning that the constraint P s p(s) = 1 must be applied when using the method of Lagrange multipliers to opti- mize entropy production

    Constraints on Forces: The forces Xs→s′(R, p) = ln p(s)Rs→s′ cannot be totally free variables. The state probability p(s) is normalized, meaning that the constraint P s p(s) = 1 must be applied when using the method of Lagrange multipliers to opti- mize entropy production

    work page

  8. [7]

    pumped.” Pumped systems of this form represent a general class of nonequilibrium systems, A <latexit sha1_base64=

    False Steady-State: While zero flux Js→s′ = 0 may sound like a steady state, this is only true when de- tailed balance p(s)Rs→s′ = p(s′)Rs′→s is satisfied. If detailed balance holds in the steady-state, then the system is in thermodynamic equilibrium, which precludes nonequilibrium behavior. A nonequilib- rium system and its corresponding NESS π will vi- ...

    work page

Show all 77 references
  1. [8]

    Each measure provides the answer to a different ques- tion

    Development of a complexity measure for CTMCs and comparison to EPR:Complexity is a deep and nuanced field with popular measures ranging from Kolmogorov-Chaitin complexity [58, 59] to statis- tical complexity [60], and beyond [61, 62]. Each measure provides the answer to a dif...

  2. [9]

    Exploring new sampling methods for CTMCs:An admittedly limiting factor in our results is the fact that our algorithms for randomly sampling CTMCs indirectly determine the statistics of entropy pro- duction. While our pumped models are rooted in pumped stochastic systems and ch...

  3. [10]

    Random matrix theory of rate equation entropy pro- duction: We anticipate that systematic predictions for EPR of large random nonequilibrium systems may be accessible through random matrix theory [63]

  4. [11]

    Rupe and J

    A. Rupe and J. P. Crutchfield, Physics Reports 1071, 1 (2024)

    2024

  5. [12]

    E. D. Schneider and D. Sagan, Into the cool: Energy flow, thermodynamics, and life (University of Chicago Press, 2005)

    2005

  6. [13]

    motivates this search. Despite in-depth investiga- tions from researchers across disciplines, there is not yet ∗ kylejray@gmail.com † alecboy@gmail.com a consensus on the universal behavior of entropy pro- duction, perhaps best illustrated by the fact that advo- cates of both ...

    2025 arXiv

  7. [14]

    E. T. Jaynes, Annual Review of Physical Chemistry 31, 579 (1980)

    1980

  8. [15]

    Seifert, Reports on progress in physics 75, 126001 (2012)

    U. Seifert, Reports on progress in physics 75, 126001 (2012)

    2012

  9. [16]

    A. Seif, M. Hafezi, and C. Jarzynski, Nature Physics 17, 105 (2021)

    2021

  10. [17]

    Batalh˜ ao, A

    T. Batalh˜ ao, A. Souza, R. Sarthour, I. Oliveira, M. Pa- ternostro, E. Lutz, and R. Serra, Physical review letters 115, 190601 (2015)

    2015

  11. [18]

    G. T. Landi and M. Paternostro, Reviews of Modern Physics 93, 035008 (2021)

    2021

  12. [19]

    Dewar, Journal of Physics A: Mathematical and Gen- eral 36, 631 (2003)

    R. Dewar, Journal of Physics A: Mathematical and Gen- eral 36, 631 (2003)

    2003

  13. [20]

    R. C. Dewar, in Non-equilibrium thermodynamics and the production of entropy: life, earth, and beyond (Springer,

  14. [21]

    Onsager, Physical review 37, 405 (1931)

    L. Onsager, Physical review 37, 405 (1931)

    1931

  15. [22]

    Prigogine, Science 201, 777 (1978)

    I. Prigogine, Science 201, 777 (1978)

    1978

  16. [23]

    A. M. Zhabotinsky, Chaos (Woodbury, NY) 1, 379 12 (1991)

    1991

  17. [24]

    Michaelian, Foundations 2, 308 (2022)

    K. Michaelian, Foundations 2, 308 (2022)

    2022

  18. [25]

    Prigogine, Introduction to Thermodynamics of Irre- versible Processes, 3rd ed

    I. Prigogine, Introduction to Thermodynamics of Irre- versible Processes, 3rd ed. (John Wiley and Sons, New York, 1968)

    1968

  19. [26]

