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REVIEW 3 major objections 6 minor 45 references

Thermodynamics of switching in multistable non-equilibrium systems

T0 review · 3 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Switching in multistable systems is governed by the entropy produced along the switching path, not by steady-state entropy.

desk verdict A solid, honest stochastic thermodynamics paper whose genuinely new relation (Eq. 20) is supported by systematic numerics, but whose headline MaxEPP claim is conditional on an approximation that fails for ~10% of toggle switch parameterizations. read the letter →

arxiv 1908.07405 v3 pith:OTFJNQLN submitted 2019-08-20 physics.bio-ph cond-mat.stat-mech

Jacob Cook , Robert G. Endres This is my paper · ORCID
classification physics.bio-phcond-mat.stat-mech
keywords stochasticthermodynamicsentropyproductionmultistabilityFreidlin-WentzellactionLangevindynamicsmaximumprinciplebistablemodelsstateswitching
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

This paper asks what sets the rates at which a multistable system switches between its stable states, far from equilibrium. By combining the fluctuation theorem for entropy along trajectories with the Freidlin–Wentzell least-action principle, the authors argue that in the large-volume limit the switching probability is controlled by the entropy produced along the switching path, not by steady-state entropy production or state entropy. Concretely, they derive a probability $P_{A\to B}\propto \exp(\Delta S^{L}_{A\to B}/2 - C_{A\to B})$ for the most probable switching path and verify on the Schlögl and toggle-switch models that differences in path entropy production track differences in action. If right, this gives a thermodynamic rule for relative stability: more dissipative switches are exponentially more probable, extending the spontaneous-reaction rule to switching in biological, ecological, and climate multistable systems.

What carries the argument

The load-bearing object is the Freidlin–Wentzell action $\mathcal{A}[x] = \frac{1}{2}\int_0^\tau (\dot{x} - f)^{T} D^{-1} (\dot{x} - f)\, dt$, which assigns an exponential cost to switching paths in the large-volume limit. The paper splits this action as $\Omega \mathcal{A}_{A\to B} = C_{A\to B} - \frac{1}{2}\Delta S^{L}_{A\to B}$, identifying the dissipative part with the entropy produced along the path, and uses the identity that every point of a minimum-action path has equal kinetic and potential contributions to justify $C_{A\to B}\approx C_{B\to A}$. This decomposition, together with the fluctuation theorem for Langevin paths, converts the action-minimization problem into a statement about entropy production, yielding Eq. (19) and the relation $\frac{1}{2}(\Delta S^{L}_{B\to A} - \Delta S^{L}_{A\to B}) \approx \Omega(\mathcal{A}_{A\to B} - \mathcal{A}_{B\to A})$.

What would settle it

Construct a two-dimensional multistable system whose saddle point is at low copy number, compute the forward and reverse minimum actions and path entropy productions, and test whether $\frac{1}{2}(\Delta S^{L}_{B\to A} - \Delta S^{L}_{A\to B}) = \Omega(\mathcal{A}_{A\to B} - \mathcal{A}_{B\to A})$; a clear violation in a slowly varying force regime would falsify the claimed universality.

Watch

Extended reading notes

Core claim

The central claim is that a reduced maximum-entropy-production principle holds for switching paths in the large volume limit: the probability of the most probable path from macrostate A to macrostate B is $P_{A\to B} = \exp(\Delta S^{L}_{A\to B}/2 - C_{A\to B})/Q_{A\to B}$, where $\Delta S^{L}$ is the entropy produced along the path and $C$ is a conservative action that penalizes deviations from the deterministic dynamics. Because the conservative actions of the two opposite switching paths are nearly equal when the deterministic force varies slowly, the difference in action between A$\to$B and B$\to$A is set by the difference in path entropy production. The paper demonstrates on random parameterizations of two bistable models that this linear relation holds, that state entropy and diffusive noise strength have negligible effect on stability, and that steady-state entropy production shows only a weak correlation with occupation probabilities. The authors state the primary conclusion as extending the rule that exergonic reactions occur spontaneously to switching in multistable systems.

Load-bearing premise

The reduced maximum-entropy-production relation rests on the approximation that the two opposite switching paths have nearly equal conservative actions; the paper finds this holds for 90% of its toggle-switch parameter sets but fails when the saddle point is at low copy number.

Editorial extensions

If this is right

  • In a bistable chemical system at large volume, the ratio of occupation probabilities is fixed by the minimum-action difference, which equals half the difference in path entropy productions; steady-state entropy production and state entropy can be dropped from the stability calculation.
  • Switches that produce more entropy along the path are exponentially more likely, so in a system with many macrostates the most dissipative transitions dominate at sufficiently large volume.
  • For one-dimensional systems, forward and reverse switching paths are time-reverses of each other, so the two switching rates are equal and the rate ratio is determined solely by the path entropy production.
  • The theory identifies when the simple rule breaks down: when the saddle point sits at low copy number, the conservative actions of opposite paths diverge and the clean relation between action differences and path entropy production no longer holds.

Reading between the lines

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

  • Eq. (20) could be used as a diagnostic on single-molecule data: forward and reverse switching rates together with per-path entropy measurements should obey the relation in slowly-varying-force regimes, and the first violations would pinpoint low-copy-number saddles where the reduced MaxEPP fails.
  • The same least-action plus entropy-production argument should extend to non-chemical multistable systems, such as ecological or climate models written as Langevin dynamics, predicting that basins reachable by more dissipative transitions are the more stable ones; this is a testable extension the paper does not carry out.
  • Should the action split generalize to spatially extended systems, relative stability of multiple coexisting attractors could be inferred from path entropy production alone, bypassing explicit minimization of the action.
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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

3 major / 6 minor

Summary. This paper asks which thermodynamic quantities control the relative stability and the switching rates between macrostates of multistable non-equilibrium chemical systems in the large-volume limit. The authors combine the Freidlin–Wentzell (FW) action of the chemical Langevin equation (Eq. 7) with the coarse-grained (Langevin) entropy production along a path, defined via the action difference between forward and time-reversed paths (Eq. 8). Splitting the action into a conservative part C (Eq. 18) and the path entropy production ΔS_L yields Eq. (19), P_A→B = exp(ΔS_L/2 − C)/Q_A→B, which is interpreted as a trade-off between minimizing the conservative action and maximizing path entropy production, i.e., a 'maximum entropy production principle (MaxEPP) for switching paths'. To obtain a criterion for relative stability of the two states, the authors introduce the approximation C_A→B ≈ C_B→A, which converts Eq. (19) into Eq. (20), an approximate proportionality between the difference of the minimum actions and the difference of the path entropy productions. The theory is tested numerically on the one-dimensional Schlögl model and the two-dimensional toggle switch model, using 100 random parameter sets for each: Fig. 3D reports strong linear correlations between action differences and path-entropy-production differences (r = 0.9445 and 1.0000), Figs.

Significance. The paper is a potentially useful contribution to stochastic thermodynamics and systems biology, provided the claims are properly qualified. The decomposition behind Eq. (19) is parameter-free given the Langevin model, and the reformulation of the FW action in thermodynamic terms (conservative action versus path entropy production) is conceptually attractive. The main strengths are the systematic numerical protocol — 100 random parameter sets per model with Gillespie simulations and geometric minimum-action path calculations — and the honest reporting of the failure statistics for the central approximation. The near-perfect linear relation in Fig. 3D, if confirmed with the fitted slope reported, would give practitioners a practical shortcut for estimating relative stability from path dissipation. On the critical side, the paper's primary novelty claim is partly a relabeling: Eq. (19) is an identity, and the falsifiable content lives in Eq. (20), whose validity is restricted by the C_A→B ≈ C_B→A approximation and its documented 10% failure rate.