    Guazzelli and J

    E. Guazzelli and J. F. Morris, A physical introduction to suspension dynamics , Vol. 45 (Cambridge University Press, 2011)

    2011

  20. [27]

    G. E. Crooks, Phys. Rev. E 60, 2721 (1999)

    1999

  21. [29]

    Rold´ an and J

    ´E. Rold´ an and J. M. Parrondo, Physical review letters 105, 150607 (2010)

    2010

  22. [30]

    Micadei, J

    K. Micadei, J. P. Peterson, A. M. Souza, R. S. Sarthour, I. S. Oliveira, G. T. Landi, R. M. Serra, and E. Lutz, Physical Review Letters 127, 180603 (2021)

    2021

  23. [31]

    Wimsatt, O.-P

    G. Wimsatt, O.-P. Saira, A. B. Boyd, M. H. Matheny, S. Han, M. L. Roukes, and J. P. Crutchfield, Physical Review Research 3, 033115 (2021)

    2021

  24. [32]

    Y. Jun, M. Gavrilov, and J. Bechhoefer, Phys. Rev. Lett. 113, 190601 (2014)

    2014

  25. [33]

    T. K. Saha, J. Ehrich, M. Gavrilov, S. Still, D. A. Sivak, and J. Bechhoefer, Physical Review Letters 131, 057101 (2023)

    2023

  26. [34]

    K. W. Tang, K. J. Ray, and J. P. Crutchfield, New Jour- nal of Physics 27, 043011 (2025)

    2025

  27. [35]

    Hayashi, H

    K. Hayashi, H. Ueno, R. Iino, and H. Noji, Biophysical Journal 98, 633a (2010)

    2010

  28. [36]

    I. A. Mart ´ ınez, G. Bisker, J. M. Horowitz, and J. M. Parrondo, Nature communications 10, 3542 (2019)

    2019

  29. [37]

    F. S. Gnesotto, F. Mura, J. Gladrow, and C. P. Broed- ersz, Reports on Progress in Physics 81, 066601 (2018)

    2018

  30. [38]

    D. J. Skinner and J. Dunkel, Physical review letters 127, 198101 (2021)

    2021

  31. [39]

    P. M. Riechers, C. Gupta, A. Kolchinsky, and M. Gu, PRX Quantum 5, 030318 (2024)

    2024

  32. [40]

    D. H. Wolpert, Journal of Physics A: Mathematical and Theoretical 52, 193001 (2019)

    2019

  33. [41]

    Van Vu and K

    T. Van Vu and K. Saito, Physical Review X 13, 011013 (2023)

    2023

  34. [42]

    A. I. Brown and D. A. Sivak, Proceedings of the National Academy of Sciences 114, 11057 (2017)

    2017

  35. [43]

    D. L. Andrews and D. S. Bradshaw, Optics communica- tions 283, 4365 (2010)

    2010

  36. [44]

    D. M. Busiello, S. Liang, F. Piazza, and P. De Los Rios, Communications Chemistry 4, 16 (2021)

    2021

  37. [45]

    A. B. Boyd, J. P. Crutchfield, and M. Gu, New Journal of Physics 24, 083040 (2022)

    2022

  38. [46]

    A. B. Boyd, J. P. Crutchfield, M. Gu, and F. C. Binder, arXiv preprint arXiv:2402.16995 (2024)

    2024 arXiv

  39. [47]

    J. M. Gold and J. L. England, arXiv preprint arXiv:1911.07216 (2019)

    2019 arXiv

  40. [48]

    Schnakenberg, Rev

    J. Schnakenberg, Rev. Mod. Phys. 48, 571 (1976)

    1976

  41. [49]

    A. C. Barato and U. Seifert, Physical review letters 114, 158101 (2015)

    2015

  42. [50]

    Shiraishi, K

    N. Shiraishi, K. Funo, and K. Saito, Physical review letters 121, 070601 (2018)

    2018

  43. [51]

    Gunawardena, PloS one 7, e36321 (2012)

    J. Gunawardena, PloS one 7, e36321 (2012)

    2012

  44. [52]