major comments (3)
  1. [Section III, Eq. (19); Discussion, first paragraph] The paper presents Eq. (19) as the derivation of a 'maximum entropy production principle for switching paths', but the decomposition ΩA = C − (1/2)ΔS_L is an algebraic identity given the definitions in Eqs. (7), (8), and (18); substituting Eq. (18) into Eq. (19) reproduces the expanded action term by term, so maximizing ΔS_L at fixed C is exactly equivalent to minimizing the FW action. Eq. (19) is therefore a rearrangement of the least-action principle rather than an independent derivation of a maximum principle. The reformulation may still be useful, but the paper should state this explicitly and locate the novel, falsifiable content in Eq. (20) and its numerical support; as written, the 'primary conclusion' overstates the status of Eq. (19).
  2. [Section III, paragraph after Eq. (19); Abstract; Discussion] The reduced MaxEPP statement — that more dissipative switches are exponentially more probable and that the relative stability of A versus B is governed by the difference in path entropy production — rests on the approximation C_A→B ≈ C_B→A that converts Eq. (19) into Eq. (20). The manuscript reports that this approximation holds for only 90% of the toggle-switch parameterizations and that 'significant divergence from the relation was generally observed in cases where the saddle point occurred at a low copy number compared to the steady states'. Since C_A→B − C_B→A is an O(Ω) quantity in the large-Ω limit, the divergence for the excluded parameter sets is not a finite-volume artifact and contributes to ln(k_A→B/k_B→A) at the same exponential order as the path-entropy-production difference. For those parameter sets, relative stability is therefore not determined by path entropy production alone, and the unqualified claims in the Abstract ('the entropy production during switching is key') and in the Discussion ('for sufficiently large volumes switches that produce more entropy will be favoured') overreach. The Discussion's secondary conclusion is already appropriately qualified, but the headline claims should be made explicitly conditional on the conservative-action approximation, or the failure regime should be characterized further (for example, by testing whether the 10% failure set persists and how it scales with Ω).
  3. [Section III, Fig. 3D and Eq. (20)] The numerical support for the central quantitative relation is reported only as Pearson correlation coefficients (r = 0.9445 for the toggle switch and r = 1.0000 for the Schlögl model). Eq. (20) predicts a specific linear relation with a definite slope (including the factor 1/2 and a definite sign), so the fitted slope and intercept with confidence intervals should be reported, and the roughly 90% of parameter sets where the approximation holds should be analyzed separately from the divergent ones. A high correlation alone does not establish the quantitative form of Eq. (20), which is the paper's main predictive statement.
minor comments (6)
  1. [Section II.B, Eq. (11)] The chain of statements 'k_A→B = k_B→A' and 'k_A→B/k_B→A = exp(ΔS_L_{A→B})' is mutually inconsistent unless ΔS_L_{A→B} = 0, which would contradict the nonzero equal-and-opposite path entropy productions shown for the Schlögl model in Fig. 2C. Please clarify whether the ratio in Eq. (11) compares the switching rate with the rate of the time-reversed path ensemble (k̄_A→B) or with the reverse-switching rate (k_B→A), and correct the corresponding sentence.
  2. [Section II.A, after Eq. (2)] The claim that the boundary term ⟨ln(P(X0)/P(XN))⟩ in Eq. (2) 'becomes negligibly small' in the large-Ω limit is not justified as stated; the logarithms of the stationary probabilities at the two macrostates generally differ by an O(Ω) amount in the WKB regime, the same order as the medium term. This statement should be derived or rephrased.
  3. [Section II.B, after Eq. (8)] The sentence 'the Langevin formalism within the steady state is equivalent to a quasi-equilibrium' is a strong interpretive claim that appears to conflict with the nonzero steady-state entropy production computed for the same models (Fig. 3A) and with the authors' own EP/EF decomposition (Figs. 2E and 2F); it should be softened or justified.
  4. [Fig. 3] Please report the values of Ω used for the simulations underlying Figs. 3B and 3D and state the range of action values sampled; the text notes that the results in Fig. 3B are coarsely discretized because a low Ω was used, and the reader needs this context to judge the large-volume relevance of the comparisons.
  5. [Section II.A, notation] The bars denoting time-reversed quantities are not rendered reliably in the text (for example, 'ln(WΓ/WΓ)' and 'kA→B/kA→B' in the paragraph after Eq. (2)); please fix the typography so the forward and time-reversed rates are clearly distinguishable.
  6. [References, novelty relative to ref. 31] Eqs. (8) and the action split leading to Eq. (19) are attributed to the authors' own earlier work (ref. 31, Endres 2017); the text should state explicitly which results are new to this paper so that the incremental contribution is clear.

Circularity Check

1 steps flagged · score 4.0 of 10

Eq. 19 is an algebraic re-labeling of the FW action; the 'MaxEPP for switching paths' is not independently derived, while Eq. 20 and the numerical tests remain non-circular.

  1. self definitional [Section III, paragraph after Fig. 3, unnumbered equations surrounding Eqs. 18-19]
    "We shall proceed with our derivation by noting that the action can be split into two parts as Ω A_{A→B} = C_{A→B} − 1/2 ΔS^L_{A→B}, where C_{A→B} is the conservative action along the path A→B and ΔS^L_{A→B} is the Langevin path entropy production (Eq. 8). ... By substituting the expanded form of the action into the expression for switching path probability, a reduced form of MaxEPP can be obtained P_{A→B} = exp(1/2 ΔS^L_{A→B} − C_{A→B}) / Q_{A→B}, (19)"

    Equation 18 defines C exactly so that Eq. 19 is the FW exponential P = exp(−ΩA)/Q rewritten as exp(1/2 ΔS − C)/Q. Consequently, 'maximizing path entropy production' at fixed conservative action is the same operation as minimizing the FW action; the MaxEPP formulation is a re-labeling of the least-action principle rather than an independent derivation. The paper itself calls this a 'reduced form of MaxEPP' and later treats it as its 'primary conclusion,' so the headline predictive statement is equivalent by construction to the input action functional. The numerical tests of Eq. 20 and the parameter surveys are not circular; they independently test the additional approximation C_{A→B} ≈ C_{B→A} against Gillespie simulations.

full rationale

The derivation of Eq. 8 from the fluctuation theorem and the Freidlin–Wentzell action is self-contained and not circular: it follows from the path probabilities and the definition of entropy production. The numerical results in Figs. 3B and 3D compare independent Gillespie simulations with FW action predictions and show strong correlations, so those are genuine empirical checks rather than fitted inputs. The only definitional circularity is Eq. 19: splitting the action into conservative and dissipative parts and presenting the resulting rearrangement as a 'MaxEPP' does not add independent content, because maximizing ΔS at fixed C is exactly minimizing A. The paper's secondary relation Eq. 20 rests on the explicitly stated approximation C_{A→B} ≈ C_{B→A}, which the authors report holds for only 90% of the toggle-switch parameterizations; that is a substantive limitation on the universality of the relative-stability criterion, but it is an assumption rather than a circular step and does not increase the circularity score. Because the central MaxEPP claim reduces to an identity while the quantitative switching-rate relations are independently tested, a moderate score of 4 is appropriate.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

The central relations rely on standard large deviation and stochastic thermodynamics results, plus two modeling assumptions specific to this paper: that the Langevin apparent entropy production can stand for the thermodynamic entropy production, and that opposite switching paths have nearly equal conservative actions. No parameters are fitted to data; model rate constants are sampled randomly to test generality.

free parameters (1)
  • model rate constants
    No parameters were fitted to data to produce the central relations. Model rate constants are drawn randomly from stated ranges to sample the model space, and none are tuned to enforce Eq. 19 or Eq. 20.
assumptions (6)
  • standard math Freidlin-Wentzell theorem: rare switching paths are exponentially suppressed with rate set by the action functional (Eq. 7) in the large volume limit.
    Invoked in Section II.B to relate minimum action paths to switching probabilities and mean first passage times.
  • standard math Steady-state fluctuation theorem for path probabilities (Eq. 2), giving total entropy production as log ratio of forward and time-reversed path probabilities.
    Used in Section II.A as the microscopic starting point; standard result in stochastic thermodynamics (Seifert, Evans and Searles).
  • standard math Chemical Fokker-Planck and Langevin coarse-graining of the chemical master equation is valid for large but finite volume Ω.
    Adopted in Section II.B following Gillespie (Ref. 22); the paper's entropy production and action are computed in this continuum limit.
  • domain assumption Switching between macrostates is rare and Poisson in the large volume limit, so occupation probabilities are determined by the ratio of switching rates.
    Stated in Section II.A: 'no significant amount of time is spent outside these macrostates' and waiting times are exponential (Fig. 1B inset).
  • domain assumption The coarse-grained Langevin entropy production ΔS_L (Eq. 8) is an adequate proxy for the thermodynamic entropy production relevant to switching, despite being an apparent entropy production.
    The paper notes in Section II.B that this is a coarse-grained apparent entropy production; its correlation with true master-equation EP is only demonstrated in the supplementary material.
  • ad hoc to paper Conservative actions of opposite switching paths are nearly equal, C_A→B ≈ C_B→A.
    This approximation is introduced in Section III to derive Eq. 20; the paper acknowledges it holds for 90% of toggle switch parameterizations and fails when the saddle is at low copy number.

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Pith. "Pith review of Thermodynamics of switching in multistable non-equilibrium systems." pith.science (2026). https://pith.science/paper/OTFJNQLN

@misc{pith2026190807405,
  author       = {Pith},
  title        = {Pith review of: Thermodynamics of switching in multistable non-equilibrium systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OTFJNQLN}},
  note         = {Machine review of arXiv:1908.07405}
}
read the original abstract

Multistable non-equilibrium systems are abundant outcomes of nonlinear dynamics with feedback but still relatively little is known about what determines the stability of the steady states and their switching rates in terms of entropy and entropy production. Here, we will link fluctuation theorems for the entropy production along trajectories with the action obtainable from the Freidlin--Wentzell theorem to elucidate the thermodynamics of switching between states in the large volume limit of multistable systems. We find that the entropy production at steady state plays no role, but the entropy production during switching is key. Steady-state entropy and diffusive noise strength can be neglected in this limit. The relevance to biology, ecological, and climate models is apparent.

Figures

Figures reproduced from arXiv: 1908.07405 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

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    author author M. Scheffer, J. Bascompte, W.A. Brock, V. Brovkin, S.R. Carpenter, V. Dakos, H. Held, E.H. van Nes, M. Rietkerk, and G. Sugihara ,\ @noop journal journal Nature \ volume 461 ,\ pages 53 ( year 2009 ) NoStop

    2009

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.