    Cabello-Garcia, R

    J. Cabello-Garcia, R. Mukherjee, W. Bae, G.-B. V. Stan, and T. E. Ouldridge, Nature Chemistry , 1 (2025)

    2025

  45. [53]

    Floyd, A

    C. Floyd, A. R. Dinner, A. Murugan, and S. Vaikun- tanathan, arXiv preprint arXiv:2409.05827 (2024)

    2024 arXiv

  46. [54]

    K. J. Ray, A. B. Boyd, G. Guarnieri, and J. P. Crutch- field, Physical Review E 108, 054126 (2023)

    2023

  47. [55]

    Dal Cengio, V

    S. Dal Cengio, V. Lecomte, and M. Polettini, Physical Review X 13, 021040 (2023)

    2023

  48. [56]

    A. B. Boyd, P. M. Riechers, G. W. Wimsatt, J. P. Crutch- field, and M. Gu, arXiv preprint arXiv:2104.12072 (2021)

    2021 arXiv

  49. [57]

    A. B. Boyd and P. M. Riechers, arXiv preprint arXiv:2501.04865 (2025)

    2025

  50. [58]

    Y. Sato, E. Akiyama, and J. P. Crutchfield, Physica D: Nonlinear Phenomena 210, 21 (2005)

    2005

  51. [59]

    Polettini, Physical Review E—Statistical, Nonlinear, and Soft Matter Physics 84, 051117 (2011)

    M. Polettini, Physical Review E—Statistical, Nonlinear, and Soft Matter Physics 84, 051117 (2011)

    2011

  52. [60]

    J. A. Owen, T. R. Gingrich, and J. M. Horowitz, Physical Review X 10, 011066 (2020)

    2020

  53. [61]

    Aslyamov and M

    T. Aslyamov and M. Esposito, Physical Review Letters 132, 037101 (2024)

    2024

  54. [62]

    A. I. Brown and D. A. Sivak, Chemical reviews 120, 434 (2019)

    2019

  55. [63]

    M. P. Leighton and D. A. Sivak, arXiv preprint arXiv:2406.10355 (2024)

    2024 arXiv

  56. [64]

    Friston, Nat

    K. Friston, Nat. Rev. Neurosci. 11, 127 (2010)

    2010

  57. [65]

    Harte, T

    J. Harte, T. Zillio, E. Conlisk, and A. B. Smith, Ecology 89, 2700 (2008)

    2008

  58. [66]

    J. M. Horowitz and J. L. England, Proceedings of the National Academy of Sciences 114, 7565 (2017)

    2017

  59. [67]

    Harte, Maximum entropy and ecology: a theory of abundance, distribution, and energetics (OUP Oxford, 2011)

    J. Harte, Maximum entropy and ecology: a theory of abundance, distribution, and energetics (OUP Oxford, 2011)

    2011

  60. [68]

    Stern, M

    M. Stern, M. Guzman, F. Martins, A. J. Liu, and V. Bal- asubramanian, arXiv preprint arXiv:2406.09689 (2024)

    2024 arXiv

  61. [69]

    Chaitin, J

    G. Chaitin, J. ACM 13, 145 (1966)

    1966

  62. [70]

    A. N. Kolmogorov, Prob. Info. Trans. 1, 1 (1965)

    1965

  63. [71]

    J. P. Crutchfield, Nature Physics 8, 17 (2012)

    2012

  64. [72]

    L¨ ohr, Journal of Systems Science and Complexity25, 30 (2012)

    W. L¨ ohr, Journal of Systems Science and Complexity25, 30 (2012)

    2012

  65. [73]

    M. Gu, K. Wiesner, E. Rieper, and V. Vedral, Nature communications 3, 762 (2012)

    2012

  66. [74]

    D. M. Busiello, J. Hidalgo, and A. Maritan, Physical Review E 96, 062110 (2017)

    2017

  67. [75]

    L. J. Fiderer, P. C. Barth, I. D. Smith, and H. J. Briegel, arXiv preprint arXiv:2504.06209 (2025)

    2025 arXiv

  68. [76]

    A. B. Boyd, D. Mandal, and J. P. Crutchfield, Phys. Rev. X 8 (2018)

    2018

  69. [77]

    Rate equation dynamics,

    K. Ray, “Rate equation dynamics,” https://github. com/kylejray/rate_equation_dynamics (2025)

    2025

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.